Types of Numbers | Summary of Algebraic Properties (Chart) |

Algebraic Properties | Proper Algebraic Notation |

## 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.

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:

### Commutative and Associative Properties

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

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:

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.

## Summary of Algebraic Properties

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

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.

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**, **2**, **3**, 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!