Types of Numbers and Algebraic Properties

This section covers:

Types of Numbers

Before we get too deep into algebra, we need to talk about the types of numbers there are out there.  We saw a few of these earlier, and you may not have seen all these types of numbers yet, but you will have to learn them in school. The letters in parentheses indicate how they are abbreviated sometimes.  Sorry, it’s not the most exciting stuff to learn….

The types of numbers we’ll talk about include Whole Numbers, Counting Numbers or Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Imaginary Numbers, and Complex Numbers:

Natural Whole Integers Rationals Irrationals Real Imaginary Complex Numbers

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.

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

There are what we call the Algebraic Properties of Equality, since they deal with two sides of an equal sign:

Properties of Equality

Commutative and Associative Properties

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

Associative Commutative Property

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”:

Distributive Property

There is one other property that is used a lot in algebra; this one is a little different.

Let’s say you are trying to take a collection for your sweet algebra teacher for an end of the year gift.  You are collecting $10 from 10 girls and 8 boys.  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 distributethrough (or “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 is another example, and one using a variable:

Distributive Property

Summary of Algebraic Properties

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

Algebraic Properties Chart

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.


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 {1, 2, 3}).  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 {1, 2, 3} and {3, 4, 5} would be {1, 2, 3, 4, 5}, since you include everything in both sets, but don’t repeat numbers.  You write union as , so {1, 2, 3}  {3, 4, 5} = {1, 2, 3, 4, 5}.

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

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 are more useful and will be used by your teacher.  These include set builder notation, inequality notation, and interval notation, as shown with examples:

Math Grammar

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” () is used, we use “[“, and if “greater than or equal to” is used (), we use “]”.  So just remember that we used brackets (also called hard brackets) if we are including the endpoint (the actual number at the end), and we use parentheses (also called soft brackets) if we are not including the endpoint.

Also note that since we never actually get to  or  , we only use soft brackets (parentheses) with them.

Remember that  means all real numbers (everything on the number line), and  means natural numbers (1, 2, 3, and so on).  Also remember that Ø means “no answer” or “no solution”; this happens sometimes in algebra.

We’re going over this now, since we’ll be talking about inequalities soon, and it will get a little more complicated on how to write our answers.  Don’t worry too much if you’re overwhelmed with all this (it’s like learning a new language!); you’ll probably give your answers in simple inequality notation or more likely interval notation.

Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Solving Algebraic Equations – you are ready!

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