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Published: Wednesday, 01 March 2023 08:06

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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

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Published: Saturday, 25 February 2023 01:30

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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. 

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Published: Tuesday, 31 January 2023 00:39

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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 as expected. However, if they were mixed in 1:1 by volume, then avg density, assuming x cm^3 of each sample, would be = (2*x + 4*x) / (2*x) = 3g/cm^3 which is exactly avg of the 2 densities. This happened because denominator is the same for both of them.

Popular Q on avg speed: A person travels from town A to town B with a avg speed of 40kmph and returns back from B to A with an avg speed of 50kmph. What's his avg speed for the trip? The answer is NOT 50kmph, as avg speed = tot_dist / tot_time. Here, person travels for more time at 40kmph and for less time at 60kmph, so avg has to be < 50kmph. If we solve, we get 48 kmph !! If he traveled for equal times at 40 kmph and 60kmph (i.e 2 hrs each), then avg speed would be 50kmph.

So, when calculating avg of averages, be careful, as the final avg may NOT be the avg of averages.

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. The widely shown proof in schools is based on drawing 4 triangles on the 4 sides of the hypotenuse aquare (c^2) to form a square of side (a+b). Equating area of bigger sq (a+b)^2  to area of c^2+ 4 right angles traingles (4*1/2*a*b) yields a^2+b^2=c^2. More proofs here: https://en.wikipedia.org/wiki/Pythagorean_theorem

Pythagoras Thm is a special case of more general triangle which isn't restricted to 90 degrees. Here hypotenuse eqn has sine term in it. Covred in high school maths.

- 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

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Published: Tuesday, 31 January 2023 00:30

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High School Geometry:

We learned simple geometry in Elementary maths. More advanced topics for geometry and trigonometry can be found at the bottom of the page on this link: https://www.mathsisfun.com/geometry/index.html

Coordinate geometry: This refers to drawing 2D shapes on a coordinate axis (horizontal X-axis and vertical Y axis). To draw 3D shpaes, we'll need 3 axis (X, Y and Z axis) which is more complex. We'll focus on 2D geometry for now, and explore simple 3D shapes.

Eqn of straight line => Eqn of straight line is y=m*x+b where m=slope of line and b=y intercept (it's also written as var c instead of var b). If you plot all points using this eqn and connect them, you will get a straight line. Reason is that slope is constant for a straight line (that is what makes it a straight line).

Distance between 2 points (x1,y1) and (x2,y2) = √[(x2-x1)^2 + (y2-y1)^2] (This can be done by using Pythagoras Thm)

Distance between a point (x1,y1) and a line (ax+by+c=0) => Draw a perpendicular from the point to the line, and find the eqn of that line (since slope and one of the points is known). Now find intersecting point for the 2 lines. Calc distance b/w the 2 points, which is the distance asked. Easier than this is to use a formula prrof of which is here: https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

2D shapes: Polygons are 2-dimensional shapes made of straight lines. These include triangles, rectangles, etc. Here we calculate area and perimeter. attach diagram FIXME

• Line/Point. These are not shapes. A line is made up of inifinite points. Lines are straight or curved. We study straight lines since they are easy.
• Square/rectangle:
• Triangle:
• Parallelogram:
• Rhombus:
• Kite:
• Polygons: (

3D shapes: Studied in Solid Geometry. It's called "solid", as we can make solid shapes only from 3D objects. Nothing we see around us is 2D. Everything is 3D. Here we usually calculate Volume and surface area of these objects. There are two main types of solids, "Polyhedra", and "Non-Polyhedra"

• Polyhedra: All surfaces are flat.
• Non Polyhedra: At least one of the surfaces is NOT flat.

Bisect Lines/Angles:

To bisect a line into 2 halves, we use  technique with compass, where we draw arcs on top and bottom of the lines from the 2 end poibts. Wherever they intersect, we draw a straight line thru it, that line divides the original line into 2 halves. Why does it work? Because, we make a rhombus and the 2 lines become the diagonal of rhombus.

