Details

Published: Wednesday, 22 March 2023 04:33

Hits: 836

Periodic Table, Atoms and Elements:

Periodic Table is one of the cornerstone of Chemistry. Before we learn periodic table, we should know a bit about atoms and how were they born.

It's believed that a "big bang" took place 14 Gya, and everything came into existence following that. Read more about "Birth of Universe" in Astronomical section.

Over there, we learned about the very first sub atomic particles, and the formation of Hydrogen and Helium atoms. Here we'll see all the atoms and the elements they form.

Atoms & Elements:

All matter that we see is atoms. There are atoms of gold, nickel, oxygen, etc. So, we have silver in pure form which consists of silver atoms. We can't divide an atom further down. Each atom has a nucleus in the center which is composed of protons and neutrons, and a cloud around it which is composed of electrons. As we saw above, neutrons and protons are formed from 3 quark particles, and an electron is a lepton. These come together to form an atom, with nucleus in the center and electron revolving around it (just like planets revolve around the sun)

Atoms of different elements have different number of electrons, protons and neutron. The characteristic thing that defines an element is the number of protons. We can't change the number of protons for a given element. If we somehow change it, than that element gets converted into another element which has that many protons. So, number of protons is called as Atomic number (Z) of an element. No 2 elements have same atomic number.

Mass, Charge and size of e, p, n:

Unified amu: amu is also written as unified amu or "u". So, you will see mass of proton written as 1u too (instead of 1amu). Mass of a proton is actually 1.007u and mass of neutron is 1.008u (so, both are slightly heavier than 1u). 1u = 1.6605x10^-27 kg

All of these electron, proton, neutron would seem like particle, but modern quantum theory states that quantum scale objects have wave-particle duality, and exhibit both wave like and particle like properties. There have been many controversies over this wave-particle nature of objects, but is now generally accepted. Bigger the size of object, it's wave characteristics become so small, that it acts like a particle fr most practical purpose. However, for very small objects, as electrons, wave characteristics are noticeable. Similarly all electromagentic radiation (i.e light, heat), etc are assumed to hold both wave and particle characteristics to help explain many phenomenon which can't be explained by assuming only a particle mode, or only a wave model.

For our purpose, we will treat all electron, proton and neutron as particles, with the understanding that they show wave properties. All of them have mass, but their mass is too small (neutron/proton have 1800 times greater mass than electron, but still small). So, gravitational forces between these masses is too small. They are held together via electrostatic forces (forces b/w opposite charges).

Electron is considered point particle with no size. Size of an atom is around 10^-10m, which is about 100,000 times larger than size of proton and neutron. So, the nuclues of an atom is very small of the order of fm, with 99.99% of the space around it formed by the electron cloud. Two atoms aren't able to get very close to each other because the repulsive force of this electron cloud prevents the nuclei from getting too close. Size of an atom is determined entirely by it's electron cloud.

A neutral atom of an element has same number of protons and electrons, as the atom is electrically neutral. It's mass number (A) is equal to the sum of number of protons and neutrons (since protons and neurons have mass of 1 amu, the atomic weight of atoms is almost same as atomic number, but not always. The man reason for this is presence of isotopes which will be explained later).

Mass number of Atom (A) = p + n

Atomic number of atom (Z) = p

Given Mass Number and Atomic number of atom, we can figure out the number of electrons, protons and neutrons.

Ex: Na has Z=11, A=23 => p=11, e=11, n=23-11=12.

Periodic Table:

We saw above how the number of protons is what defines an element. This wasn't known to mankind, and a lot of thinkers were trying to come up with some theory on what makes an element unique. The first periodic table to become generally accepted was that of the Russian chemist Dmitri Mendeleev in 1869: This wikipedia link has details of all elements in Periodic Table: https://en.wikipedia.org/wiki/Periodic_table

This is a pictorial periodic table: https://periodictable.com/

This is another periodic table that shows certain properties of all atoms in periodic table: https://www.periodic-table.org/

The very first thing to know in a periodic table is that Elements are organized by number of protons they have, which start from 1 proton for Hydrogen, all the way to 110+ protons for some heavier elements. This is known as "Atomic Number". However what determines the properties of any element is NOT the number of protons, but the number of electrons. However, it's not just the raw number of electrons, but electrons in the outer shell (and also the subshell they are in) that mostly determine most of the electrical and chemical behavior of the element. Since elements are neutral in charge, number of electrons is bound to be the same as the number of protons. We can make these atoms lose or gain electrons or share electrons via chemical reactions, in which case they may become charged.

Today, 118 elements are known, the first 94 of which are known to occur naturally on Earth at present.

Periodic Table trend (this will be more relevant once you cover the contents in this section) => https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Descriptive_Chemistry/Periodic_Trends_of_Elemental_Properties/Periodic_Trends

Electron Orbit:

Atoms have energy shells, which are the shells or space around the nucleus where electrons are to be found. The reason electrons can only be found in these distinct spaces around the nucleus is because electrons can only have certain distinct energies. These distinct spaces are called electron orbital. Pauli's Exclusion principle states that no 2 electrons can have the same energy state, implying each orbital (which is a energy state) can have a max of 2 electrons, one with +ve spin and one with -ve spin. If we had only single spin for electrons, then there could have been only 1 electron per orbital.

Bunch of electron orbitals are grouped into sub shells which are further bundled into shells. There are 4 sub shells named as s,p,d and f. s has only 1 orbital, so can have max of 2 electrons, p has 3 orbitals and so a max of 6 electrons, d has 5 orbitals, and so a max of 10 electrons, and f has 7 orbitals and so a max of 14 electrons. "s" is spherical orbital, "p" is dumb bell shaped, d and f are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found with a 90% probability. When Scientists in early 1900 started studying atoms, they became aware of the shells (Bohr's model), but didn't know about the existence of sub shells. They knew by looking at spectral lines emitted by these atoms that some spectral lines were more intense than others, but didn't know the reason. Reason turned out to be these sub-shells that were present within a single shell. The designations s, p, d, and f result from those early historical attempts to classify atomic spectral lines. (The letters stand for sharp, principal, diffuse, and fundamental, respectively.) After f, the letters continue alphabetically as g, h, i, j, etc. So, sub-shells can be thought of as finer granularity of a shell => When you look at a shell closely, you see it's not one but rather is composed of bunch of sub shells.

The shells are numbered from shell=1 to shell=7. Each shell can have max of 2*n^2 electrons. So shell1 can have max 2 electrons, shell2=8 electrons, shell3=18 electrons, shell4=32 electrons. shell5=50 electrons and so on. It's not the total number of electrons in an atom, but rather the number of electrons in outer shells (rather outer sub shell) that determines the properties of elements. Hence all elements with same number of electrons in outer sub shell are put into same column, to indicate that they have similar properties. Rows are automatically formed due to this, as the elements are put in increasing order of their Atomic number. So, if an element has 3 electrons in n=2, then elements with 3 electrons in n=3, n=4, n=5 etc will all have similar properties.

Shells are numbered as follows:

• 1n => 1st shell has only 1 sub shell = 1s (max 2 electrons)
• 2n => 2nd shell has 2 sub shells s,p = 2s (max 2 electrons)  and 2p (3 of these have max 6 electrons)
• 3n => 3rd shell has 3 sub shells s,p,d = 3s (max 2 electrons), 3p (max 6 electrons) and 3d (5 of these have max 10 electrons)
• 4n => 4th shell has 4 sub shells s,p,d,f = 4s (max 2 electrons), 4p (max 6 electrons), 4d (max 10 electrons) and 3f (7 of these have max 14 electrons)
• 5n to 7n => 5th to 7th shells are filled in slightly intricate ways which are shown in diagram below.

Electrons are filled in orbitals starting from lowest energy orbital to highest energy orbital. It turns out that our numbering system above is not in same order as increasing energy level, i.e we would expect 3d to be lower in energy than 4s based on numbering, but in reality, 4s is lower in energy than 3d. The simple reason is that orbitals for n=3 and higher tend to get closer to each other, and so the energy levels are very close. This results in some subshells from lower "n" having higher energy than subshells in higher "n". Below is the correct order of increasing energy:

1s ≪ 2s < 2p ≪ 3s < 3p ≪ 4s < 3d < 4p ≪ 5s < 4d < 5p ≪ 6s < 4f < 5d < 6p ≪ 7s < 5f < 6d < 7p ≪ ... (Here the sign ≪ means "much less than" as opposed to < meaning just "less than"). The way to remember this sequence is very easy, and is in the wiki link above for periodic tables. It's called the "Aufbau Principle/rule".

Behaviour of electron's is described by 4 quantum numbers (QN):

• Principal QN (n) => This is the shell number n shown above. This is the main shell that determines the distance from nucleus.
• Angular Momentum QN (l) => This ranges from 0 to n-1, and relates to shape of orbital (i.e denotes sub shells s, p, d, f, etc). So for n=2, we will have 2 values of l=0, 1. l=0 => s orbital while l=1 => p orbital, and so on.
• Magnetic QN (m) => this ranges from -l to +l. So for l=1, it has 3 values -1, 0, +1. This determines the orientation of sub shell. Sp, p subshell above can have 3 orientations as m can have 3 values. These 3 orientations are in x, y and z direction.
• Spin QN (s) => This is the orientation of electron spin, and can have only 2 values => +1/2 and -1/2.

 Pauli's exclusion principle => No 2 electrons can have the same QN. i.e all n. l, m and s can't be same for 2 electrons within an atom.

The electron cloud diagram shows the region where an electron has 90% probability of being found.  S is spherical, p is dumb bell shaped, d and f are more complex shapes.The shapes grow bigger with inc "n" as sub shells are further away from nucleus. The shapes are very important, as that is where the bonding takes place with other atoms. So, the net shape of the compound is dictated by orbitals participating in bonding.

This video has further description => https://www.youtube.com/watch?v=Ewf7RlVNBSA

Below is the Periodic Table showing electron distribution in various shells and sub sells. As can be seen, the order of increasing energy is followed. So, electrons start filling up in above order starting from Hydrogen (first element in periodic table) with only 1 electron, all the way to Oganesson (last element in periodic table) with 118 electrons. There are some elements for which the electrons are filled in higher level sub-shell, before lower level sub-shell are filled (ex: copper, silver, gold, etc have only 1 electron in their "s" sub shell, even though they should have 2. Instead that 1 electron moves to higher "d" subshell). This exception happens because 2 half filled sub shells are more stable than "1 full sub shell and one close to half subshell". This is called "exception to Aufbau Principle" and is particularly seen in higher "n", where diff in energy b/w sub shells is smaller, and hence increasing tendency for this to happen.

Elements are organized in rows and columns. There are 7 rows for 7 shells (n=1 to n=7). Elements in a row don't share any properties in common. There are 18 columns (total number of columns is 32, but they are usually numbered from 1 to 18. The higher columns from 19 to 32 are are not important as they are "f block" elements  and hence not numbered). The 1st column has 1 electron in outermost shell, 2nd column has 2 electrons in outermost shell and so on, until we reach the 32nd column which should have 32 electrons in the outer most shell. But because of the abnormal reordering of energy levels of sub shells, we get to a higher level shell before we are able to completely fill a lower level shell.

