Introduction to Algebra

The_Vast_Void · February 16, 2021, 8:57am

So, as I said earlier in a post, I will be trying to help people here better understand basic algebra. Now, I’m not a professional teacher, I am not educated in educating people, but I think I may be able to help some of you.

Now, algebra isn’t really a complicated subject. It is essentially the study of more complicated arithmetic equations, with some new concepts like variables, parameters, more operators, and functions. This is how Wikipedia defines algebra: " In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols". And I guess this is a decent explanation, but you can’t really understand what algebra is before learning it.

So, here’s a brief overview of what I’m going to try and review:

Introduction to algebra: A review of the structure of an equation
Introduction to variables: The concept of replacing numbers with non-numeric symbols
Parameters
Systems of equations
Introduction to polynomials: Factoring and multiplying polynomials (this may take several posts)
Solving second degree polynomials
Inequalities with polynomials
Logarithms and other algebraic functions

This isn’t final, and I may add and remove topics. If you have any suggestions, don’t hesitate to post them!

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The_Vast_Void · February 16, 2021, 10:35am

Here’s the first post. Hopefully everything is understandable, and if you have any questions, feel free to ask! Everything here should be a review, but ask if something doesn’t make sense to you. I wanted this to get out of the way we could get to the more interesting bits, so tell me if I rushed something too much. Thanks for reading!

Topic 1: Arithmetic review

Topic 1

Studying the structure of an equation (a review of arithmetic)

Now, all of you should have a good understanding of arithmetic before attempting to understand algebra. If you don’t, go finish that first. Arithmetic is the core of all math, and it is of utmost importance that you understand it before continuing on to more complicated subjects.

The equals sign

So first off, what is an equation? Well let’s look at the etymology of the word equation. Just by dissecting the word, you can see that it is similar to equal + tion. For two things to be equal, they are essentially equivalent. So basically what an equation is mathematical statement that two values or equal, equivalent, and the same. The mathematical symbol that denotes equality is the equals sign, =.
So, the most simple equation is:
0 = 0
Zero is equal to zero. Or:
7 = 7
Seven is equal to seven.
This is important to understand. Unlike arithmetic, where the equals sign is usually used to say “what’s the answer?”, for example:
2 + 2 =
In algebra, the equals sign is commonly used to denote that two things are equal, and it is up to you to figure out how. That doesn’t mean that the equals sign in arithmetic is somehow a different symbol than the one in algebra. They are very much the same. Algebra just unveils the complexity of this seemingly simple symbol.

Arithmetic operators

If you have indeed listened to my advice, you should be familiar with these fundamental operators:

Plus (+) - this is the symbol for addition. It takes two numbers and combines their values. Eg. 4 + 2 = 6. What exactly is this saying? If you take 4 of anything, and add 2 more of that something, you will end up with 6 of that something. For instance, if you have 4 sticks, which we can represent as ||||, and 2 more sticks (||), in total you will have 4 + 2 sticks, which is |||||| or 6 sticks.
Minus (-) - this is the symbol for subtraction. It takes two numbers and tells you the difference between them. For example. Eg. 4 - 2 = 2. This means that if you have four of anything, and you take away 2 of that something, you’ll be left with two of that something. Let’s use our stick example. Our four sticks look like this: ||||, now if we take away two sticks (||), we’ll be left with || or 2.
Times (×) - this is the symbol for multiplication. It takes two numbers and gives you what you would have if you had some number of times the other value. This is essentially repeated addition. If you have 4 sticks, and someone else has 3 times that many sticks, that means they have 4 × 3 sticks, or 4 + 4 + 4 sticks, which is 12.
Divide (divided by) (÷) - this is the symbol for division. If you have two values, it tells you how many times one value can fit into another. If you have 6 sticks and you give separate them equally among two of your friends, each friend will get 6 ÷ 2 = 3 sticks.

There are a few more points we must go over about these operators. First off, if you haven’t noticed, addition and subtraction or what we call “inverse” or “opposite” operations. If you add 2 to something and then subtract 2, you’ll end up with the same value you started with. (This happens no matter which operation you use first). For example: 10 + 6 - 6 = 16 - 6 = 10.
The same is true with multiplication and division. If you, for example, multiply something by 2 and then divide that by 2, you will end up with what you began with. (This too, is not affected by the operation you decide to use first). For example: 10 × 6 ÷ 6 = 60 ÷ 6 = 10

Another important note is that addition and multiplication are commutative. This means that if you add 2 and 7, that is the same as adding 7 and 2. This is pretty intuitive if you think about what exactly addition is doing. Think about it with the stick example I was using earlier. 2 + 7 can be written as || + ||||||| = ||||||||| = 9. 7 + 2 = ||||||| + || = ||||||||| = 9
Now, let’s think about why multiplication would be commutative. Let’s say you have 7 × 6. This can be written like 7 + 7 + 7 + 7 + 7 + 7. So let’s say we’re dealing with little pebbles, it can be written as:

.   .   .   .   .   .    . . . . . .
.   .   .   .   .   .	 . . . . . .
.   .   .   .   .   .	 . . . . . .
. + . + . + . + . + . =  . . . . . .
.   .   .   .   .   .	 . . . . . .
.   .   .   .   .   .	 . . . . . .
.   .   .   .   .   .	 . . . . . .