Circles:

We learned about Circles in "Elementary Maths", but there are a lot of properties of circles and triangles inside or outside the circles that yields a lot of interesting theorems.

• Incenter: A circle which touches the 3 sides of a triangles is completely inside the triangle and is called the incircle. The centre of incircle is called the incenter and can be found by dividing angles of each vertex in half and finding the intersecting point of these 3 angle bisectors. 2 adjacent triangles turn out to be similar, which proves that such a point is incenter of the triangle. There's also 3 excircles defined which touch the 3 exterior or extended sides of the triangle. Excircles are not usually discussed in high school geometry.
  • Link: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
  • Law of Cotangent explained below can also be used to find the radius of Incircle.
• Circumcenter: A circle which passes thru the 3 vertices of a triangle is called the Circumcircle, and the center of such a circle is called Circumcenter. Circumcenter can be found by drawing perpendicular bisector of the 3 sides. The intersecting point is the circumcenter. It can be proved that such a point is equidistant from the 3 vertices by observing that the 2 triangles on each side of triangle are congruent.
  
  • Link: https://en.wikipedia.org/wiki/Circumscribed_circle
  • Law of Sines (explained below) can be used to find the radius of the circumcircle.
• Orthocenter: Orthocenter of a triangle is the point where the 3 altitudes of the triangle coincide. The perpendicular is drawn by drawing altitude from each of the 3 vertices on to the opposite side. Orthocenter doesn't have any special property as Incenter or circumcenter have.
  
  • Link: https://en.wikipedia.org/wiki/Altitude_(triangle)#Orthocenter
  • One important property of orthocentre for a right trangle is: square altitude (h) from right angle to hypotenuse = product of the lengths of hypotenuse segments divided by the altitude, i.e h^2 = √(p*q)
  • The orthocenter lies inside the triangle iff the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle.
  • 3 altitude of the triangle can be found by using heron's formula below (by using 1/2*base*height=Area)
  • Both inradius and circumradius of the traingle are related to the height of the 3 sides via inradius and circumradius theorem. See wikilink above for the relation.
  • Orthic or Altitude triangle: The feet of the altitudes of an oblique triangle form the orthic triangle. Also, the incenter (the center of the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.
• Centroid: Centroid also known as geometric center or center of figure, of a figure is the arithmetic mean position of all the points in the surface of the figure. For a object with uniform mass, it's also the center of gravity. The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). Mathematicaly, centroid is the mean of the 3 coordiantes, i.e coordinates of centroid (x,y) = ((x1+x2+x3)/3, ((y1+y2+y3)/3) where (x1,y1), (x2,y2) and (x3,y3) are the 3 coordiantes.
  
  • Link: https://en.wikipedia.org/wiki/Centroid

Trignometry (Triangle Geometry):

Trignometry is a branch of Geometry that deals exclusively with triangles. You may wonder how come triangles have a whole branch of Mathematics dedicated to itself !! There are lots of things that's possible with triangles, and knowing triangles well forms the basis of Geometry.We'll talk more about right angled triangles in separate section on "Trignometry",as they are the most interesting ones. Here we just go thru few basics.

Finding Sides and angles of any Generic triangle:

So far, we looked at right angle triangles. We figured out that given 2 sides of a right angled triangle, we can find out 3rd side by using Pythagoras theorem. Then we can find out the 2 angles of the triangle by using trignometric tables (by finding out sin,cos or tan ratio and then seeing what angle corresponds to that ratio). We also figured out that given 1 side and 1 angle, we can again create a unique triangle. Here we can find remaining sides by using trignometric tables. So, right angles are easy.

