[Guide] The Largest Math Guide in History

The Largest Math Guide in History

Written by Unb4nn3d and MatthewGor123
This Guide Was Written In Response To -> http://www.sythe.org/showthread.php?t=490925​

Status : 6/50 Complete

To quickly find a section of our guide, please use the Ctrl + F (Find) function for assistance.
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I. Arithmetic

In this section, we will cover many different topics in this branch of mathematics, including:

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A. Types of Numbers

In arithmetic, it is important to recognize the many different types of numbers, as they are all crucial in understanding more advanced mathematics. The most basic of the classifications of numbers is the distinction between positives, negatives, and zero. The positive numbers are all the numbers greater than 0, expressed as being to the right of 0 on the number line (generally), while the negative numbers are all the numbers less than 0. However, the classification of numbers gets much more specific than this.

There is a “hierarchy” to these numbers, as you can see from this diagram:



It may be easier to understand the hierarchy from top to bottom, or from bottom to top. Just try to make sense of it.

A1. Complex Numbers. Any number you can possibly imagine is considered a complex number. It is the highest rank of the hierarchy. All complex numbers are expressed with the formula: a + bi, where i is an imaginary number that is explained shortly. a and b are both real numbers, but a is called the real part, while b is called the imaginary part. The multiplication and addition identities are 0, so to form a real number, b would be 0, and to form an imaginary number, a would be 0.

A2. Real Numbers. Real numbers are the set of all rational and irrational numbers. These include pi, root 2, -(e), 0, etc.

A3. Imaginary Numbers. On the same level of the hierarchy as real numbers are the imaginary numbers. An imaginary number is a number that, when raised to an even exponent yields a negative number. This will be explained in greater depth later. All imaginary numbers are expressed with the formula bi where b is some real number, and i is the square root of (-1).

A4. Rational Numbers. As mentioned earlier, real numbers are the set of all rational and irrational numbers. The rational numbers are the set of all the numbers that can be expressed as a ratio of two integers. Rational numbers are often expressed as fractions, but can also be expressed as a terminating decimal. For example, 8/3, 7.142857142857…, -156, 0, and -5/3, and 2.75 are all rational numbers.

A5. Irrational Numbers. These numbers are the ones whose value is clearly defined, but cannot be expressed using a ratio of integers. These are a subset of real numbers. Numbers in this set are commonly expressed as letters and are always rounded when calculation is necessary. Numbers in this set include π, root 2, -(e), etc.

A6. Integers. These are the next step in the hierarchy after rational numbers. The integers are the set of all numbers that can be expressed as a ratio of two real numbers where the numerator is a multiple of the denominator. These numbers include the negatives, positives, and 0. Members of this set include -15, -3, 0, 1, 2, 3, 25, etc.

A7. Whole Numbers. The whole numbers are the set of all the positive integers as well as 0. The numbers in this set include 0, 1, 2, 15910, and 5238629.

A8. Natural (or Counting) Numbers. These are probably the first numbers everyone was introduced to. They are the set of just the positive integers. 0 is not a counting number.

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B. The Four Basic Operations

As you should probably all know, there are four basic operations in arithmetic. The four are addition, subtraction, multiplication, and division. Each of these operations has specific properties, and works differently with the different types of numbers. In this part of the guide, you will learn about the properties of each of the operations, as well as a refresher on how to do the operations with each of the types of numbers. The properties that will be discussed are: commutativity, associativity, identity elements, and inverse elements.

A commutative operation is an operation that does not depend on the order.

An associative operation is an operation that satisfies the rule (a#b)#c=a#(b#c) for any values of a, b, and c where # is the operative symbol.

An identity element of an operation is an element in the domain for the operation that satisfies the rule a#n = a = n#a, where # is the operative symbol, n is the identity element, and a is any element in the set.

Not all operations have all, or any, of these properties. Domains and Ranges of functions and operations will be touched upon, but substantially later in the guide.