• 1st row: For n=1, max # of electrons is 2, so only 2 entries in 1st row. subshell=1s
• 2nd row:For n=2, max # of electrons is 8, so only 8 entries in 2nd row. sub shells=2s, 2p
• 3rd row:For n=3, max # of electrons is 18, so there should be 18 elements, but only 3s and 3p fill up. Before 3d fills up, 4s starts filling up, so only 8 elements are here with sub shells = 3s, 3p
• 4th row: For n=4, max # of electrons is 32, so there should be 32 elements, but we have 3d left over from n=3. So, we fill 4s, 4p and 3d.
• 5th row: For n=5, max # of electrons is 50, so there should be 50 elements, but we have 4d and 4f left over from n=4. So, we fill 5s, 4d and 5p.
• 6th row: For n=6, max # of electrons is 72, so there should be 72 elements, but we don't ever get there as there are already so many sub shells left unfilled from n=4 and n=5. We fill in order of 6s, 4f, 5d and 6p
• 7th row: For n=7, max # of electrons is 98, so there should be 98 elements, but we don't ever get there as there are already so many sub shells left unfilled from n=5 and n=6. We fill in order of 7s, 5f, 6d and 7p

Atom Formation:

As we saw above, atoms were formed from protons, neutrons and electrons coming together.

Hybrid Orbitals:

A very simplistic explanation was shown above where s has single orbital with spherical shape, p has 3 orbitals in 3 dimension of x,y,z, while d has 5 orbitals along 5 orientations, etc. For some elements in periodic table, it was observed that when these atoms came close to each other, the shape of the lattice structure formed didn't match our above theoretical expectation. These newer shapes prompted concept of hybrid orbitals => Some of these s and p orbitals mix up to form hybrid orbitals which are a mix of s and p orbitals. This happens only for 5 elements in 2nd row => Be, B, C, N and O. Of these, C, N and O comprise most of the compounds found in nature, so we'll look at these 3:

C => 2s2 and 2p2 hybridize to 4 2sp3 hybrid orbitals. It forms a tetrahedral shape. Tetrahedral shape allows the 4 vertices in a 3D shape to be equidistant from each other.

Proof of tetrahedron 109⁰ angle here: https://www.ctralie.com/Teaching/Tetrahedron/

hybrid orbitals => https://www.youtube.com/watch?v=vHXViZTxLXo

Water's tetrahedral shape => Above vid explains why water has 109 degree angle b/w the 2 H atoms. It's because of hybridization.

Bohr's Model:

A very simplistic explanation of various observed properties of Hydrogen like atoms was proposed by Neil Bohr's model in 1913, which matched exactly with what was observed in Experiments. Bohr's model is not correct, nor does it explain behaviour of other atoms, but it was a big step in understanding atoms. It was based on Classical Physics (or Newtonion Physics). In 1926, Schrodinger explained properties of all atoms based on his famous Schrodinger Equation which was a departure from Classical Physics into Quantum Physics. Schrodinger Equation is the one that is the accepted one, however it's very complex and not solvable for larger atoms.

Here's the derivation of Eqn based on Bohr's model: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Quantum_Chemistry_(Blinder)/01%3A_Chapters/1.07%3A_Hydrogen_Atom

Force b/w proton and electron of hydrogen atom = q1*q2/(4*pi*ε₀*r^2) = -e^2/(4*pi*ε₀*r^2)

Potential Energy of system at distance r is d(E) = - ∫F.dr => E = - _∞^R∫F.dr = - (-(-e^2/4*pi*ε₀*r))]∞^R = --e^2/(4*pi*ε₀*R) (potential energy is -ve meaning it takes energy to separate them. The energy at infinity is 0, so -ve energy means energy is going lower, i.e sytem is losing energy. So, +ve energy has to be supplied or work has to be done to bring it back to 0 energy).

Since electron is moving in a circle, a centripetal force is needed as per Newton's law, i.e m*v^2/R = F => m*v^2/R = e^2/(4*pi*ε₀*R^2) => Eqn (1)

Total energy of the Atom = T(Kinetic Energy of electron) + V(Potential Energy of electron-proton) = 1/2*(e^2/4*pi*ε₀*R) - e^2/4*pi*ε₀*R = -1/2*(e^2/(4*pi*ε₀*R)) => (Total energy is -ve, meaning energy was lost in bringing the electron-proton system together. So, energy has to be supplied if we want to break this sytem and take charges to infinity. Energy needed will be +1/2*(e^2/(4*pi*ε₀*R). So, where did this energy get lost? Well, there must have been electromagnetic waves that were emitted out when nucleus with proton and electron came close by to form a system of electron orbiting around the proton. This resulted in some energy lost out. This is exactly the energy that has to be provided to this system to break apart this electron-proton system. Usually it's provided by absorbing some light wave.)

Total Energy of atom is -ve => Energy has to be supplied to bring it to 0, i.e break it apart.

There are 2 variables, speed v and Radius R, and we need 1 more eqn to solve this. At this point, Bohr postulated that angular momentum of electron can't be continuous value, and can only take discrete values, i.e mvR = n*h, where h=planck's constant, and n=0,1,2,3,... There were no physics involved in this assumption, nor did he know why it would take only discrete values.

However 10 years later, De Broglie figured out the real reason. Electrons are both particle and wave. For a wave to form constructively in a Radius R, the distance 2*pi*R can only take values of λ, 2λ, 3λ, .. etc. Only then can standing waves be formed in that distance. De Broglie wavelength is also defined as  λ=h/p where p=momentum=m*v, h=planck's constant.

Link: https://phys.libretexts.org/Bookshelves/College_Physics/Book%3A_College_Physics_1e_(OpenStax)/30%3A_Atomic_Physics/30.06%3A_The_Wave_Nature_of_Matter_Causes_Quantization

This yields: 2*pi*R=n*h/(m*v), or mvR=h/2*pi => Eqn (2) same as Bohr's postulate for angular momentum.

Using this 2nd Eqn along with 1st Eqn, we can solve for R. Use v=h/(2*pi*m*R) from 2nd Eqn and substitute in 1st. We get R=n^2*a₀ where a₀ is the radius of 1st shell for n=1, a₀ = (h/2*pi)^2/(m*e^2) = 0.5Å. Angstrom is the unit that is used for size of atoms, as that's the order of size for most of the atoms. An Angstrom is 1/10 of a nm.

Size of atoms:

How does the size vary for larger atoms? For Helium, we have 2 proton and 2 electron. Since the 2nd electron is also in n=1 shell, the radius of atoms doesn't increase, but at the same time, the force b/w the electrons and protons is doubles, since there is charge=+2(e) on 2 protons. So, we would expect size of Helium atom to be smaller. In fact, since going across the periodic table in a given row, the largest shell number remains same, it implies that sizes should decrease, as we get more protons attracting the electrons. This is what we see in reality. Going from 1st group to 18th group, size decreases to almost 1/3.

Also, as we move down the table, sizes should increase as an extra shell gets added. However, at the same time more protons are also getting added, resulting in greater attraction. There's also a shielding effect, that shields outer shell electrons from full charge of nucleus being seen (as inner shell electrons can be considered as part of nucleus, hence effective charge is lowered by the number of electrons). As a result of these 2 opposing effects, we can consider charge remaining almost same as we add extra shells. So sizes end up increasing solely due to extra shell being added. Going from n=1 to n=2, size increases by 3X, but going from n=2 to larger shells, size increases are more modest at 20% per shell added.

In reality, radius of these orbits is not well defined. electrons have a probability cloud around the nucleus, and can be found anywhere, not at a definite radius. So, it's hard to define what is the size of atom. Below link explains various ways, size of atoms are defined and measured.

Size of atoms: https://chem.libretexts.org/Courses/Howard_University/General_Chemistry%3A_An_Atoms_First_Approach/Unit_1%3A__Atomic_Structure/Chapter_3%3A__The_Periodic_Table/Chapter_3.2%3A_Sizes_of_Atoms_and_Ions

Size of Hydrogen is about 50pm=0.5A, while largest element in 6th row has size=300pm=3A. For most of the atoms, size lies in between 1A and 2A, except for atoms in top right and bottom left of the periodic table.

Ionization energy (IE):

We can solve for Energy for any shell (n) as E_n = -13.6eV/n^2 where n=1,2,3...

So, energy increases (becomes less -ve) as n increases. Electrons can move from n=1 to n=2 by absorbing energy (from light waves). If enough energy is supplied, electrons may jump to very large "n" shell, which is effectively close to "0" energy. "0" energy implies that electrons and protons are at infinite distance with no attraction b/w them, i.e no force to hold them together, at which point electrons are free to move out from the atom and wander anywhere. So, 13.6eV is the energy needed to free the electron, or in other terms, ionize the Hydrogen atom. You have to supply that energy via some light wave which is absorbed by the atom and frees the electron. This is known as the IE for1st electron to ionize H to H⁺.

This Bohr's theory explained why each element had it's unique spectral lines. However, it didn't talk about sub-shells s, p, d ,f and so his formula assumed same energy for all electrons in a given shell (which is incorrect).

Energy of any EM wave for a given frequency = h*ν where h=planck's constant, ν=frequency of wave. So, as freq of EM wave goes up, energy goes up. That's why X-ray, Gamma ray are considered harmful to human body, as they have very high freq, resulting in higher energy which can damage human cells.

Energy (in eV)  = h*c/λ = 1240 eV-nm / (λ in nm) => Energy of a photon in visible light wave (λ=700nm to  λ=400nm) = 2eV - 3eV

To convert eV to Joules, we can use defn of Joules which is defined as 1J=1N-m. When 1 coulomb charge moves thru Electric field of 1V/m, then work of 1J is performed. So, 1J=1Coulomb*1V

1 electron's charge = 1.6*10^-19C => 1eV = 1.6*10^-19C*1V = 1.6*10^-19J. => 1eV = 1.6*10^-19J

For any element's atom to absorb or emit photon, it's electron has to move from 1 energy state to another energy state. Since energy states are discrete based on E_n = -13.6eV/n^2, it implies that

h*ν= -13.6eV*(1/n1^2 - 1/n2^2) where n1 and n2 are two energy states where the electron jumps from/to, and ν is the frequency of wave emitted or absorbed. This matched exactly with the spectral lines observed for Hydrogen atom.

Below link shows how spectral line can be calculated based on energy of electrons jumping from 1 shell to another: https://chem.libretexts.org/Courses/University_of_Arkansas_Little_Rock/Chem_1402%3A_General_Chemistry_1_(Belford)/Text/6%3A_The_Structure_of_Atoms/6.3%3A_Atomic_Line_Spectra_and_Niels_Bohr

Ionization energy is usually expressed in J/mole (defn of mole is explained later below). Energy of 1 electron moving under Voltage 1V = 1 eV =1.6*10^-19J. 1 mole = 6.02*10^23 electrons, so energy = 6.02*1.6*10^4J=96.3KJ/mole.

So, 1eV/atom = 96.3KJ/mole.

IE of H->H⁺ is = 13.6eV = 13.6*96.3=1310 KJ/mole. This is what we see in the IE table of elements in link below for Hydrogen.

IE Concepts => https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/07%3A_Periodic_Properties_of_the_Elements/7.04%3A_Ionization_Energy

However, as we move to other elements, Bohr's simplistic model no longer works for IE. For next element Helium, the Ionization energy is lot higher at 2400 KJ/mole as He has full shell (1s2) . Same is the case with other noble gases. As we go down the periodic Table, IE goes down as electrons are in shells further away from the nucleus, so less energy is required to free that electron. As we move across a row from left to right, IE increases since effective nuclear charge increases, as well as the radius is smaller due to electrons being pulled closer in.

