Types of Numbers and Algebraic Properties

Types of Numbers

Before we get too deep into algebra, we need to talk about the types of numbers there are out there, since you’ll have to be familiar with them. These are Whole Numbers, Counting Numbers or Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Imaginary Numbers, and Complex Numbers, as shown in the table. The letters in parentheses indicate how they may be abbreviated.

Type of Number Examples Hints/Notes

Natural Numbers (Counting Numbers)

 ($ \mathbb{N}$)

Numbers you use for counting: $ 1, 2, 3,…$ It’s “natural” to count on your fingers: $ 1, 2, 3,…$
Whole Numbers The natural numbers, plus $ 0: 0, 1, 2, 3, …$ The word “whole” has an “o” in it, so include 0.
Integers

 ($ \mathbb{Z}$)

Whole numbers, their opposites (negatives), plus $ 0: … –2, –1, 0, 1, 2, …$ Integers can be separated into negative, 0, and positive numbers.
Rationals

($ \mathbb{Q}$)

Integers and all fractions, positive and negative, formed from integers. These include repeating fractions, such as $ \displaystyle \frac{1}{3}$ or $ .33333…$ or $ .\overline{3}$. The word “rational” is a derivation of “ratio”, and rational numbers are numbers that can be written as a ratio of two integers. “Q” stands for quotient.
Irrationals Numbers that cannot be expressed as an integer or fraction; irrationals include $ \pi ,\,\sqrt{2},\,e.$ (We’ll learn about these later). If something is “irrational”, it’s not easy to explain or understand.

Real Numbers

($ \mathbb{R}$)

Rational Numbers and Irrational Numbers. The real number system can be represented on a number line, for example:

If a number exists on a number line that you can see, it must be “real”.

Note that the “smallest” real number is negative $ (-)$ infinity $ (-\infty)$, and the largest real number is infinity $ (\infty) $. We can never really get to these “numbers” ($ -\infty $ and $ \infty $), but we can indicate them as the “end” of the real numbers.

Complex Numbers

($ \mathbb{C}$)

Real numbers, plus imaginary numbers (concept only, such as $ \sqrt{{-2}}=\sqrt{2}i$). “Imaginary” numbers are difficult to imagine, since they are so “complex”.

Here’s a Venn Diagram that shows how the different types of numbers are related. Note that all types of numbers are considered complex. And don’t worry too much about the complex and imaginary numbers; we’ll cover them in the Imaginary (Non-Real) and Complex Numbers section.

Algebraic Properties

Before we get into algebra, we also need to talk about some of the properties we’ll use to solve equations. We’ll need these to get the variable all by itself on one side of the equal sign – which is the basis of algebra at this point. Let me show you this with just plain numbers; since these work with plain numbers, they also work with variables (letters)!

Algebraic Properties of Equality

Here are the Algebraic Properties of Equality, since they deal with two sides of an equal sign:

Example

Property of Equality

 $ \begin{array}{c}5=5\\5+1=5+1\\\,\,\,\,\,\,\,6=6\,\,\,\,\,\surd \end{array}$ Additive Property of Equality: adding the same thing to both sides of an equation. In the example, we start with $ 5$ on each side, and then add $ 1$ to each side to get $ 6=6$. Try it for other numbers!
 $ \begin{array}{c}5=5\\5-3=5-3\\\,\,\,\,\,\,\,\,2=2\,\,\,\,\,\surd \end{array}$ Subtraction Property of Equality: subtracting the same thing from both sides of an equation. Again, if we start out with $ 5$ on each side, and then subtract $ 3$ from each side, we get $ 2=2$.
 $ \displaystyle \begin{array}{c}5=5\\5\times 4=5\times 4\\\,\,\,\,\,\,\,\,\,20=20\,\,\,\,\,\surd \end{array}$ Multiplicative Property of Equality: multiplying by the same thing on both sides of an equation. If we multiply each side by $ 4$, it works, too!
 $ \displaystyle \begin{array}{c}6=6\\6\div 2=6\div 2\\\,\,\,\,\,\,\,\,3=3\,\,\,\,\,\surd \end{array}$ Division Property of Equality: dividing by the same thing on both sides of an equation. If we divide each side by $ 2$, it works too!

Commutative and Associative Properties

There are two more properties that will be very useful in solving algebra equations:

Example

Property

 $ \begin{array}{c}(5+4)+2=5+(4+2)\\9+2=5+6\\\,\,\,\,\,\,\,\,\,11=11\,\,\,\,\,\,\surd \end{array}$ Associative Property: grouping numbers differently (using parentheses).

This only works for addition and multiplication. Try it yourself with subtraction and division to see that it doesn’t work.

I remember this since you associate yourself with different groups.

 $ \begin{array}{c}\,8\times 4\times 2=4\times 2\times 8\\32\times 2=8\times 8\\\,\,\,\,\,\,\,\,\,\,\,\,\,64=64\,\,\,\,\,\,\surd \end{array}$ Commutative Property: changing the order of numbers or variables.