B1. Addition

Addition is probably the most fundamental operation in all of mathematics. The operation "addition" is represented using the "plus" sign (+). When we want to verbalize (a + b), we can say it in a few ways. We can say "a plus b", "the sum of a and b", or sometimes just "a and b." We learn it at a very young age, and tend not to forget. Addition is commutative and associative; this is taken as a given; it is a postulate. The identity element of addition is 0, because x + 0 = x. This is another postulate that is very easily visualized. Keep in mind that adding a negative is the same as subtracting a positive, and vice versa; (a + (-b)) = (a – b). We will review how to add integers, decimals, and fractions. So, let’s add two large counting numbers, because if we can add counting numbers, we can add (or, I guess, subtract) all integers. Here’s a quick animation showing how to add counting numbers that I just kinda quickly threw together.



Adding decimals is a very similar process, except for you have to line up the decimal points for the top and the bottom values, and then you just do normal addition. It’s not un-normal to have something like:

1234.56

+ 2.251623

1236.811623

It’s the same addition as you do with counting numbers, but make sure you line up the decimal points!

Where people generally run into some problems is the addition of fractions.

Here's a simple, step-by-step guide on how to do this:

Well, you now know the basic rules of how to do arithmetic calculations without a calculator! Congrats

B2. Subtraction

After addition, the next simplest operation is subtraction. When you can add stuff, you can also take stuff away. Subtraction is the inverse function of addition; you take 3 + 4 = 7, then 7 - 4 = 3, and 7 - 3 = 4. It is represented with the "minus" sign (-). When you verbalize (a - b), you can say "the difference of a and b" or "a minus b." Subtraction does not contain any of the three basic properties. You may at first think that 0 is an identity element of subtraction, but it's not. Look back at our basic definition:

. While it is true that a - 0 = a, what do you get when you do 0 - a? You get -(a), the opposite of a. That is, of course, not equal to a for all inputs a, and therefore 0 is not an identity. Subtraction is not commutative, because (a - b) =/= (b - a). This should be pretty logical, but a proof may be shown later during the ALGEBRA sections; for now, just pick any two numbers, and see if they work. For almost all values, they don't (unless you pick the same two numbers). And is subtraction associative? No. (a - b) - c =/= a - (b - c). Again, this will be proven later probably, but just for now, test three different values. If there is any one set of values that does NOT work, the operation is not associative.

Let's begin subtracting

If you can remember that subtracting a negative is the same as adding a positive: (a - (-b)) = (a + b), that will help a lot. Another key rule to remember is if you're doing (a - b), and b is greater than a, your answer will be negative. To subtract it more conveniently, you could say that the answer is - (b - a), and that way you don't have to fumble with stuff less than 0. You only need to know how to subtract positive counting numbers, numbers with decimals, and fractions. I put together another animation showing how to subtract two large counting numbers:



One thing that is not included in this is animation is what happens if you have, let's say, 100 - 85, and you need to borrow 10 from "0" - what you would do change the two 0's to 9's, and then the 1 at the start to a 0. This is because you're essentially borrowing the 100, and just converting it to a 90 and a 10. It made sense in my head .

Subtracting decimals is an identical process as adding decimals; you line up the decimal points, then you just subtract normally. You would convert anything past the decimal point to 0's on the top/bottom. For example;

123.45670000

- 7.65432198

115.80237802

Then, you just subtract normally, ultimately borrowing 10000 from the 700000.

Again, subtracting fractions is really where people have the most difficulty.

And that is how you subtract two numBeRs!

B3. Multiplication

Multiplication is the third of the four basic operations that we are going to talk about. The uses of multiplication are endless - you use it to find areas of items, to calculate expenses, and much, much more. In math, there are three ways to represent multiplication. One is using the simple "x" sign; (a) x (b). You can also use the asterisk (*) when you want to represent it on the computer. A third way of representing multiplication is the interpunct; (·), such as a a·b).

Multiplication is basically the process of repeated addition. Multiplication is commutative and associative, because (a * b) = (b * a), and a * (b * c) = (a * b) * c. Again, this can be proven later, but for now, just test values if you don't believe me. The identity element of addition is 1; 1 * n = n = n * 1, and this is another one of the postulates. Also, keep in mind that anything multiplied by 0 is going to be equal to 0. When you're multiplying, if you can just remember: positive * negative = negative, negative * positive = negative, positive * positive = positive, and negative * negative = positive, you'll be all set. Therefore, it's only necessary to know how to multiply positive numbers.