What we see until now is called the First IE, as it's the very 1st electron that's removed to give the atom a +ve charge. If we remove the 2nd electron, it's called the 2nd IE, and so on for 3rd IE, 4th IE, etc. As expected IE1 < IE2 < IE3 .... as nucleus is already +vely charged after the removal of 1st electron, so removing electron from more +ve species takes more energy. On top of this, if we hit a full shell, and next electron is to be removed from completely full shell, then 10X more energy is required. This excessive energy is not available in chemical reactions, that's why we see atoms losing electrons only upto when they hit the full shell.

IE of diatomic molecules like H₂ , O₂ , N₂  etc is usually higher compared to their atomic parts as energy of molecule is lower (as covalent bond formed more strongly ties the electron to proton), IE1 of H₂ is 1488 kJ/mole, while IE1 of single H atom was 1310 kJ/mole. IE2 for H₂ is higher than IE1, but is not directly measurable, as H₂⁺ is highly unstable and readily breaks into individual protons. Below link shows IE for few diatomic atoms.

IE of diatomic molecules => https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Atomic_Theory/Ionization_Energies_of_Diatomic_Molecule

Interestingly, we see that IE of F Molecule is lower than that of F atom. There are many such cases, so we need to develop theory of hybrid orbitals.

Bond Energy (BE)

For atoms, total energy is the energy required to break apart the atom into nucleus (protons+neutrons still intact) and individual electrons. For molecules, this total energy isn't that meaningful, as we are not interested in disassociating molecule into "nucleus" and individual electrons. Rather we are only interested in to breaking that bond and forming individual atoms with their electrons intact. i.e If we are forming H₂O from H₂ and O₂ , we are interested into energy required to break apart H₂O into H₂ and O₂ . This is known as bond energy or bond dissociation energy, and it reflects the energy present in that bond. If we are further interested in bond energy or H molecule or O molecule, then we break apart H₂ and O₂ into H and O atoms. Bond energy is totally different than IE.

When a molecule forms from individual atoms, usually some energy is released as the total energy of the molecule is lower than the sum of energies of it's atoms. This is because the formation of a molecule itself implies that energy would be lower for the molecule, hence the reason the molecule formation took place. Or seen from other perspective, the bond will require energy to break it, hence energy was released when forming it, implying lower energy of the molecule. Such reactions are called exothermic reaction which are chemical reaction that release energy (so the total energy of the newly formed compound is lower). In contrast the opposite reaction, where the compound is broken into individual atoms needs energy to break the bond, and is called endothermic reaction. For some compounds, even the forming of compounds from it's constituent atoms may need energy or breaking it may release energy. When few compounds react to form few other different compiunds, then it's hard to say whether energy is released or absorbed. It depends on the amount of bond energy present in each compound in the reactants as well as in the final products. We'll discuss how to find that in a separate section later.

Bond dissociation energy of H₂ molecule is 436KJ/mole.

H₂ (+Energy) -> H + H (this requires 436KJ/mole of energy for this reaction to happen). This is endothermic reaction

H + H -> H₂ (+Energy) (this releases 436KJ/mole of energy when this reaction happens, no external energy is required). This is exoothermic reaction

So, the total energy of H₂ molecule is energy of each H atom + energy released. Energy of single H atom is -1310KJ/mole, so 2 H atoms have -2620KJ/mole (or -1310 KJ for half the mole, since 2H atoms combine into one molecule as H₂. However, formation of H₂ molecule releases 436KJ/mole energy as the bond formed is lower in energy. So, energy of H₂ molecule is = -2620 - 436 = -3056 KJ/mole.

Molecules with triple bond have higher BE than those with double bond, which in turn have higher BE than molecule with single bond. BE of most common bonds shown below.

BE link => https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Chemical_Bonding/Fundamentals_of_Chemical_Bonding/Bond_Energies

Energy bands:

We saw above that isolated Hydrogen atom will have energy levels as -13.6eV, -13.6/4eV, etc. However, when 2 of such Hydrogen atoms come close together, they start interacting and affecting the fields. As such 1s, 2s, 2p energy levels from different atoms cannot co-exist due to Pauli's exclusion principle. It states that no 2 electrons can occupy the same energy level. In response, these energy levels split into 2 => So, 1s will split into 1s(low) and 1s(high) energy levels. When we have 4 atoms, they will split into 4 very closely spaced levels. For large number of N atoms, these levels will split into N levels, which forms a band. For atoms with multiple orbitals, the outermost orbitals split more than the inner ones. When large number of atoms are brought together in forming a solid, there are so many levels from splitting, that a continuous band of energy is formed. Within this continuum of energy band, there are some regions called as "forbidden energy band" or "band gaps" where no orbitals exist no matter what the temperature is. These band gaps and their widths are determined from the nature of the atoms. Explained below:

Energy bands => https://eng.libretexts.org/Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Energy_bands_in_solids_and_their_calculations

Thermal and Electrical conducting properties of solids are determined from these Energy Bands. The topmost band with completely filled electrons is called valence band (VB) (since it's filled with valence or outer shell electrons) while the band right above it is called conduction band (CB). Conduction band may be partially or not filled. Number of electrons in Conduction band determine the conductivity of that element. If the conduction band is empty to start with, and if somehow thru thermal excitation, electrons can be made to jump to this band, then that material will start conducting. This gives rise to metals, nonmetals and SemiC.

Supplement (along with sim) => https://eng.libretexts.org/Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Energy_bands_in_solids_and_their_calculations

• In Metals, valence band and conduction band overlap each other. VB is filled with electrons while CB always has empty states. In presence of Electric field, VB electrons jump to CB causing it to conduct.
• In Insulators, valence band and conduction band are separated by large energy gap of several eV. Electrons from VB can't get excited enough to jump to CB, and hence no conductivity.
• In SemiC, the bandgap is smaller at 1eV, and so with enough Electric field, electrons can get excited to jump to CB, and hence starts conducting a little. Except for this BG difference, SemiC are similar to Insulators.

Fermi Level:

The energy band in the materials is filled with electrons based on a Fermi Dirac distribution curve. This gives the probability of finding an electron at Energy level E, and at Temperature T. using this, we can find the exact distribution of electrons in any energy band. Fermi Energy level (E_F) is a hypothetical level within this band, where the probability of finding an electron is 50%. More the electrons above this Fermi level, more the electrons in CB and higher the conductivity. As Temp inc, more and more electrons occupy higher Energy band, and hence are in CB, and start conducting even more. However, as we see in the section on R,L,C, conductivity is determined both by number of electrons in CB as well as the mobility of electrons. In metals, inc in number of electrons in CB is almost negligible with Temp, as most of the electrons are already in CB. However, mobility decreases resulting in lower conductivity. With SemiC, situation changes as exponentially more electrons move into CB with inc temperature. Mobility still decreases but only as i/T^2 while electron conc in CB goes exponentially. So, conductivity of insulators and SemiC increase exponentially with inc in Temp.

Link => https://eng.libretexts.org/Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Fermi_Energy_and_Fermi_Surface

Isotopes:

We saw that number of protons is unique to an element. But how about number of neutrons, is that unique too? Turns out, that number of neutrons for an atom of an element can change, but the chemical properties are almost the same. Such variation of elements which have atoms with different number of neutrons are called isotopes of that element. It's very hard to distinguish b/w Isotopes of an element, as spectroscopy (which depends on electron's energy levels) will give same spectral lines for diff isotopes. Their chemical and physical properties are almost indistinguishable, except for their mass. Isotopes may be stable (don't decay due to radioactivity) or unstable (decay over time due to radioactivity).

Mass Spectrometry: To separate out isotopes of an element, a special equipment/ called Mass spectrometer is used (mass is used in definition implying it measures mass via spectrometry). Lower mass isotopes are deflected more than ones with higher mass.

Nuclide is a term similar to isotope where it refers to atom whose nucleus has specified number of protons and neutrons. For ex: Carbon-13 is a nuclide or isotope of Carbon with 6 protons and 7 neutron (number 13 refers to atomic mass number A). Carbon-13 is also written as ¹³
₆C
where 13 on top refers to A and 6 on bottom refers to Z. We need both numbers A and Z in defining an element uniquely, since if we omit A, then we can't diff b/w isotopes. The term nuclide is used since nuclear properties of these isotopes vary a lot, depending on the number of neutrons in nucleus. A total of 3339 nuclides is known of which 339 are naturally occurring, and remaining 3000 have been created artificially. Of these 339, 286 have existed since creation of solar system,  with 251 being stable nuclides and remaining 35 with very long half life of > 100M years. So, that leaves us with 53 (339-286=53) naturally occurring isotopes which are radioactive. Stable isotopes are the only ones we talk about at lower Z. Unstable isotopes occur at high Z (starting from polonium with Z=84). They are studied in detail in nuclear chemistry that we'll study separately.

Wiki: https://en.wikipedia.org/wiki/Isotope

Hydrogen is the only element whose isotopes show most diff in chemical properties primarily due to the fact that the 3 isotopes of Hydrogen differ by weight in ratio of 1:2:3, so that starts affecting the chemical reactions, as much heavier isotopes of an atom have lower reactivity than lighter isotopes of same element. When we go to high Z elements, the extra number of neutrons don't cause weight to go up by 2X or 3X, so their extra weight effect is much more muted in chemical reactions.

When we look atomic weight (NOT atomic mass number) of elements in periodic table, we see that none of them are integer values, even though they are expressed in amu. How is that possible, since number of protons or neutrons is an integer number? The reason is that atomic weight of any element is the weighted avg of all naturally occurring isotopes of that element (since that is how atomic weights were determined historically. Pure samples of that element were taken and weights calculated, which contained isotopes of that element in same proportion as what was found in nature). About 80 elements in periodic table have stable isotopes. There's a chart in link above that shows the plot of N(# of neutrons) vs Z (# of protons). For lower Z, N is usually same or 1 more than Z. Generally the ratio of N:Z keeps going higher as Z increases, since more neutrons are needed to bind the nucleus together when more protons are present (needed to keep protons further apart and lower their repulsive forces, which is possible if we insert more neutrons b/w them which attract not only other neutrons but other protons too).

Ex: Berillium has 2 isotopes with N=3 and N=4 (i.e atomic mass=6 amu with 7.5% concentration and 7 amu with 92.5% concentration). This gives avg amu as = 6*0.075+7*0.925=6.92 which is close to the amu reported in the periodic table. Similarly for all other elements.

Magic Numbers (MN): Whether an isotope is stable or not is determined by the number of protons and neutrons in nucleus. We saw the bank of stability in above wiki link. There is a trend observed with elements Helium, Oxygen, Calcium, Nickel, Tin, Lead and hypothetical Unbihexium. Their isotopes are highly stable because they have magic number of protons or neutrons in their nucleus. MN is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the nucleus. The seven most widely recognized magic numbers as of 2019 are 2, 8, 20, 28, 50, 82, and 126. 

Doubly Magic: Nuclei which have neutron numbers and proton numbers both equal to one of the magic numbers are called "doubly magic", and are generally very stable against decay. The known doubly magic isotopes are Helium-4, Helium-10, Oxygen-16, Calcium-40, Calcium-48, Nickel-48, Nickel-56, Nickel-78, tin-100, tin-132, and lead-208.

Wiki => https://en.wikipedia.org/wiki/Magic_number_(physics)

Lbretext => https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Nuclear_Chemistry/Nuclear_Energetics_and_Stability/Nuclear_Magic_Numbers

Nuclear power => https://www.nuclear-power.com/nuclear-power/reactor-physics/atomic-nuclear-physics/nuclear-stability/

All isotopes try to get closer to the band of stability by undergoing decay. This decay is called radioactive or nuclear decay and the unstable atom loses energy and emits out energy in form of ionizing radiation.