6 × 7 can be written as:

.   .   .   .   .   .   .    . . . . . . .
.   .   .   .   .   .   .    . . . . . . .
.   .   .   .   .   .   .    . . . . . . .
. + . + . + . + . + . + . =  . . . . . . .
.   .   .   .   .   .   .    . . . . . . .
.   .   .   .   .   .   .    . . . . . . .

Which you may notice is just 7 × 6 flipped on its side! So there are the same number of pebbles!

Subtraction and division are not commutative, though. For instance: 9 - 6 = 3, but 6 - 9 = -3. 4 ÷ 2 = 2, but 2 ÷ 4 = 0.5 (I will discuss fractions, decimal points, and negative numbers soon.)

There are two more operators you should be aware of:

Power - this can be written as something like 6³ or 4². What this means is that you take the big number on the bottom and multiply it by itself the number of times the number on the top says. For example: 5³ = 5 × 5 × 5 = 25 × 5 = 125. Sometimes this will be called “squared” or “cubed”. If you have some number squared, it is multiplied by itself twice (ie. 5 squared is 5² = 5 × 5 = 25). If a number is cubed, it is multiplied by itself three times (ie. 4 cubed is 4³ = 4 × 4 × 4 = 16 × 4 = 64)
Root (√)- this can be written as something like ²√9 or ³√64 (if no number is given, like √25, then it is the “square” root, the number is assumed to be 2). What this means is you’re looking for a number that multiplied by itself the number of times the small number says, you’ll get the number inside the √. For example:
³√64 = 4 because 4³ = 64. √25 = 5 because 5² = 25.

Both powers and roots are what we call “exponents”.

The order of operations

The order of operations is the universally accepted order in which operations are performed. It goes:

Parentheses (sometimes called brackets)
Exponents
Multiplication and division (multiplication does not always come before division. They are performed from left to right.)
Addition and subtraction (again, they have the same priority, and are performed from left to right.)

A common acronym for this is PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction. BUT REMEMBER: multiplication and division have the same priority and are evaluated from left to right, and so is addition and subtraction.
For instance:

6 + 2 × 4 ÷ (9 - 7) = 6 + 2 × 4 ÷ 2 = 6 + 8 ÷ 2 = 6 + 4 = 10

Now, here’s a trick one:
4 ÷ 2(3 - 1)
Is it 4 ( 2 × 2 = 4 )
Or 1 ( 4 ÷ (2 × 2) = 4 ÷ 4 = 1)?
The answer is actually 1. I’ll explain why this is later when we discuss variables and parameters.

Fractions and decimals

Fractions are really just another notation for division. They can be written like: ²/₉ which is really just 2 ÷ 9. The upper part of the fraction is called the “numerator” and the lower half is the “denominator”. If you have a fraction like: ⁴/₁₆, you can divide (or multiply) both halves of the fraction. So ⁴/₁₆ = ^{4 ÷ 4}/_{16 ÷ 4}.
Decimals are just numbers written in terms of powers of 10. So like: 0.2 = 2/10 = 1/5, 0.34 = 34/100 = 17/50 and so on.

Negative numbers

Negative numbers are numbers that come before zero on the number line. They are written with a minus sign before them, like -1. -1’s distance from 0 is 1, but it comes before 0. Subtraction can be written as the addition of negative numbers. For instance: 4 - 2 = 4 + (-2). Notice the parentheses around -2. These aren’t strictly necessary, but I suggest you use them.

Topic 2

Introduction to variables

One of the main concepts of algebra (it may actually be the main concept) is the idea that numerical values can be represented by anything. I can use the greek letter alpha (α), the latin letter “a”, the hebrew letter alef (א), a question mark (?), anything. These are called “variables”.
Many questions you will get in algebra will require you to find the values of these variables. For example if I gave you the question:
x + 2 = 3
What would x equal? The answer should be pretty obvious, it’s 1. This is pretty simple to solve, and it requires almost no knowledge of algebra. But this is important to understand. Numbers can be represented by anything, not only the symbols that you’re used to. Variables can be anything, both their symbol and their value. For instance, instead of x, I could’ve written:
α + 2 = 3
(α is the greek letter alpha.)