This only works for addition and multiplication. Try it yourself with subtraction and division to see that it doesn’t work.

I remember this since the word “commutative” has an “o” for the second letter; this reminds me of “order”.

As an example of why the Associative and Commutative properties are important, we may need to use these to show that $ 5+4+2=2+2+7$:    $ \displaystyle \begin{align}5+4+2&=2+2+7\\5+4+2&=7+2+2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Commutative}\\\left( {5+4} \right)+2&=\left( {7+2} \right)+2\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Associative}\\9+2&=9+2\\11&=11\end{align}$

Distributive Property

There is one other property that is used a lot in algebra; this one is a little different, but it has to do with “distributing things through”. Let’s say you are trying to take a collection for your algebra teacher for an end of the year gift. You are collecting $10 from 10 students in Class A and 8 students in Class B. You can see that you will collect $180, but there are two different ways to solve this problem:

$ \$10×(10 + 8)=\$10×18=\$180$

You add up all the students in both classes first to get 18 people, and then multiply by $10.

OR

$ (\$10×10)+(\$10×8)=\$100+\$80=\$180$

You first collect the money from the students in Class A to get $100, then collect the money from the students in Class B to get $80, and then add the two amounts together to get $180.

This is called the Distributive Property, since we can either leave the 10 on the outside of the parentheses, or distribute through (“push it through“) to both the numbers on the inside of the parentheses. We can do this when there is addition or subtraction inside the parentheses.

Here are other examples, including one using a variable, which we’ll learn about in the Solving Algebraic (Linear) Equations section.

Example

Distributive Property

$ \displaystyle \begin{align}4\times (2+3)&=4\times 5\,\,=4+4+4+4+4\\&=(4+4)+(4+4+4)\\&=4\times 2+4\times 3\,\,\,\,\,\surd \end{align}$ I just randomly divided up 5 into 2 and 3. When you multiply 4 times 5 ($ 2+3$), it is just like adding 4 five times. Because of the Associative Property above, you can lump together the first two 4‘s first and the last three 4‘s second.

This “proves” the Distributive Property.

 $ \displaystyle 2n+3n=n(2+3)=n(5)=5n$ Here is an example of the Distributive Property with variables; you will see this a lot in algebra. This is an example of “combining like terms”, or putting together the same variables.

Summary of Algebraic Properties

Since these are pretty important, here’s another table with these properties (and a couple more) with examples:

Name Hints Examples Notes

Associative

“Grouping”

You “associate” with different groups.

$ \begin{array}{l}5+\left( {15+4} \right)=\left( {5+15} \right)+4\end{array}$

Works with addition and multiplication, not subtraction or division.

Commutative

“Ordering”

Since Commutative has an “o” in it, think “order”.

$ \begin{array}{l}5+4+3=4+3+5\end{array}$

Works with addition and multiplication, not subtraction or division.

Distributive

“Distributing or Pushing Through Parentheses”

Think of “distributing” something to your friends.

$ \matrix{
5 \times \left( {3 + 4} \right) \cr
= \left( {5 \times 3} \right) + \left( {5 \times 4} \right) = 15 + 20 = 35 \cr} $

 

$ \begin{array}{l}5-2\left( {x-3} \right)=5-2x+6=11-2x\end{array}$

 

$ \begin{array}{l}5x+7x=\left( {5+7} \right)=12x\end{array}$

When negatives are on the outside of the parenthesis, make sure you distribute the negatives to second number, too. (Multiplying two negatives results in a positive.)

Identity

“Staying the Same”

You always come back to your “identity”.

$ \begin{array}{c}9+0=9\\9\times 1=9\end{array}$

Additive identity is 0.

Multiplicative identity is 1.

Inverse

“Undoing”

When you put your car in “inverse”, you go backwards.

$ \displaystyle \begin{align}9+-9=0\\9\times \frac{1}{9}=1\\\frac{8}{9}\times \frac{9}{8}=1\end{align}$

Additive inverse is $ -a$, since $ -a+a=0$.

 

Multiplicative inverse is $ \displaystyle \frac{1}{a}$, since $ \displaystyle \frac{1}{a}\times \frac{a}{1}=1$; the multiplicative inverse of $ \displaystyle \frac{a}{b}$ is $ \displaystyle \frac{b}{a}$, since $ \displaystyle \frac{a}{b}\times \frac{b}{a}=1$. Multiplicative inverses are reciprocals.

There are actually other properties used in algebra that you’ll be learning, but these are the main ones you’ll be using to solve algebra problems. Remember that your goal in solving algebra problems is to get the variable or unknown to one side all by itself! We’ll see this in the Solving Algebraic Equations section.

Proper Algebraic Notation

One more boring thing we must talk about before we solve equations is proper algebraic notation, or “grammar”. Just like English has proper “grammar”, math does to! The proper way to write the solutions of equations (and inequalities, which we’ll learn shortly) is shown below.

Sets

A set of numbers (or anything!) is a collection of items that are called elements. A set can be finite, such as the numbers 1, 2, and 3 (written as $ \left\{ {1,2,3} \right\}$). A set can also be infinite (with an unlimited set of numbers), such as the set of real numbers, including all the fractions, between 0 and 1.