• Alpha decay: Nuclides which have Z>82 are at top of plotted kine and unstable. which have  decay by emitting a alpha particle which is just a He Nucleus. This reduces N by 2 and Z by 2, effectively changing the element to something else, but making it more stable in the process (as the new element gets closer to stable band). Short range and low energy.
• Beta Decay: Nuclides which have very high number of neutrons Beta decay have more energy and more range than alpha decay. There are 2 types of beta decay.
  • Beta β^- Decay: Nuclides which have very high number of neutrons compared to protons (lying above the band of stability) decay by accepting protons. The increases Z by 1 and decreases N by 1. This also changes the element.
  • Positron β⁺ Decay:  Nuclides which have very low number of neutrons compared to protons (lying below the band of stability) decay by releasing protons (i.e positron). The decreases Z by 1 bringing Z closer to N. Electron capture is another way to decay where an electron is captured in the nuclei, basically reducing Z by 1. This results in release of X-ray, and is different than positron decay, but achieves the same result.
• Gamma decay: This is where high energy photons are released from an excited nucleus in high energy state to bring it to a low energy state. Gamma rays are most energetic and have high frequency.

Mole and Avagadro Number:

Since the number of atoms is very large even in small amounts of any element, we need to come up with a unit for number of atoms. Mole is such a unit. It's the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of "carbon-12 ¹²
₆C
". We take 12 g of ¹²
₆C
 in the defn of mole, but we could have easily taken 1g of "Hydrogen-1 ¹
₁H
" in the defn. 1 atom of "Carbon-12" has 12 amu of weight, while 1 atom of "Hydrogen-1" has 1 amu of weight.

So, 12g of C-12 has = 12g/12amu = 1g/1amu atoms, while 1g of H-1 has 1g/1amu atoms. So, all these elements have same number of atoms for a weight that is normalized depending on how many amu of weight each atoms has for that element compared to H-1 atom. To keep it simple, we say 1 mole is the number of atoms in 1 g of H-1 (or 12g of C-12 to be precise). It turns out 1 mole = 6.02*10^23 (approx), and that is the number of atoms in 1g of Hydrogen. This is also the number of atoms in 4g of He (since each He atom is 4 amu), and 16g of Oxygen (since each Oxygen atom is 16 amu).

The exact number 6.02214076 * 10^23 is also known as "Avagadro number" or "No". The exact number of atoms in a given mass of an element is impossible to determine, since it will vary considerably depending on the shape, crystalline structure, purity, temperature, pressure, etc. However, for sake of definition, we chose this somewhat arbitrary number to 8 decimal places as our reference number for "Avagadro Number".

Details

Published: Wednesday, 01 March 2023 08:06

Hits: 489

Before we start with real Verilog coding, we should use an editor which recognizes Verilog mode and formats it appropriately.

We have "verilog mode" in emacs that not only allows formatting, but allows us to write succinct code which can be expanded to real Verilog code. We'll talk about this feature below.

Verilog mode for Emacs:

Verilog-mode.el is the extremely popular free Verilog mode for Emacs. It can be downloaded from link over here (It's not included by default in standard emacs download). https://www.veripool.org/wiki/verilog-mode

Once downloaded and installed, emacs will start showing verilog specific highlighting, indenting, etc. To verify if verilog mode is there in your emacs, open emacs. You would see tabs on top for File, Edit, Options, etc. If verilog mode is installed, there should be a tab for "Verilog". If you click on "verilog" tab, you should see a pull down showing various action tabs as compile, Recompute AUTOs, etc. You can do more customization by reading help section.

Verilog has a lot of redundant typing built into the syntax. Verilog mode for emacs provides "AUTO" keyword that we can put in verilog code within comments. Putting these keywords within comments /* */ guarantees that this code will work seamlessly with any other tool, which has no knowledge of these "AUTO" keywords as they are in comments.Only verilog mode in emacs looks for these special keywords to expand your verilog code. We can always manually edit the code further or make changes that we didn't like that were introduced by AUTO.

Info about AUTO keywords here: https://www.veripool.org/wiki/verilog-mode/verilog-mode_veritedium

To make changes after inserting AUTO* keywords in verilog code, we can use shortcut "Ctrl+c , Ctrl+a", or goto Verilog tab on emacs and click on "Recompute AUTOs". This expands AUTO keywords (still keeps the commented AUTO keyword so that we know ehere the changes happened).

For older designs which don't have AUTO keywords in them, we can use emacs to inject AUTO in them, so that next time if there are more changes, we can just do "Recompute AUTOs". For Inject AUTOs, we can use shortcut "Ctrl+c , Ctrl+z". All of these AUTO keywords would be injected within comments, so they won't affect if used with other tools.

verilog mode also allows us to expand system verilog .* port connections, so it's easy to check and debug in source code.

Few AUTO cmds:

NOTE: there should be no space between /* and AUTO keyword, i.e /*AUTOARG*/ will work, but /* AUTOARG */ may not work due to spaces)

1. AUTOARG: This is to add argument list for any module. It looks at i/p and o/p ports and adds those in port list of module.

ex: module (/* AUTOARG */); input a; output b; ... => gets transformed into => 

module (/* AUTOARG */

//outputs

out_1, out_2, ..., out_n,

//inputs

in_1, in_2, ..., in_n);

input a; output b; ...

NOTE: Even if we declare no input/output ports. AUTOARG expand will look for undriven input signals and floating output signals, and assign them as input/output ports. This is useful to detect ports automatically.

ex: module (/* AUTOARG */); //This will automatically infer all I/O ports and put them here after expansion

NOTE: To distinguish ports inserted manually by the user, vs ports inferred from AUTOARG expansion, verilog adds comments in auto inserted ports. i.e //From mod_1 of mod_1.v or //To mod_1 of mod_1.v

2. AUTOINPUT, AUTOOUTPUT: This looks for undriven input signals and floating output signals, and assign them as input/output ports. This is useful to detect ports automatically.

ex: module (/* AUTOARG */); /* AUTOINPUT */ /* AUTOOUTPUT */ //This will automatically infer all I/O ports and put them here after expansion (along with AUTOARG expansion)

3. AUTOINST: This automatically adds port connections for a submodule by reading module defn. It creates net connections with same names as port names for module defn. NOTE: AUTOARG is for adding ports to module, while AUTOINST is to add port connections to sub-module.

ex: submod I_submod (/* AUTOINST */); => gets transformed into => submod I_submod ( .out(out), .in(in) ...);

However, sometimes we may want connecting wires to be named differently than port name. In that case we can define exceptions by coding those connections ourselves. AUTO won't overwrite those.

ex: submod I_submod ( .a(a_1), /* AUTOINST */); => gets transformed into => submod I_submod ( .a(a_1), .out(out), .in(in) ...); => NOTE: port "a" remain connected to "a_1"

NOTE: To distinguish ports connected manually by the user, vs port connections inferred from AUTOINST expansion, you can put your own comments for manually connected ports (while auto port connections will show no comments)

4. AUTO_TEMPLATE: NOTE: this has a underscore in it. We use this keyword for same situation as AUTOINST, but when there are multiple instantiations of a module. In such a case, writing exceptions for port connections for each submodule become cumbersome, so we use other reserved special char (as @, $, etc) to make complicated renaming possible.

ex: /* submod AUTO_TEMPLATE (.z(out[@]), .a(invec@ [])); */ => NOTE: here whole thing is in comments as opposed to just the keyword AUTO_TEMPLATE. @ in template takes the leading digits from instantiating module's name. empty [] takes the bit range for the bus from module defn. More complicated lisp expressions possible. Regular expressions also supported. @ suffices for most of the cases. AUTO_TEMPLATE is used in conjunction with AUTOINST. AUTO_TEMPLATE does all complicated connections with renaming, and anything left over is taken care of by AUTOINST. However, AUTOINST always overrides AUTO_TEMPLATE connections in case of conflict, so hardcode connections in instantiation of module if you are unsure what the AUTO_TEMPLATE or AUTOINST is going to do (generally it's used where we want to tie port connections to constants, so we hard code those connections in actual inst of module, as you see in ex of AUTOINST above).

ex: submod I_0 (.z1(out_z), .i2(), .. /* AUTOINST */); => This is regular instantiation of submod. Here AUTOINST is used in regular way. However, since AUTO_TEMPLATE is also defined for this module "submod", emacs will attempt to make connections for port "z" and port "a" (port z connected to out[0] and port a connected to invec0[31:0], since @ takes leading digits from I_0 which is 0. If there are more instances like I_BL_1, then connections will be named out[1], invec1[31:0], etc))

ex: /* submod AUTO_TEMPLATE (.sense_\(.*\) (sense@_\1), ..) */ => \( \) is emacs basic regular expression (see in linux regexp section). Anything inside it is put in var \1. So, here for inst I_0, sense_in4 would be connected to sense0_in4, for inst I_1, it would be sense1_in4 and so on.

NOTE: To distinguish ports connected by AUTO_TEMPLATE, verilog adds comments in auto inserted ports. i.e //Templated. Any user comments put in AUTO_TEMPLATE section are not shown in expanded connections.

5. AUTOWIRE, AUTOREG, AUTOLOGIC => AUTOWIRE takes o/p of submodules and declares wires for them. AUTOREG saves having to duplicate reg stmt for nets declared as outputs. AUTOLOGIC declares logic instead of wire or reg.

ex: module ... output y; /* AUTOWIRE */ /*AUTOREG */ wire a; ... => gets transformed into => ... output y; /* AUTOWIRE */ /*AUTOREG */ wire b; reg y; wire a; ...

6. AUTOSENSE (or shortcut AS) => replaces everything with sensitivity list. System Verilog supports @* for sensitivity list, so AS is not needed any more.

ex: always @ (/* AS */) begin ... end => gets transformed into => always @ (/* AS */ a or b or sel) begin ... end

Running emacs in batch mode:

Instead of opening emacs and doing it on gui, we can run emacs in batch mode (i.e as cmd line w/o invoking editor) as follows:

emacs -batch my_design.v -f verilog-auto -f save-buffer => same as doing "compute AUTOs" and saving in same file

Details

Published: Saturday, 25 February 2023 01:30

Hits: 733

Classification of Life:

The very first thing that we should know when studying biology is what is a living being. This is what Biology is all about. Since there are so may different kinds of living things, scientists have come up with grouping all of these living things into different buckets. Cell is the fundamental thing that is common to most of the living things.

Wikipedia article on Life: https://en.wikipedia.org/wiki/Life

Origin of Life:

We talked about Starting of the Universe and subsequent formation of Earth in "Astronomical Science" section. Here we study about when life started forming on Earth.

Timeline of Evolution : https://en.wikipedia.org/wiki/Timeline_of_the_evolutionary_history_of_life

The formation of planet where life could form started around 4 Gya. Some fossils indicate life formed between 4Gya and 3.5Gya. 

The last universal common ancestor (LUCA) is the most recent population from which all organisms now living on Earth share common descent. It is believed to have lived around 4Gya in high temperature water near ocean floor, was single celled and had genes or genetic code. There has never been any fossil evidence confirming it's presence. However based on the tree of life, there had to be some common ancestor, and that last common ancestor is named LUCA. There may have ancestors to LUCA, but we are limiting ourselves to the last common one since that is what matters for our future evolution.