So, how do you go about solving equations with variables? Well, equations (and inequalities, for the most part, but we’ll get to them later) have this really handy property: you can do anything to one side of an equation, as long as you do it to the other. For instance, if I have:
5 + 5 = 10
If I add 2 to the left side of the equation, I get 5 + 5 + 2 = 10, which just simplifies to 12 = 10. This is obviously incorrect. But if I add 2 to the right side of the equation, I get: 5 + 5 + 2 = 10 + 2, which simplifies to 12 = 12, which obviously is correct.
Why is this? Well, what an equation is telling us is that both sides of the equation are the same. 5 + 5 is the same thing as 10. And in order for two things to stay the same after undergoing some change is for both of them to undergo the same change. Let’s say I have two pieces of paper. If I draw a line on one, the only way for both of my pieces of paper to be the same is if I draw the exact same line on the other piece of paper.
So, using this property, let’s solve our equation x + 2 = 3. In order to get the answer, we want to end up with something like x = __ (for example, x = 2 or x = 3.3). So we need to get all of the xs on one side, and all of the numbers on the other.
So, let’s begin. We start off with:
x + 2 = 3
Now, in order to get x by itself, we subtract 2 from each side:
x + 2 - 2 = 3 - 2
Simplifying, we get:
x = 1

This was a relatively simple equation. Let’s try a harder one:
x × (2 + 3) - (2x - 4) ÷ 2 = x + 3
First, let’s simplify:
x × 5 - (2x - 4) ÷ 2 = x + 3
5x - (2x - 4) ÷ 2 = x + 3

Now, when you have something like (2x - 4) ÷ 2, you must do the operation on all elements in the parentheses. For instance, if you have (3 + 6 + 9) ÷ 3, we can simplify and then get 18 ÷ 3 = 6, or dividing each element, we get: 3 ÷ 3 + 6 ÷ 3 + 9 ÷ 3 = 1 + 2 + 3 = 6. Why is this? Let’s think about this in terms of sticks. We can think about whatever is inside the parentheses as a bundle of some number of sticks. If we divide it, we find the number of sticks if we evenly split them into that number of groups, so we can split them up into groups of whole sticks, or we can split each stick into that number, and group each part together. Like, if we’re dividing the number of sticks by 3, we can split each stick into 3, and group the top thirds together, the middle thirds together, and bottom thirds together. Then we count the number of whole sticks in each group. For instance, if we have 9 sticks:

|||||||||
|||||||||
|||||||||

If we split each third of a stick:

Group 1: |||||||||

Group 2: |||||||||

Group 3: |||||||||

Now we count the number of sticks in each group. There are 9 thirds of sticks, so that means 3 whole sticks. Let’s say you have (3 + 6 + 9) ÷ 3 the value in the parentheses is equivalent to a group of 3 sticks, 6 sticks, and 9 sticks. We then divide each stick into thirds, so that splits it into three groups. Each group has 3 thirds from the first group (3 + 6 + 9), 6 thirds from the second group (3 + 6 + 9), and 9 thirds from the third group (3 + 6 + 9). So we get 1 + 2 + 3.

Let’s continue simplifying our equation. We have:
5x - (2x - 4) ÷ 2 = x + 3
Now, we know: (2x - 4) ÷ 2 = 2x ÷ 2 - 4 ÷ 2 = x - 2, so we get:
5x - (x - 2) = x + 3
When we have - (x - 2), that’s the same thing as saying: -1 × (x - 2) = -1 × x + (-1) × (-2) = -x + 2
5x - x + 2 = x + 3
Simplify:
4x + 2 = x + 3
Subtract x from both sides:
4x - x + 2 = x - x + 3
3x + 2 = 3
Subtract 2 from both sides:
3x + 2 - 2 = 3 - 2
3x = 1
Divide both sides by 3:
x = ¹/₃
So x is one third.

That’s it for now! If you have any questions, feel free to ask!

Topic 3

Parameters

Parameters are like variables, but are viewed as constant. Parameters take the place of variables, but usually you don’t need to find their values. They are used to find the values of other variables. Like so:
Find x:
2x + 3 = 4a + 7
We simplify:
2x = 4a + 4
x = 2a + 2
So, here you can see that we didn’t find the parameter a, instead we used it to find x. There’s no real distinction between parameters and variables, their real distinction is only really obvious in terms of functions, which we will cover later on. Essentially, all you need to understand is the concept that you can find one variable in terms of another (variable or parameter).
Let’s take a look at another example:
(x + 2)(a + 3) = 4 + a
Now, remember, when you have something like (x + 2)(a + 3), you can simplify this to: x(a + 3) + 2(a + 3) (you are multiplying each element in the first parentheses by the second parentheses) which then expands to: ax + 3x + 2a + 6, but this may actually not be the smartest way to do this. Why don’t we just divide both sides by a + 3? Then we get:
(x + 2)(a + 3) / (a + 3) = (4 + a) / (a + 3) (/ means division)
x + 2 = (4 + a) / (a + 3)
x = (4 + a) / (a + 3) - 2
Now we need to simplify the right side. Remember that when subtracting or adding fractions, they must have the same denominator (let’s see if you can figure out why). So we can multiply and divide 2 by (a + 3), so we get:
x = (4 + a) / (a + 3) - 2(a + 3) / (a + 3)
2(a + 3) / (a + 3) is just 2, but now that it has the same denominator as (4 + a) / (a + 3), we can add them. Like so:
x = (4 + a - 2(a + 3)) / (a + 3)
x = (4 + a - 2a - 6) / (a + 3)
x = (-a - 2) / (a + 3)
We can simplify this to:
x = - (a + 2) / (a + 3)
This may not be the prettiest answer, but you’ll see a lot of answers like this.

That’s it for now, if you have any other questions, feel free to ask.