An example of Multiplication would be the following:
2 x 4 = 8.
8 is the solution because 2 x 2 x 2 = 8, by factoring 4 into (2 x 2).

Another example may be this:
2 x 5 x 6 = 60.
-This can be factored into 2 x 2 x 3 x 5
-2 x 2 = 4 ; 4 x 3 = 12 ; 12 x 5 = 60

A Multiplication Table is shown below:


To use this table, simply follow the numbers from the top and left side until your fingers meet.

For example: On the top, put your finger on the number 6, and onto the number 8 on the left side.
Now slowly slide your fingers across the table and your fingers should touch at the number 48.
Simple memorization of these charts shall help you out greatly, and will remain with you for the rest of your life.

Multiplication isn't always this simple; when you have a two+ digit number multiplied by another two+ digit number.
This is called Complex Multiplication.
Complex Multiplication includes more than 2-3 numbers being multiplied at one time.

One is presented with a problem:



To solve this problem, first multiply the top number by the number on the most right.
You will have (2,843 x 2), which comes to 5,686.
-Put this number directly below.

Now multiply the next number to the left, which is (2,843 x 8), and you will get 22,744(x).
*NOTE* When moving to left, an X must be placed on the right for each row you move down from the first.

So now, add 227,440 and 5,686, which will 233,126, your solution.

Multiplying fractions is probably the easiest part of fractions. Here's what you do:

That's it for multiplication

B4. Division

Division is the fourth and final basic operation that will be included in our Basic Operations section. It is the inverse of Multiplication. To record division, it is common to write a / b, or a ÷ b. Sometimes, division is represented by placing one number over the other, with a bar (known as the vinculum) separating them; a/b

Division is not commutative, because (a/b) =/= (b/a). You can test this using numeric examples. It is also not associative; the placement of parenthesis matters - (a/b)/c =/= a/(b/c). Finally, there is no identity because while a/1 = a, 1/a is the reciprocal of a, not a, thus not satisfying the rules. When you speak of division, you generally say "a divided by b," or the "quotient of a and b." In the equation a / b = c, a is known as the dividend, b is the divisor, and c is the quotient."

One is presented with the following problem.

The inverse would equation would be a = c/b

Plug in numbers.

4 * 2 = 8

Divide 2 from both sides.

You will end up with 4 = 8/2

*NOTE*

You can NEVER divide by 0.

If your Denominator in a fraction is equal to 0, then your solution will be "undefined"

Also, if you have 2/√3, you may have to rewrite it to make the fraction rational.

-To do this, multiply the numerator and denominator by √3

You will end up with (2√3)/3.

If you have (2/(5-√3)), you will multiply by (5+√3)/(5+√3) to rationalize it. We will discuss why in the algebra section, but basically tou end up with 2(5+√3)/(25-3) = (10+2√3)/22. This is just about proper notation.

Next, we will discuss long division. Long division is naturally the most confusing for students.

You are given the equation 10260 / 15

Write it and solve by using Long Division:



Dividing by Decimals is quite different.

Dividing fractions is also relatively simple, so here's how you do it:

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C. More Advanced Operations

The next part of our guide will cover a few types of more advanced operations.

C1. Exponentiation

After learning the four basic operations in detail, it is very common to learn the operation of "exponentiation." Exponentiation and logarithms are very closely tied together, but more detail about this will be explained in that section. Exponentiation is basically the process of repeated multiplication, just like the operation multiplication is repeated addition. To record exponentiation, one generally writes aⁿ or a^n, where a is known as the base, and n is known as the exponent or the power. The domain and range for exponentiation is not always all real numbers, as you will see later in this section. To verbalize this, one would say: "a to the ("to the" is sometimes replaced with "raised to the") n-th power." If it is implied what you are talking about, you can omit the words "to the" or "raised to the" and "power," thus leaving "a to the n-th" or "a to the n." Exponentiation involves multiplying the number, a, by itself n times. As you can probably figure by your common sense, exponentiation is not commutative, associative, and does not have an identity element. yⁿ is obviously not equal to nʸ for most values of x and n. It should also be quite logical that it's not associative, and has no identity (it is rate that operations that are non-commutative have identities). Here are some crucial, fundamental rules of exponents:

1. xʷ * xʸ = x⁽ʷ ⁺ ʸ⁾
   
2. (x&#695ʸ = x⁽ʷ ʸ⁾
   
3. (x&#695/(x&#696 = xʸ⁽ʷ ⁻ ʸ⁾
   
4. x⁰ = 1, x¹ = x (these rules are easily shown through the other properties of exponentiation)
   
5. x⁻⁽ʷ⁾ = 1/(x&#695
   
6. ʸ√(x&#695 = x(ʷ/ &#696 (with this property, we can see why the domain of exponentiation is limited under some circumstances; if x is negative, w is odd, and y is even, we are asked to find the even root of a negative).

If you are given aⁿ = b, if b is the unknown, you can just multiply a by itself n times (or use a calculator). If a is the unknown, you can radicals. Let's say:

a⁵ = 1024.

We can find the 5th root (or, by rule #6, the 1/5 exponent) of both sides, thus yielding:

(a&#8309¹/⁵ = 1024¹/⁵. By using rule 2, we get a = 1024^(1/5), and by punching this into a calculator, we get a = 4.

However, what if n is the unknown? What if you're given 5ⁿ = 125? Apart from trial and error, the only way to solve this is through the use of logarithms. Read section C4 to find out how!

C2. Absolute Value

Though the concept of absolute value is extraordinarily simple, we felt like it deserves a separate section in the guide, prior to algebra. In arithmetic, absolute value is literally just the number, disregarding the sign of this number. It is recorded |x|, and is pronounced "the absolute value of x" or "the cardinal value of x." The properties are all relatively logical, so they won't even be introduced until the algebra section. A few examples of problems involving absolute value:

2 * |-5| = 10
-|-2*5| = -10
|-2|*|-5| = 10
10 - (-|2*5|) = 10 - (-10) = 20
-|10-2|*|5|= -8 * 5 = -40.

Absolute value is considered a grouping symbol, so you always want to do what is inside the absolute value prior to doing any operations attached to it.

Can you guess what the graph of y= |x| would look like? It'd resemble a V. We will learn more about graphing absolute value (and its similarity with parabolas) in the algebra section.

C3. Factoring in Detail

The first step of the majority of factoring involves finding the prime factorization. This is actually surprisingly easy. The easiest way, in my opinion, is to just keep dividing a large number by primes you know work, and create a "Prime Factor" tree. Let's pick a number: 6720. To find the prime factorization, just make a tree (if you're uncomfortable with just writing it out):

Here is a way to factor the number "6720". Unb4nn3dz0r is GOD at Paint!


You can use the "tree" method any time you can't easily recognize the prime factorization of a number. Remember, ANY integer can be expressed as the product of prime numbers.

You might be wondering; what the hell will I ever need to know the prime factorization of a number for? Well, a simple answer: to do stuff with fractions. One occasion where you need the prime factorization of a number to work with fractions is to find the Least Common Denominator of two numbers. Finding the GCF and LCM was already explained in the "Addition" section of Fractions. [If anyone feels this explanation is not sufficient, feel free to say so!]

C4. Logarithms (C4?! Oh noes! Dyno-mite!)

Welcome to the section on Logarithms.

Logarithms are the inverse function to exponentiation. However, unlike exponentiation, logarithms only require 1 input, unless otherwise specified. It is recorded log(subscript base) x, where x is known as the exponent (we will explain why shortly), and the subscript symbol is known as the base of the log. If no base is specified, it can be assumed that the base is 10. Another type of log is the natural log, which is actually log(base e), and is recorded ln (x). As most subscript letters are not available in Unicode, assume that letters font recorded in size 1 implies the base of the log, and will not be in parenthesis. For example; log2 (8) = 3 would imply log(base 2) of 8.

On most calculators, you have two types of log keys.
These keys include one for common logarithms (base 10) and one for natural logarithms (base e).

log(x) - common log

ln(x) - natural log

In most instances, LOG standing alone has the base of "10". This number is naturally the base of 10, e, or 2.

One is presented with a problem:
2^x = 8

Rewrite:
log₂ 8 = x

Using the last property of logs, we have that:

log₂ 8 = (log₁₀ 8) /(log₁₀ 2). We can now use a calculator and get 3. x = 3.