How about finding sides and angles of a unique triangle which is not right triangle? It's possible to do it via 2 laws (both laws are actually same law, but written differently. One can be derived from the other. Below are the 2 laws:

1. Law of Sines:

This is an important theorem to find length of sides or angles of a triangle given it's 2 sides and an angle or an angle and 2 sides (i.e given some combination of length of sides and angles that makes it unique). It is used to find the radius of circumcenter too. Stated mathematically, this is the law of Sines:

Sin(A)/a = Sin(B)/b = Sin(C)/c =1/2R where a,b.c are the lengths of 3 sides, and A,B,C are 3 angles opposite the 3 sides (i.e angle A is angle opposite side a, meaning angle between sides b and c), R is the circumcenter.

In other words, a:b:c = Sin(A):Sin(B):Sin(C) => i.e sides are proportional to the sine of the respective angles.

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_sines

Proof: Proof is very simple. If you find area of any triangle by multiplying base and height in 3 different ways, then we can get this equality. There is one more proof based on inscribing a triangle around a circle. That proof is also given in the wikipedia link above.

2. Law of Cosines:

This is a variation of the Law of Cosines. Here given 2 sides and the angle between them, we can find the 3rd side much easily. Law of Sines won't give us the 3rd side that easily. Law of Cosines is a more generic case of Pythagoras theorem, where it applies to angles other than 90 degrees. Stated mathematically, this is the law of Cosines:

c^2 = a^2 + b^2 -2abCos(C) where a,b.c are the lengths of 3 sides, and C is angle between sides a and b, i.e opposite side c)

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_cosines

Proof: Proof is remarkably simple if we use coordinate system, with one vertex of triangle placed at (0,0). We make a right triangle out of the given triangle, with the extra 2 legths of the right angle triangle being bCos(C) and bSin(C). Now we apply Pythagoras thm, so that c^2=(a+bCos(C))^2 + (bSin(C))^2. Rearranging terms and using identity sin²(Θ) + cos²(Θ) = 1, we get the formula above.

NOTE: There is -ve sign on the last term. A very simple way to remember this is as follows =>If angle C was right angle, then we get back to Pythagoras thm via Law of Cosines (by using Cos(C)=0). If angle C is acute angle, then "c" would be smaller than the one for a right angle triangle, so c^2 has to be less than a^2+b^2. So, we need to subtract some term from this, which is done by having a -ve sign to 2ab. Cos(C) is +ve for acute angle. For obtuse angle, Cos(C) is -ve, which makes the 3rd term +ve (-2ab*(-ve value) = +ve value). So, c^2 becomes greater than a^2+b^2 which is what is expected.

3. Law of Cotangent (or Cot Theorem):

Though not so common, Law of Cot provides a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. It is used to find the radius of inscribed circle too. It states as follows:

Cot(A/2)/(s-a) = Cot(B/2)/(s-b) = Cot(C/2)/(s-c) = 1/r where s is the semiperimeter of the triangle i.e s = 1/2(a+b+c), and r is the radius of the inscribed circle.

Furthermore inradius is also given by (only interms of sides and no angles) => r = sq root ((s-a)(s-b)(s-c)/s)

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_cotangents

Proof: Proof is on wiki link above.

Heron's Formula:

Using the law of cosines, or by using pythagoras theorem, we can find out the area of a triangle given it's 3 sides. Law of Cot can also be used to derive this. This is known as Heron's formula as stated below.

Given 3 sides of a triangle as a,b,c, it's Area = √(s(s-a)(s-b)(s-c)) where s is the semiperimeter of the triangle i.e s = 1/2(a+b+c)

Wikipedia link proves it here: https://en.wikipedia.org/wiki/Heron's_formula

Heron's formula is special case of Brahmagupta formula, which is a special case of Bretschneider's formula for finding out area of any quadrilateral: https://en.wikipedia.org/wiki/Bretschneider's_formula

Triangle Theorems:

1. Menelaus's Thm: It relates the ratios obtained by cutting 3 sides of a triangle. Wiki: https://en.wikipedia.org/wiki/Menelaus%27s_theorem

Circle Theorems:

1. Thales's Thm and others related to triangles inside circles: Link => https://www.mathsisfun.com/geometry/circle-theorems.html