Photosynthetic organisms appeared between 3.2 and 2.4 Gya and began enriching the atmosphere with oxygen. More complex unicellular cells called Eukaryotes appeared capable of sexual reproduction appeared around 2 Gya. Earliest multicellular cells appeared around 1.5Gya. Life remained mostly small and microscopic until about 580 million years ago, when complex multicellular life arose, developed over time, and culminated in the Cambrian Explosion about 538.8 million years ago. It's called the "Biological Big Bang". This sudden diversification of life forms produced most of the major phyla known today. It is estimated that 99 percent of all species that ever lived on Earth, over five billion years ago have gone extinct. Estimates on the number of Earth's current species range from 10 million to 14 million of which about 1.2 million are documented, but over 86 percent have not been described. However, it was recently claimed that 1 trillion species currently live on Earth, with only one-thousandth of one percent described.

First life must have arose from non living material. Biology, abiogenesis (from a- 'not' + Greek bios 'life' + genesis 'origin') or the origin of life is the natural process by which life has arisen from non-living matter, such as simple organic compounds. The prevailing scientific hypothesis is that the transition from non-living to living entities on Earth was not a single event, but a process of increasing complexity involving the formation of a habitable planet, the prebiotic synthesis of organic molecules, molecular self replication, self-assembly, autocatalysis, and the emergence of cell membranes.

Cells:

Cell is the structural and functional unit of life. Every living thing is made up of cells, with simplest life made of 1 cell, and more complex ones made of trillions of cells. Look in "Human cell" section for details.

Classification of life:

Earliest classification of life was conducted by the Greek philosopher Aristotle (384–322 BC), who classified all living organisms known at that time as either a plant or an animal, based mainly on their ability to move. However as more complex plats and animals were found, new classification was needed. In 1740s, Carl Linnaeus introduced his system of binomial nomenclature for the classification of species. Linnaeus attempted to improve the composition and reduce the length of the previously used many-worded names. The Linnaean classification has eight levels: domains, kingdoms, phyla, class, order, family, genus, and species. Only cellular life are included in life's classification. Virus being non cellular are not considered in this classification.

Below is the classification:

1. Domain: A domain, superkingdom, or empire, is the highest rank of all Organisms taken together. There used to be 2 empire system: Prokaryota and Eukaryota. However, Carl Woese made a revolutionary breakthrough in 1977, and realized that Prokaryota domain that grouped Archaea and Bacteria into one were actually genetically different. They arose separately from an ancestor with poorly developed genetic machinery. This gave rise to 3 domain system whose taxonomy was devised by Carl, Otto and Mark in 1990.
   
   1. Archaea: These are prokaryotes. The first observed archaea were extremophlies, living in extreme environments such as hot springs and salt lakes with no other organisms. Improved molecular detection tools led to the discovery of archaea in almost every habitat. Archaea are particularly numerous in the oceans, and the archaea in plankton may be one of the most abundant groups of organisms on the planet. Archaea were initially classified as bacteria, because of numerous similarity b/w them. Archaea have membrane lipids that are branched hydrocarbon chains attached to glycerol by ether linkages. The presence of these ether linkages in Archaea adds to their ability to withstand extreme temperatures and highly acidic conditions,
      
      • Link: https://en.wikipedia.org/wiki/Archaea
   2. Bacteria: Bacteria are prokaryotic cells just like Archaea, but their cell membranes are instead made of phosopholipid bilayers. They characteristically have none of the ether linkages that Archaea have. Internally, bacteria have different RNA structures in their ribosomes making it a domain by themselves. They constitute a large domain of prokaryotic microorganisms.  Most bacterial species exist as single cells; although they can sometimes group to form larger multicellular structures. These multicellular structures are often only seen in certain conditions. Humans and most other animals carry millions of bacteria. Most of the bacteria in and on the body are harmless. However, several species of bacteria are pathogenic and cause infectitious disease. Antibiotics are used to treat those bacterial infections.
      
      • Link: https://en.wikipedia.org/wiki/Bacteria
   3. Eukaryota: This includes eukaryotes explained above. Archaea and Bacteria are Prokaryotes.
2. Kingdom: A kingdom is the second highest taxonomic rank, just below domain. There are 6 kingdoms as per US books, while some countries only refer to 5 kingdoms.
   1. Empire Prokaryota is divided into 2 kingdoms: Eubacteria (Bacteria domain) and Archaebacteria (Archaea domain). This was originally just 1 kingdom called Monera, but Carl Woese discovery caused it to separate out into 2 kingdoms.
   2. Empire Eukaryota is divided into 4 kingdoms: Plantae (plants), Animalia (animals), Fungi and Protista. Protista is kingdom of primitive forms, which doesn't fall in other kingdoms. Protista consists mostly unicellular and simple multicellular organisms. Fungi don't photosynthesize like plants, and don't move like animals (growth is their way of mobility). They had to be classified separately mainly because of difference in nutrition. These are 3 modes of nutrition for organisms:
      1. Autotroph: An autotroph is an organism that produces complex organic compunds using carbon from simple substances such as CO₂ generally using energy from light (photosynthesis) or inorganic chemical reactions (chemosynthesis) They convert an abiotic source of energy (e.g. light) into energy stored in organic compounds, which can be used by other organisms (e.g. heterotrophs). Autotrophs do not need a living source of carbon or energy and are the producers ina food chain (plant or algae)
      2. Heterotroph: A heterotroph is an organism that cannot produce its own food, instead taking nutrition from other sources of organic carbon, mainly plant or animal matter. In the food chain, heterotrophs are primary, secondary and tertiary consumers, but not producers.
      3. Saprotroph: A saprotroph is an organism involved in the processing of decayed (dead or waste) organic matter. Sapro means "decayed matter". Saprotrophic nutrition is a process of chemoheterotrophic extracellular digestion, mostly seen in fungi.
3. Phylum: A Phylum is below Kingdom. Phyle means tribe or clan. Phyla can be thought of as groupings of organisms based on general specialization of body plan. At its most basic, a phylum can be defined in two ways: as a group of organisms with a certain degree of morphological or developmental similarity, or a group of organisms with a certain degree of evolutionary relatedness. The minimal requirement is that all organisms in a phylum should be clearly more closely related to one another than to any other group. Link: https://en.wikipedia.org/wiki/Phylum. Following are current Phyla classification:
   
   1. Archaea Kingdom = contains 2 phyla
   2. Bacteria Kingdom = contains 40 phyla
   3. Animalia Kingdom = contains 40 phyla. Arthropods (vertebrates with segmented body and jointed limbs) phyla account for over 80 percent of all known living animal species. Humans belong to Chordata phylum, which includes birds, amphibians, mammals, etc.
   4. Plantae Kingdom = contains 14 phyla
   5. Fungi Kingdom = contains 8 phyla
   6. Protista Kingdom = contains 19 phyla
4. Class: A class has historically been conceived as embracing taxa that combine a distinct grade of organization—i.e. a 'level of complexity', measured in terms of how differentiated their organ systems are into distinct regions or sub-organs—with a distinct type of construction, which is to say a particular layout of organ systems. Not too many details available. Not important. Link: https://en.wikipedia.org/wiki/Class_(biology)
5. Order: Order is one level below class. Not too many details available. Not important. Link: https://en.wikipedia.org/wiki/Order_(biology)
6. Family: Family is one level below Order. Not too many details available. Not important. Link: https://en.wikipedia.org/wiki/Family_(biology)
7. Genus: Genus is an important level just below Family and above Species. There are several criteria to group organisms into one genus. Link: https://en.wikipedia.org/wiki/Genus
8. Species: Species is the basic unit of classification. It is often defined as the largest group of organisms in which any two individuals of the appropriate sexes can produce offspring. The most recent rigorous estimate for the total number of species of Eurokytes is between 8 and 8.7 million, with many more discovered every year. "Animalia" kingdom has the largest number of species at around 300K, with "Plantae" kingdom a distant second with 30K species. Link: https://en.wikipedia.org/wiki/Species

Binomial Nomenclature: All species are given 2 part name in biology. The first part of a binomial is the Genus to which the species belongs. The second part is the specific name or specific epithet to which they belong within that species.

• Ex: "Boa Constrictor" is one of the species of the genus "Boa", with "constrictor" being the species' epithet.
• Ex: "Gray wolf's" scientific name is Canis lupus, with Canis (Latin for 'dog') being the generic name shared by the wolf's close relatives and lupus (Latin for 'wolf') being the specific name particular to the wolf. 

Details

Published: Tuesday, 31 January 2023 00:39

Hits: 683

Elementary School Maths :

This is Maths starting from grade1 and continuing grade8. Grade 6 to Grade 8 is known as middle school and not elementary school, but until 8th grade, maths being taught is still elementary in nature, so we'll club it all in Elementary maths. In fact all the maths until 8th grade can be grasped by a kid by his 5th grade very easily, and is a 6-12 month effort in elementary school. I would highly advise to complete all 8th grade maths by 5th grade, so that the kid can start learning real high school maths by start of his 6th grade. If your kid is practicing maths sample papers for grade 5, I would highly suggest practicing 8th grade maths papers, as the content is almost the same from 5th grade to 8th grade, it's just that the difficulty level increases a little bit.

This is the sequence of study material from grade 1 to grade 8. Basic Algebra, Basic Trignometry and Basic Plotting are the main areas covered here.

Basic Algebra:

From mathsisfun website, we can start algebra from this pre algebra section: https://www.mathsisfun.com/algebra/index-pre-algebra.html

Then we can move on to algebra 1 section: https://www.mathsisfun.com/algebra/index.html

Numbers:

Natural numbers: numbers starting from 1 to infinity. We don't include 0 since earliest humans didn't know 0 when they started counting. They also didn't know about decimals, fractions, etc. So you can think of natural num as those occuring naturally, i.e complete num with no decimals or fractions. ex: 1, 2, 3, ....

Whole numbers: numbers starting from 0 to infinity. They are whole, no decimals or fractions. ex: 0, 1, 2, 3, ....

Integers: Same as whole numbers except they can be negative. ex: -2, -1, 0, 1, 2, ..

$\text{Rational numbers: N}$start color #7854ab, start text, R, a, t, i, o, n, a, l, space, n, u, m, b, e, r, s, end text, end color #7854ab: : N umbers that can be expressed as a fraction of two integers.

ex: 4/5, 1.45, 7/9, 7/9, 0.777777, √36, etc.

When a number has digits that repeat for ever, we put a bar on the top of repeating numbers to indicate it's repeating. So, 0.77777... repeating for ever is rep as 0.7 and a line on top of 7.

NOTE: For number 0.7777, it looks like it can't be represented by a fraction, as it goes on forever. However, it can be proved to be a fraction as follows:

x=0.7777, 10x=7.77777 => 10x-x=7 => 9x=7 => x=7/9

In fact, any number of form where set of numbers is repeated infinitely can be rep as fraction.

ex: x=2.3475757575... => 100x=234.757575... => 100x-x=234.7575 - 2.347575 => 99x=234.75-2.34=232.41 => x=232.41/99 => x=23241/9900 => This fraction can be reduced further if it has common factors.