If you are given the equation "log₈ 512 = x", then x = 3 (using same method).

Check the equation: 8^3 = 512

You are given log(2) = .3010
Calculate log(32) + (log 16) without a calculator with logs.

As we know, 32 = 2 ⁵, and 16 = 2⁴. Therefore;
log(2 &#8309 + log(2&#8308 is equivalent to log(32) + log (16).

By the fourth property presented:
5 log 2 + 4 log 2 = 9 log 2 = 9 * .3010 = 2.7090 [correct to the nearest 1000th, a calculator yields 2.7093]

C5. Order of Operations

The order of operations can be simply shown by the anagram PEMDAS.

Here is what PEMDAS stands for:

P - Parenthesis

E - Exponent

M - Multiplication

D - Division

A - Addition

S - Subtraction

If you don't remember PEMDAS easily, you can think of it as an acronym for Please Excuse My Dear Aunt Sally. Some people say that sexual thoughts stay in the mind the longest. So, if you're one of these people...Pussy Eaters May Die After Sex.

You would perform these functions in order, according to PEMDAS, from left to right.

Here is a complex equation:
[2 + 4(6 + 3 * 2)² - 4 / 3 + 8]

You would solve the problem according to the anagram.
-First solve the equation in the Parenthesis, by using Multiplication and then Addition.
[2 + 4(12)² - 4 / 3 + 8]

Next you would solve the exponent, and include the multiplication.
[2 + (4 * 144) - 4 / 3 + 8]

Simplify Parenthesis:
[2 + 576 - 4 / 3 + 8]

Next do Division:
[2 + 576 - 1.33 + 8]

Now group all of the Addition, and then do the subtraction.
Final Answer = 584.67

And now you know the Order of Operations!

C6. Decimals and Fractions

One of the most important concepts of arithmetic is the understanding of decimals and fractions. Why? They are *everywhere* and the ability to work with them is a very helpful life-skill. Decimals are just written as normal numbers, but typically when one thinks of decimals, they think of something like 10.5, or 5.14285714... This is very true as well. Significant figures are also related to decimals, but this is another discussion entirely. For now, let's just stick to the basics. Fractions are always written in the form x/y. Sometimes, x or y can be irrational, and while the resulting fraction may not represent a rational number, it is not uncommon to record something like: (pi + 4)/3 in fractional form. A fraction that, when simplified to lowest terms, has a prime number in the denominator will always produce a repeating decimal, but the inverse of this is not always true (ie; 1/6). Here, what I was hoping to do was to teach you how to convert fractions -> decimals and vice versa.

Fractions -> Decimals are very simple. Literally, all you do is divide one by the other, and write this down. If the fraction is represented by an integer/integer (all rational numbers will be able to be represented this way), then the decimal will either be terminating or repeating/recurring. A terminating decimal, such as 10.3560000... or 10.50000 is a decimal which ends with a string of 0's (obviously, this can be infinitely long, depending on certainty). The string of 0's is generally omitted in mathematical calculations, except for in certain sciences where significant figures are very important.

However, sometimes there are these pesky buggers that are repeating decimals. If the decimal is rational, it will always be either terminating or repeating, so you must determine where the repeat happens. Basically, if you are doing this without a calculator (reference Long Division), you will clearly see the where the pattern begins to repeat. However, on paper, it is sometimes more difficult to visualize. You're going to have to find a way, there's really no other way to phrase it - just find out where the repeat begins. If you're given 15/7, you will find that it equals 2.142857142857142857, etc. As you can probably recognize, the 142857 are going to be continuously repeating. You can record this as 2.142857142... [signifying the repeat], or you can place a vinculum over the repeating portion.



2.142857

Going from Decimals -> Fractions takes a bit more getting used to, but is also very easily acquired. If you are given a terminating fraction, you can literally just place the decimal in the numerator [omitting the decimal point], and then count the number of digits you have to place in the denominator. Then, you can just factor out like terms (see section C3).

For example, converting: 5.25125 into a fraction. The numerator will be 525125, and the denominator will have 6 digits; 10⁶. So, you have 525125/1000000. You can simplify this and find it's equal to 4201/8000.