Union and Intersection

The union of two or more sets includes everything in either of the sets. For example, the union of the sets $ \left\{ {1,2,3} \right\}$ and $ \left\{ {3,4,5} \right\}$ is $ \left\{ {1,2,3,4,5} \right\}$, since you include everything in both sets, but don’t repeat numbers. You write union as $ \cup$, so $ \left\{ {1,2,3} \right\}\cup \left\{ {3,4,5} \right\}=\left\{ {1,2,3,4,5} \right\}$.

The intersection of two or more sets includes only those things that are in both sets. For example, the intersection of the sets $ \left\{ {1,2,3} \right\}$ and $ \left\{ {3,4,5} \right\}$ is $ \left\{ 3 \right\}$, since you include only the numbers in both sets, but don’t repeat the numbers. You write intersection as $ \cap $, so $ \left\{ {1,2,3} \right\}\cap \left\{ {3,4,5} \right\}=\left\{ 3 \right\}$.

The way these are written (with the brackets) are called roster notation, since you have a “roster” or list of numbers.

Set Builder, Inequality, and Interval Notation

Other notations that are useful include set builder notation/inequality notation, and interval notation, as shown with examples. You can see how there may be many ways to show set builder notation.

Example Words or Equation

Set Builder/Inequality Notation

Interval Notation (used most often)

 $ x=4$

 $ \left\{ {x\,|\,x=4} \right\}$ or $ \left\{ {x:x=4} \right\}$

“the set of all $ x$ such that $ x$ equals 4”.

 $ \left[ 4 \right]$

(Not typically used)

$ x\ne 6$

 

All real numbers except 6

$ \left\{ {x\,|\,x\ne 6} \right\}$   or

$ \left\{ {x\in \mathbb{R} \,|\,x\ne 6} \right\}$   or

$ \left\{ {x\,|\,x\,\in \mathbb{R},\,x\ne 6} \right\}$

“the set of all numbers $ x$ such that $ x$ is not 6 and $ x$ is a real number”.

$ \begin{array}{c}\left( {-\infty ,6} \right)\cup \left( {6,\infty } \right)\\\left( {-\infty ,6} \right)\,\text{or}\,\left( {6,\infty } \right)\end{array}$

 

(Either way can be used)

All Real Numbers

 $ \left\{ {x\,|\,x\in \mathbb{R}} \right\}$  $ \left( {-\infty ,\infty } \right)$

No Solution

{ }

Ø (symbol for null set or “nothing”)

 $ x=4,5,6$

$ \{4,5,6\}$

  $ \left[ 4 \right]\cup \left[ 5 \right]\cup \left[ 6 \right]$

(Not typically used)

 $ \displaystyle -2\le x<4$

($ x$ is greater than or equal to –2 and less than 4)

$ \left\{ {x\,|\,-2\le x<4} \right\}$   or

$ \left\{ {x\in \mathbb{R}\,|\,-2\le x<4} \right\}$

Inequality notation: $ -2\le x<4$

$ [2,4)$

(The 2, with the “hard” bracket, is included, but the 4, with the “soft” bracket, is not.)

 $ \displaystyle x\le -2\text{ or}\,\,\text{ }x>2\text{ }$

($ x$ is less than or equal to –2 or greater than 2)

$ \left\{ {x\in \mathbb{R}\,|\,\,x\le -2\,\,\text{or}\,\,x>2} \right\}$   or

$ \left\{ {x\,|\,x\le -2\,\,\text{or}\,\,x>2} \right\}$

Inequality notation: $ x\le -2\,\,\text{or}\,\,x>2$

$ \left( {-\infty ,-2} \right]\text{ }\cup \text{ }\left( {2,\infty } \right)$

$ \left( {-\infty ,-2} \right]\,\text{ or }\left( {2,\infty } \right)$

Positive even numbers

$ \left\{ {x\in \mathbb{N}\,|\,\,\,x=2n} \right\}$    or

$ \left\{ {x|\,\,x=2n\,\,\,\text{and}\,\,\,x\in \mathbb{N}} \right\}$

Not applicable; not an interval

Note that for interval notation, if a “less than” ($ <$) is used, we use “(“, if a greater than ($ >$) is used, we use “)”. If a “less than or equal to” ($ \le $) is used, we use “[“, and if “greater than or equal to” is used ($ \ge $), we use “]”. Thus, we use brackets (also called hard brackets) if we are including the endpoint (the actual number at the end), and parentheses (also called soft brackets) if we are not including the endpoint. Since we never actually get to $ \infty $ or $ -\infty $, we only use soft brackets (parentheses) with them.

$ \mathbb{R}$ means all real numbers (everything on the number line), and $ \mathbb{N}$ means natural numbers (1, 23, and so on). Also remember that Ø means “no answer” or “no solution”; this happens sometimes in algebra.

Learn these rules, and practice, practice, practice!

On to Solving Algebraic Equations – you are ready!