Brain Teaser: One very weird and common sense defying number is 0.99999. If we take x=0.9999, then 10x=9.9999 => 10x-x=9 => 9x=9=> x=1. So, the fraction to give 0.999 is 1. But dividing 1/1, we never get 0.999. So, how did 0.999 become 1? We know 0.999 is not equal to 1. In limiting case, it becomes 1. But it's formally proved to be 1, which looks incorrect as dividing 1 by 1 will never give you .9999. Does that mean that .9999 is irrational (see below for irrational numbers), since we couldn't find 2 integers p and q such that p/q=0.99999. Go figure it out !! There is a link here (look at the bottom of the page in the link): https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html

Irr$\text{ational numbers: N}$start color #7854ab, start text, R, a, t, i, o, n, a, l, space, n, u, m, b, e, r, s, end text, end color #7854ab: : N umbers that can't be expressed as a fraction of two integers. Thousands of years back, it was thought by the smartest mathematicians that all numbers are rational, as they could not see how a number can't be represented by a fraction. Their thinking was that if any decimal number existed, it can be put on a number line and by choosing higher numerator and denominator for the fraction, they can get closer and closer to it. However they were proved wrong later. These new set of numbers were so unorthodox, they they called them irrational. Number pi was such an example, and was proved rigorously in last 200 years or so to be irrational. Similarly many square roots as √2, √3, etc are irrational. Irrational numbers are totally different class of numbers. All whole numbers and integers fall under the umbrella of Rational numbers.

One quality of irrational numbers is that the decimal numbers don,t ever repeat in any pattern (for ex, if the decimal numbers repeated, we could employ the process for rational numbers to find such a fraction)

ex: Π (pi), √2, √3, √6, etc

There is very simple proof of why √2 can't be rep as a fraction. The proof assumes that √2 = p/q, and then by contradiction concludes that it can't be fraction. Proving pi is irrational is much more difficult.

Basic Arithmetic: 4 kinds of operations allowed on numbers => addition, subtraction, multiplication and division.

3 ways to represent numbers => Integers (both +ve and -ve), Decimal, Fraction (we won't talk about irrational numbers for now). The table below shows all possible combination of 4 operations on these numbers. These operations form the basic of Maths education, so make sure your kid is comfortable with all the operations below. It's not necessary to learn the multiplication table, but it does help to know multiplication table from 1 to 10.

Once all above arithmetic is understood well by the kiddo, there's not much left in Maths algebra.

add/sub rules: -(+ve) = -ve, +(-ve)=-ve, +(+ve)=+ve, -(-ve)=+ve

mult/div rules: +ve mult/div +ve = +ve,  -ve mult/div +ve = -ve, +ve mult/div -ve = -ve, -ve mult/div -ve = +ve, I haven't found a good explanation of why -ve multiplied or divided by -ve becomes +ve ??

distributive property: This is one of the most important properties and difficult one to grasp for kids. A*(B+C) = A*B + A*C

Successive Distribution: An extension of distributive property is applying it successively. Ex: (A+B)*(C+D) => Here we treat (A+B) as one variable "P". then this thing becomes P*(C+D). We can apply distributive property to get P*C+P*D. Now we substitute P back We get (A+B)*C+(A+B)*D. Now we can again apply distributive property to get A*C+B*C+A*D+B*D.

So, (A+B)(C+D) = A*C+B*C+A*D+B*D => This can be extended to any number of variables and any number of nested parenthesis.

Order of operations: Given a complex arithmetic as 2*5+7+ (4+5)/9 + 2, student should be able to know the BODMAS (aka PEDMAS) rule and Left to right rule. Basically first look for precedence of operators, and if operators have same precedence then use left to right rule (i.e when * and / appear in an equation, then left to right rule used to do computation, since * and / are at same precedence). This is just a convention, so that different people don't interpret the same equation differently and come up with different answers. The prudent thing to do with equations is to enclose them in parenthesis, so that it's always clear, which ones you want to do first.

ex: a-(b+c).d+e = a-bd-cd+e

operation on fractions: As explained in the table above, add/sub, mult/div on fractions are same as on decimals. However, few points to keep in mind:

A. (a+b)/c => is same as (a+b)*1/c => same as a/c + b/c (distributive property)

B. c/(a+b) => This cannot be reduced any further. Distributive property applies to numerator, not to denominator. i.e c/(a+b) is not the same as c/a + c/b

• c/a + c/b = c*(1/a+1/b) = c(a+b)/ab = c/(a+b) [(a+b)^2/ab)] => My multiplying both numerator and denominator by (a+b). We see the extra term in sq bracket which is never 1, so c/(a+b) ≠ c/a + c/b

C. (a+b)/(c+d) => is same as a/(c+d) + b/(c+d) => Distributive property applies to numerator here. Denominator remains intact.

Percentage, ratio can now be introduced. They are just different way of writing fractions or decimals. No new concept here. Make sure the student understands that x% is just x * 1/100 (i.e % is just other way of writing out of 100 parts).

Prime numbers: Prime numbers (PN) are numbers which are only divsible by 2 numbers, 1 and itself. 1 is not a PM as it has only divisor: 1 (while PN as per defn needs 2 divisors). 2 is the only PN which is even. PN are very important concept to develop, and a kid should be able to figure out prime numbers for all numbers less than 100. Make sure they know all the prime numbers correctly between 1 to 100. We write all numbers b/w 1 to 100, and then start striking off numbers which are multiples of 2, 3, 5 and 7. This will yield PN as shown below (Maybe ask kids in 5th grade to write a simple program in python to test if a number is prime or not).

All PN < 100: (there are 25 such numbers)

• 2, 3, 5, 7
• 11, 13, 17, 19
• 23, 29
• 31, 37
• 41, 43, 47
• 53, 59
• 61, 67
• 71, 73, 79
• 83, 89
• 97

Composite number: Composite number is defined as number which has more than 2 factors. Basically, it's numbers which ar not prime (with the exception of 1, which has only 1 factor). So, 1 is neither prime nor composite. All other numbers which are not prime are composite.

Co prime numbers: Co prime numbers are numbers that have only one common factor which is 1, i.e they are prime wrt each other. i.r 8 and 15 and coprime since factors of 8 are 1,2,4,8 while factors of 15 are 1,3,5,15. So, by themselves 8 and 15 are not prime, but thay are coprime as a pair as only common factor b/w them is 1. 

Factorization: Factorization refers to expressing a integer as a multiple of 2 or more integers. Important thing to note is that we say integers, which means +ve and -ve numbers can be factors of a integer. Also, 1 and the number itself are always factors of a given number, since 1*number=number.

ex: 36=2*18, here 2 and 18 are factors of 36. To find all factors of 36, we find out all integers which completely divide the number (without leaving a remainder). So, in this case, 1,2,4,6,12,18 and 36 are all factors of 36. On top of this, all -ve counterpart of these numbers are also factors of 36, since 36=(-2)*(-18). So, -1,-2,-4,-6,-12,-18,-36 are also factors of 36, though we don't usually write -ve numbers as factors (not useful). But strictly speaking, they are factors.

Prime factorization: prime factorization can also be grasped well by a kid in 5th grade. We saw how to find factors of a integer. Of al the factors of a integers, the factors which are prime are called prime factors. It should be stressed that any number can be factored into prime numbers, and there is one and only one unique way of factorizing a number into prime numbers. Most of the smaller numbers can be factorized into prime numbers 2, 3 and 5.

ex; 36 = 2*2*3*3 = 2^2 * 3^2 (integers 2 and 3 are prime factors here)

We can find all factors of a given number from it's prime factors, since all other factors are formed from these prime numbers. Let' say a number is factored into prime numbers as follows:

N = p1^e1 * p2^e2 * ... * pn^en where N is a given +ve integer whose prime factors are p1,p2,..,pn, and their corresponding powers are e1,e2,..,en.

All possible factors with p1 prime number are 1, p1, p1^2, p1^3, ...., p1^e1

Similarly all possible factors with p2 prime number are 1, p2, p2^2, p2^3, ...., p2^e2

All the way to pn are 1, pn, pn^2, pn^3, ...., pn^en

We can line up all of these prime factors into a column, with 1st col being 1, p1, p1^2, ..., p1^e1,  2nd col being 1, p2, p^2, ..., p2^e2, and so on. Let's try to figure out all possible factors of N (not just prime factors):

• From first row of all col except the last one,we get (en+1) factors: 1st factor of N is 1st row of all col = 1*1*1...*1 (N times) = 1, 2nd factor = 1*1*1...*1*pn = pn, 3rd factor = 1*1*1...*1*pn^2 = pn^2, ... en th factor= 1*1*1...*1*pn^2 = pn^en = pn^en
• Now if we take second last col, we get (e_n-1 + 1) factors for each factor we got above.
• If we continue this way, the very first col has (e1+1) factors.
• So, total number of all factors possible is (e1+1)*(e2+1)*....*(en+1)

As an ex: 162=2^1*3^4 => This has total (1+1)*(4+1)=10 factors which are 1*3, 1*3^2, ..., 1*3^4 for 1st row (total 5 factors possible), then 2*3, 2*3^2, ..., 2*3^4 for 2nd row (total 5 factors possible). There are no other factors possible, as every factor is accounted for. None of the factors are repeated here, as they are all made from prime numbers with different exponent to each of the prime numbers. So, total number of unique factors possible is 10.

This is all the reason, why prime factors so important in factorizing any number. First they allow us to compute all possible factors, and secondly they allow us to have a unique representation of any number as multiple of other numbers.

HCF, LCM: HCF can be taught to help the kid learn how to reduce fraction to it's lowest ratio, while LCM can be taught to help him learn how to add or subtract fractions when denominators are different. There are techniques based on prime factorization that help them determine HCF and LCM. Look up on Khan Academy. Of course what I've seen is that kids don't really use LCM HCF for these fractions. They use it only when they are directly asked HCF or LCM of 2 numbers. My son still reduces fractions the long way (by dividing it by a small number as 2,3 etc and then repeatedly dividing it), instead of using HCF !!

Fraction reduction: Fraction reduction can be done by repeatedly dividing numerator and denominator by same number, until they are prime wrt each other. This is an an important area that kids will need to be comfortable with, since they will need to identify equivalent fractions. One simpler way to reduce fractions is to do prime factorization of numerator and denominator, and then cancel the common terms.

ex: 52/72 = 2*2*13/2*2*2*3*3 = 13/18

Other way to reduce fractions by using HCF.

ex: 52/72 => Find HCF of (52,72). Then divide both 52 and 72 by their HCF.

Mean, Median and Mode:

Mean, Median and Mode are simple concepts.

Mean: It is the avg value of a given sample. It's calculated by summing the value of all the samples divided by the number of samples.

ex: If kids in a class have scores of 70, 90, 10 and 30, then Mean = (70+90+10+30)/4 = 200/4 = 50.

Quantities which are formed division of other quantities can't be averaged by just dividing it by number of samples.

Ex: density = weight/volume.  Let's say we have sample A with density of 2g/cm^3 and sample B with density of 4g/cm^3. If we mix them in ratio 1:1 by weight, what is the mean density of the mixture. It seems like mean density should be the avg of 2 densities, so mean should be 3g/cm^3.

Mean density = Total_weight/Total_volume => That is how any mean is defined.

Let's we take x g of sample A, then sample B is also x g. So, total weight=2*x g. Total volume = x/2 cm^3 + x/4 cm^3 = x(1/2+1/4)=3*x/4, So mean density = 2*x/((3/4)*x) = 8/3 = 2.66g/cm^3 and 3g/cm^3 as expected. However, if they were mixed in 1:1 by volume, then

Median: Median is the mid point of a sample where half the samples values are below that number, and the other half are above that number.

ex: For above ex, our median has to be a number which is greater than 10, 30 but less than 70 and 90. We may choose such number to be any number as 40, 50, 60, etc and they will all be median. Generally we choose the avg of 2 middle numbers as median, so here median=(30+70)/2= 50.