Now, to convert repeating fractions to decimals, follow the steps in the diagram.


You already know how to do the arithmetic with decimals and fractions, so this section is officially done

C7. Other
..::Will Add Upon Request::..



Sources of Information:
http://id.mind.net/~zona/mmts/miscellaneousMath/typesOfNumbers/typesOfNumbers.html
http://en.Wikipedia.org/wiki/Arithmetic
http://mathforum.org/dr.math/

WE HAVE FINALLY COMPLETED THE FIRST TWO SECTIONS TO OUR GUIDE ON ARITHMETIC

THIS GUIDE IS OFFICIAL 6/50 COMPLETE, AND IT WILL BE EDITED OCCASIONALLY!.

*Additions*
11-08-08 Added Arithmetic Section.
11-11-08 Added Part C. More Advanced Operations

 

YesIneed2ndpost

 

I'm Wowed already :X

 

Nice guide, probably got inspiration from my math guides . Anyways, like the pictures and you got some pretty basic stuff down. If you could, add some advanced stuff other than the operations, like fractions and decimals. 8.5/10

 

Yes. We need the 2nd post reserved incase we go over the limit for the post, if there is one. And thank you very much Matthew for helping write this whole thing, as we've done half and half of the work so far.

I hope we can add a new section either tonight or tomorrow, and continue to expand on our Guide to make it the best that will ever exist.

Hehe, and it's nowhere near finished!

Nope, no inspiration at all.

My inspiration came from Venom1, R33l2r3al, and The Dark.
They have all been my true inspiration in the UE section.

I've been wanting to make a Math Guide for a month now, but had no clue how to go about doing it.

MatthewGor123 and I teamed up to make this guide, which has made it a lot easier.

And btw.. there are Fractions and Decimals in there, and they have been.
Apparently you didn't take long enough to read through it.

And if you noticed, the Guide is 1/50th complete. Look at the Index.

__________________________________________________________________________

Thanks again for all the comments! =D

 

Here is the edit so far, I'll finish when my brain works again.

 

Very well done, I like how you broke all the different subjects apart with nice color. And used very good / easy to understand detail. Good work, 9/10.

 

xD We haven't even really started If you didn't see, it's 4/50th of the way done ^_^. But thanks, we've worked pretty hard Credit Unb4nn3d for how awesome it looks

 

Wow, amazing job.
This is over the top, detailed, nice formatting and a ton more!
9/10, you get the other .1 vote when you finish.

Great job.

 

Best guide I've ever seen :O
10/10
Great use of pictures.
Great use of information.
Great use of colours.

I can't wait till it's finished.

 

Thanks! You can thank UnB4nn3dz0r for the pwnage use of colors We're gonna be working on it soon; today really wasn't a good day for either of us. We should have C. More Advanced Operations done tomorrow.

 

Schoolwork has unfortunately slowed our additions to this Guide.
MatthewGor123 and I, however, have finally been able to find and devote enough time to be able to implement Part:

C. More Advanced Operations

-This update was made on 11-11-2008 at 10:57 PM EST

 

Nice guide, not going to read it all because it's too hard to find somewhere where it's stuff I care about, a.k.a number theory, but cool guide.

 

I hate precalc

 

Wow, really nice guide.

Would be helpful to add some things on parabolas and quadratics including the factorised forms.



Can't wait until it's finished. 10/10 so far on what you have done.

 

Very nice guide Alex, <3

seeing as im failing math this will be useful to me.

nice amount of pictures, well written and very in depth.

9/10

 

nice my brain hurts 8/10

 

How about, very nice guide Matthew and Alex.

It should. Wait until tomorrow... *Hangover*

 

pemdas?
We're taught bedmas in school, it is esentially the same thing but maybe an honorable mention to that as well?

Also consider rewording logarithms, I'm really confused(never heard of them until today, and I'm in grade 11 precalc, normal?) most concepts come to me quite easily, but that made no sense whatsoever (to me). 9/10 to the other finished parts though.

 

Your, logarithms are quite a difficult concept to grasp at first. More will be explained as we move forward with this guide. We've both been extremely busy of late, but we promise to further update this guide ASAP!