Mode: Mode is the easiest. It's the number in the sample that is repeated the most times.

ex: In above ex, since each number is repeated only once, each number is a mode. However, we has sample, 70,90,10,30,70, then 70 will be the mode since it's repeated 2 times.

Powers:

Powers are just an extension of multiplication, atleast for powers of integers (i.e where exponent is an integer, and base can be any decimal). Any number in form x^y = x*x* ... *x (i.e x repeated y times). (x^y)^z is same as x^(y*z) as x^y is repeated z times. x^(y^z) is different than (x^y)^z as in x^(y^z), we first calculate a=(y^z) and then do x^a.

Below table shows various x^y possibilities where x, y can be any real number. Here x is called the base, and y is called the exponent. Any fraction can be treated as integers with a division or as decimals. So, there is no separate table for fraction base or fraction exponents.

When y is a decimal (i.e exponent is decimal), it's hard to understand what it means. We'll understand it in the table below.

 NOTE: whenever exponent is a real number, it's not easy to solve such powers, and we resort to trial and error. So, the 3rd col of above table where exponent is real is not really expected to be solved by students w/o a calculator.

Operations: All basic operations of add, sub, mult and div can be done on powers too.

Add: ex: 2^4 + 2^5 = 2^4 *(1 + 2^1) = 3*(2^4) => explain distributive property in solving these

Sub: ex: 3^5 - 2^4 => These can't be solved by factoring out common terms as 2 and 3 are prime wrt each other.

Mult: 2 kinds: 1 with same base, and other with same exponent

Same base: ex: 2^5 * 2^6 = 2^(5+6) = 2^11 => exponents add up during multiplication, of same base

Same exponent: ex: 2^5 * 3^5 = (2*3)^5 = 6^5 => bases an be multiplied, if same exponent. Converse also true: 6^5 = (2*3)^5 = 2^5 * 3^5

Div: ex: 3^6/3^5 = 3^(6-5) = 3^1 => exponents subtract during division, of same base. this is the reason why x^0 is always 1, as X^n / X^n = 1 => X^(n-n) = 1 => X^0 = 1

Also -ve exponents can be understood the same way. 3^4 / 3^6 = 3^(4-6) = 3^(-2). However 3*3*3*3/(3*3*3*3*3*3) = 1/(3*3) = 1/(3^2) => 3^(-2) = 1/(3^2)

Exponent: Here, we raise an exponent to further exponent. In this case, the exponents multiply.

ex: (2^3)^5 = (2^3) multiplied 5 times = 2^(3*5)=2^15.

So, (x^y)^z = x^(y*z)

Brain Teaser: What is 0^0. This is a question which has found various different answers from Mathematicians. We know 0^y where y≠0 is 0. We also know that x^0 where x≠0 is 1.

In limiting case y->0 for the eqn 0^y, we get 0^y->0, For other eqn x^0, in the limiting case x->0, we get x^0->1. So, we get 2 different answers which makes our task difficult. What if we take limit x->0 and y->0 simultaneouly for x^y. then we see that it approaches to 1. this video shows it: https://www.youtube.com/watch?v=r0_mi8ngNnM&t=701s

However, it looks very weird, since 0 raised to something like 0 gets us to "1" and NOT "0". Defies common sense, doesn't it. The problem is if we take limit some other way, then we get different answer of either 0 or 1, depending on whose limit we take, x or y. This link tries to explain on what the value is: https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html

In nutshell, looks like for all practical purposes 0^0 =1, but if you want to be accurate, then it's undefined or indeterminate.Just like 0/0 is undefined or indeterminate. We may say that x/y, where we start taking limit of x->0 and y->0, and prove that x/y=1, but that's incorrect. We can argue the same way that 0^0 is not 1, but indeterminate. The answer is nobody knows !!

Expressions:

So far, we talked in terms of constant numbers, but we can also have all arithmetic operations on variables. i.e var x multiplied by var y = x.y. We can have multiplication/division of variables (x.y/z) or addition/subtraction (x+y-z). These equations written in terms of variables are called algebraic expressions. They have constants and coefficients too. A variable can take any value, it is not fixed but a constant is a fixed value.

ex: xy + 2ab + 4 => This is an algebraic expr. x,y,a and b are var here. 4 is a constant. 2 is also a constant and is called the coefficient of the term a.b.

There are many therems and other cool maths properties based on poly and their degree. More complex ones are part of high school maths (shown in "high school maths" section). Some basic eqn with poly are shown next.

Basic equations: If we have 2 variables x,y, we can write reduced form equations for powers of 2.

1. (x+y)^2 = x^2 + 2*x*y + y^2 => trinomial with deg=2 (degrees are explained in high school maths section)
2. (x-y)^2 = x^2 - 2*x*y + y^2 => trinomial with deg=2
3. (x+y)*(x-y) = x^2 - y^2 => binomial with deg=2
4. (x+y)^3 = x^3 + 3*x^2*y + 3*x*y^2 + y^3 => Quadnomial with deg=3
5. (x-y)^3 = x^3 - 3*x^2*y + 3*x*y^2 - y^3 => Quadnomial with deg=3

Square root: square root of any number is such a number which when squared gives that number, i.e square root of 9 is 3, since 3*3=9. square root is denoted by √ .This is same as where exponent is 1/2. i.e √9 = (9)^(0.5) = (9)^(1/2)

square roots are encountered very often in solving many kinds of equations, not so much for cubic root or higher powers of root. So, we try to keep a table for square root of numbers handy. We should be able to figure out square root of any number under 100. This is a good exercise for student. First we start with square root of numbers which have integer square root.

ex: √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5 and so on

Then square root of prime numbers. We will keep a table handy. The way to find square root of a prime number is difficult, but we can use indirect way to verify square root of prime numbers, by squaring the answer and checking if it comes close to the prime number itself.

ex:

√2 = 1.414 (if we square 1.414, we should get 2, i.e 1.414 * 1.414 gets us very close to 2, so the answer is correct)

√3 = 1.732

√5 = 2.23

√7 = 2.64

and so on

Now, we can calculate square root of any non prime number by factoring it into prime numbers or numbers which have a integer square root.

ex:

√6 = √(2*3) = √2 * √3 = 1.4 * 1.7 =

√8 = √(2*2*2) = √2 * √2 * √2 = 1.4 * 1.4 * 1.4 => However there is an easier way by noting that 8 can be factored into a number which has an integer square root. So, 8 = √(4*2) = √4 * √2 = 2*1.4=2.8

√10 = √(5*2) = √5 * √2 = 2.2 * 1.4

and so on for larger numbers.

Best way to find square root of any number is to write it as prime factors, and then group them in pairs. Ones without any pair, remain as square root, while others being in pair come out of square root.

Cubic root and other higher nth root:

These are usually not expected to be solved by hand, and a calculator can be used. However for simple integer ones, they should be solved by student.

ex: cubic root of 8 = ³√(8) = 8^(1/3) = a number which when multiplied 3 times gives 8. such a number is 2. so, cubic root of 8 is 2.

ex: 4th root of 16 = ⁴√(81) = 81^(1/4) = a number which when multiplied 4 times gives 81. such a number is 3. so, 4th root of 81 is 3.

Equations:

solving equations is one of the skills that should be taught in elementary school.

ex: Find a number such that when it's multiplied by 3 and added with 7, it becomes 19. The student may first try to solve it by trial and error, and come with an answer 4. Then introduce concept of variable "x", and show him how this equation 3*x + 7 = 19 can be solved to give an answer of 4.

Equations of 1 variable: Here there is only 1 variable "x" that we are solving for.

Linear equations: Here x doesn't have higher powers to it, and is relatively easy to solve. These are expected to be solved by 7th or 8th grade student. You add/subtract or mult/div by same thing on both sides, until you get "x" by itself on one side. It's called linear, because if you draw it on a plot, it turns out to be linear or a straight line. Linear equations have only 1 solution for x. However, you can also have cases where there is no solution or there are infinite solutions.

ex: 4*(x+2) +7 = 3*x - 10 => gives 1 solution for x, which is x=-25. No other x will satisfy this equality

ex: 4*x + 2 = 4*x + 3 => Here 4*x cancels on both sides, and for this equality to be true, 2 has to be equal to 3, i.e 2=3. Since that's not possible, there is no solution

ex: 4*x + 6 = 2*(2*x+3) => Here 4*x cancels on both sides, and we get 6 on both sides, i.e 6 =6. since this is always true, this equation is satisfied for any x, so it has infinite solutions.

Inequality:

As we have equality in equation, we can have inequality (i.e < or >) in equations too. These can be solved the same way as above, just that it has infinite solutions, as compared to equality which usually has just one solution.

ex: 6*x+7<2*x+11 => 4*x<4 => x<1 => This implies that any real number less than 1 is the solution to this inequality. We can plug in x=0 to check if our answer is correct. We that we get 7<11, which is correct. Next we put x=2 which is not a solution., so our inequality should be incorrect. We get 19<15 which is incorrect as expected. So, our solution looks correct.

NOTE: one weird case with inequalities happens when you multiply or divide the 2 sides by a -ve number. i.e if x<3, and we multiply both sides by -1, then it becomes -x<-3. However, this incorrect, as x<3 => x can be 2,1,0,etc. So, for x=2, -x=-2, but -2 is not less than -3, so -x<-3 is incorrect. This happened because we multiplied both sides by a -ve number. Whenever, we multiply or divide both sides by a -ve number, we need to change the direction of inequality sign. So, if x<3, then if we multiply both sides by -1, we need to do -x>-3 (i.e < became >).

CAUTION: We don't change the direction of inequality sign when doing + or -. We do it only when doing * or / by a -ve number. If we find this confusing, then we should do + or - to achieve. Let's see this by an example:

ex1: -2*x+5<7 => -2*x<2 => x>-1 (by reversing the direction from < to > when dividing by -2)

ex2: -2*x+5<7 => -2*x<2 => -2*x+2*x<2+2*x => 0<2+2*x => -2<2*x => -1<x => x>-1 (here we got the same answer but we used + and - instead of divide by -ve number, so we didn't have to do any reversal of inequality sign. sometimes this is easier to understand, and I recommend solving inequalities this way)

Equations of 2 variable: Here there are 2 variables "x" and "y" that we are solving for. Here too, we can have higher powers of x and y.

Linear equations: Here x and y have powers of 1, i.e they are both linear.

ex: find 2 numbers whose sum is 5. this can have infinite solutions, as one of the answers in 2 and 3, while other solution is 4 and 1, and many more decimal and integer solutions. However, if we add 1 more constraint to the numbers that their difference has to be 1, then there is only 1 solution which is 2 and 3. So, for solving an equation in 2 variables, we need at least 2 equations to solve it uniquely.

There are 2 ways to solve these kind of equations:

1. Find y in terms of x, by using 1st eqn and then substitute for y in 2nd eqn.

ex: x+2y=7, 2x+5y=15 => Here from 1st eqn, y=1/2*(7-x). Now substitute y in 2nd eqn. i.e 2x+5*1/2(7-x)=15 => solve for x in this eqn, and then find y by using either 1st or 2nd eqn

2. Here we try to cancel x or y by multiplying 1st or 2nd eqn such that one of the variables has the same coefficient, and then add or sub the 2 eqn to cancel the variable out.

ex: x+y=7; x-y=3 => Add both RHS and LHS, which cancels out y, giving x=5, then solve for y=2.

More Equations: We'll learn solving more complicated eqn in high school maths section

Basic Geometry:

Good material on geometry is on this link: https://www.mathsisfun.com/geometry/index.html

There are different kinds of 2D and 3D figures. We'll look at 2D figures here => Line, triangle, quadrilateral, circle, pentagon, etc.

Quadrilaterals => Any 2D figure enclosed with 4 sides which are straight lines is a quadrilateral. Ex are: square, rectangle, etc.

Link showing different kind of Quadrilaterals => quadrilaterals

There are very few fundamental concepts in Geometry. All known theorems are derived from these few fundamental theorems. We'll learn these fundamental ones, and derive everything else from these fundamental theorems or concepts.

1. concept of angles: straight line has 180 degrees, perpendicular lines have 90 degrees, and around a line, the total angle is 360 degrees. This can be thought of as 1st concept or theorem.

2. Concept of Lines: These are straight lines.

• Intersecting lines: They have opposite angles the same. They are known as opposite angles. The angles next to each other add up to 180 degrees and are known as complimentary angles. This can be thought of as 2nd concept or theorem.
  
• Parallel lines: They have transversal line crossing them. In this case, corresponding angles are equal. This can be thought of as 3rd concept or theorem.
  

3. Triangle:

Theorem for sum of angles: Sum of 3 angles of triangles is 180 degrees: proof by drawing parallel line. This can be thought of as 4th concept or theorem.

Find remaining angles of a triangle, given 2 interior angles or outside angles. This is a good exercise.

Rght angle triangle: Right angle triangles are an important category of triangles, as they have many special properties.

pythagoras theorem (c^2 = a^2 + b^2) and it's proof (where a, b and c are sides of a right triangle). 

Proof: Drop a line at right angle from vertex to hypotenuse. It may be proved that 3 triangles formed are similar to each other (if one angle is X, other angle is (90-x). This is true for all 3 triangles). This similarity property may be used to calculate ratios of sides, and find height of triangle. Rearranging it yields Pythagoras Thm. Also calculating area of triangle using all 3 sides as base, and multiplying it by height yields eqn for Pythagoras Thm.  More proofs here: https://en.wikipedia.org/wiki/Pythagorean_theorem

- Sine, cosine and tangent are advanced topics for high school. We'll learn these later in "high school maths" section.

Before we start with any problems on triangles, we should learn how to draw triangles using ruler, compass and protractor. There are many kinds of triangles possible:

Based on above table, only the cells in green are the ones where we have just the right info to draw a unique triangle. In all other cases, we have either insufficient info, or extra info.

Impossible triangles: Triangles which are not possible to draw is something that students should know. Third side of a triangle should always be less than the sum of other 2 sides of triangle, and it should also be more than the difference of the other 2 sides of triangle.

ex: Given triangle ABC, |AB-BC| < AC < AB+BC => Here we take the +ve diff of the sides, hence modulus

4. Perimeter: Perimeter of square, rectangle, triangle, parallelogram, etc is just the sum of the sides. Perimeter of circle is called circumference and is shown to be 2*Π*R, where R is the radius. The proof is based on calculus.

5. Area: Area of square, rectangle, triangle, parallelogram (explain how area of triangle is 1/2 of area of rectangle, and how area of parallelogram, is same as that of a rectangle with same height). Area of Circle is Π*R^2, where R is the radius. The proof is based on calculus.

6. Volume: Volume of cube, rectangular pyramid, (volume cylinder, cone, sphere are high school topic. Proof of these require calculus).

3D Shapes:

3D shapes are more complicated than 2D, so finding area and volume requires more work. Below are few easy shapes (3D geometry is high school, but put here anyway):

• Cylinder:
  • Area = area of circle *2 (on top/bot) + area of lateral side = 2*Π*R^2 + 2*Π*R*H
  • Volume = Π*R^2*H
• Cone:
  • Area = area of circle (on bot) + area of lateral side. Finding area of lateral side looks hard. However, if you draw any circle on piece of paper, cut it out and then cut any angular segment of it, then it will always form a cone. So, basically, you want to find the area of this segment that forms cone. We need to find out angle of segment that's cut out, but turns out we don't need that. We know the total length of the segment to be 2*Π*R, and radius of this circle is S=sq rt (R^2+H^2). So, lateral area =  1/2*2*Π*R*S = Π*R*S. Proof can be found by cutting infinitesimally small segments which are all isoceles triangles and then adding them. (it's same prrof as we used in finding area of circle by using it's circumeference where Area=1/2*2*Π*R*R=Π*R^2). So total area = Π*R^2 + Π*R*√(R^2+H^2)
  • Volume: Volume is 1/3rd of a cyliner. Not so obvious to see.
• Pyramid:
• Sphere: Most complex

Basic Plotting:

some introductory material on plotting is on this link: https://www.mathsisfun.com/data/index.html

Equation of Straight Line:

Introduce X axis and Y axis, and how to find where a coordinate lies.

Then introduce students to an equation of line, which is of form y = m*x+c. Here m is the slope and c is the y intercept. Show him how y=m*x is the same line as y=m*x+c, but just shifted by c units up or down. y=m*x line always passes thru origin, as x=0, yields y=0. In many textbooks, this eqn is rep as y=m*x+b (i.e y intercept is written as "b" instead of "c").

NOTE: slope "m" is taken as positive for going up a hill left to right (i.e push up a hill or push means +ve). Line going down the hill from left to right is treated as -ve slope. When 2 points are used to find a slope, then you don't have to worry about -ve or +ve slope, as using (y2-y1)/(x2-x1) will automatically give you the correct sign of the slope.

Eqn of staright line can be written in different forms as shown below. One form is more convenient than the other depending on what is given (i.e m, c, coordinates (x1,y1), etc).

1. Slope intercept form: This is the form we use when m and c are given. This is what we saw above: y=m*x+c. This is the most widely used form, and easiest to visualize.
2. Point Slope form: This is used when we have slope m given, as well as one of the coordinates given. Then (y-y1)=m(x-x1). If m is not given, but instead, other coordinate (x2,y2) given, then we can use the 2 coordinates to find m=(y2-y1)/(x2-x1). This is also referred to as 2 point form.
3. Standard form: This is of the form Ax+By=C. This eqn is actually derived from other form called "intercept form", where the x-intercept and y-intercept are given as (a,0) and (0,b). Then we can use 2 point form to write: y-0=(x-a)*((b-0)/(0-a) => y=(x-a)(-b/a) => x/a+y/b=1. We write this in form b*x+y*a=ab or Ax+By=C which looks simpler than writing in x/a+y/b=1.

Most important is to show that the linear equations of 1 variable that we introduced in "equations" section above, can also be solved by plotting the straight line, and looking for the coordinate where y=0. That X coordinate is the solution of that linear eqn.

The next section on "high school maths" has plots for straight line, quadratic functions and exponential functions.

Basic Transformations: Translation, Rotation and Reflection:

One very popular topic in Maths and IQ/Gifted Talented tests or any general Brain test is Translation, Rotation and Reflection. A figure is rotated, reflected and /or translated, and kids are asked on how would the final figure look like. This can get very complicated depending on axis across which the required operation is done. Fortunately, for elementary and high school maths, very simple operations are done, which can easily by figured out by using formula below

Translation:

Translation is the easiest. We just move the figure horizontally left or right by said number of units. We just need to add or subtract that number of units to x coordinate. Sometimes the figure is moved vertically up or down. in this case, we add or subtract that number of units to y coordinate. if we translate in some other direction (i.e not horizontally or vertically, but in a slanted direction, then we have to first move into x direction horizontally and then y direction vertically.

IMP: For any function f(x), we can obtain a plot for f(x+b), where x is replaced by "x+b" in the function f(x), the new plot of f(x+b) will be shifted by "b" towards the left compared to the original function f(x). This is easy to see, as whatever f(x1) was for a given x1, now we get the same value of f(x+b) for x=x1-b, so that x+b becomes x1-b+b=x1, so the whole function is shifted.

Similarly given f(x), the plot of f(ax) will be expanded/compressed version of f(x) where x axis is expanded/compressed, where whatever value of f(x1) was, will now be at x=x1/a. NOTE, the y values of function don't change/ Where they occur on x axis changes.

So, for any modified function f(ax+b), we can just use the above 2 observations to convert it to f(a*(x+b/a)). Now we use the shift rule to shift functio left by b/a, and then scale the x-axis by a, to get the final plot of f(ax+b).

Reflection:

Reflection is next easy one after translation. Easiest way to solve reflection problem is to look at vertices of figure to be reflected. Start with vertex 1, drop a line perpendicular to the reflecting surface starting from that vertex, and then extend that line by the same length inside the reflecting surface (on the other side). That way you get mirror reflection of that one vertex. Now repeat the process with other vertices. Finally connect all these reflected vertices to get the reflected figure. It doesn't matter whether reflection is across x axis, y axis or any slanted line. Same procedure gets applied.

Most common reflection: Reflection around y=x line. Here points (x,y) get reflected to become points (y,x)

Another teaser question: Reflection around line: y=mx+c. This is more general case of y=x case above (i.e m=1, c=0). This has to be solved to figure out what the new coords will be after reflecting.

Rotation:

Rotation is the hardest of all transformations. It requires a lot of visualization, and easily gets confusing. You can rotate any object by any degree around any point. Rotation by 90 degrees (clockwise and anticlockwise), and 180 degrees (clockwise and anticlockwise rotations for 180 degrees are the same) are very common, and those are the ones we'll discuss below. It's very important to know which point are we rotating the figure around. The rotated shape will appear in different places depending on which point is it rotated around.

1. Rotation by 90 degrees around origin: Let's consider a point (x1,y1) in 1st quadrant (i.e where both x and y are +ve). Let's make a right angle triangle for that point, with base=x1 and height=y1. If we rotate this triangle 90 degrees clockwise, then the triangle rotates to a new position, and it's base now has length y1 and height has length x1. So, the new coordinate after 90 degrees rotation is (y1, -x1). Note the sign changed since y coordiante of new triangle is now -ve. Similarly when we rotate 90 degrees anticlockwise, the result is the same except that x coordinate of new triangle becomes -ve, so new coordinates are (-y1, x1).

We repeat the same process with other vertices, and then connect them to get the rotated figure.

2. Rotation by 180 degrees around origin: This is very simple. You consider a point (x1,y1) in 1st quadrant (i.e where both x and y are +ve). If we rotate this point 180 degrees clockwise or anticlockwise, the the new point becomes (-x1, -y1).

3. Rotation around any point (X0, Y0): So far we looked at rotation around origin. Rotation around any other point looks complicated, but it's actually very simple. You consider a rigid bar from point (X0, Y0) which is attached to the figure to be rotated. Now we rotate this bar by 90 degrees or 180 degrees, and with that the figure also rotates. This becomes the new coordinates of the figure.

To find out new coordinates for any point (x1,y1), we repeat the same exercise as before. However, now we shift our origin to (X0, Y0). Then we get new coordinates for (x1,y1) as (x1-X0, y1-Y0). Since we know how to do rotation around origin, we can now rotate all the vertices of the figure and get the new vertices.

NOTE: We treated (x1, y1) or (X0,Y0) as positive, but above formula are true irrespective of whether (x,y) are +ve or -ve.  So (-3,2) with 90 degree clockwise rotation will become (2, -(-3))=(2,3) while for 180 degree rotation, it will become (-(-3),-2)=(3,-2)

That's the end of Elementary Maths. That wasn't too hard for you, was it