The ideas of equality, expressions, and evaluation are vital concepts when working with equations.** **An

*equation*is a mathematical statement that tells you that two things have the same value — in other words, it’s a statement with an equal sign. The equation is one of the most important concepts in mathematics because it allows you to boil down a bunch of complicated information into a single number.

Mathematic equations come in lots of varieties, with the two most common being arithmetic and algebraic equations.

## Three properties of equality

The three properties of equality are called reflexivity, symmetry, and transitivity:

Reflexivity says that everything is equal to itself. For example,

1 = 1

23 = 23

1,000,007 = 1,000,007

Symmetry says that you can switch around the order in which things are equal. For example,

4 · 5 = 20, so 20 = 4 · 5

Transitivity says that if something is equal to two other things, then those two other things are equal to each other. For example,

3 + 1 = 4 and 4 = 2 · 2, so 3 + 1 = 2 · 2

Because equality has all three of these properties, mathematicians call equality an *equivalence relation*.

Here are a few examples of simple arithmetic equations:

2 + 2 = 4

3 · 4 = 12

20 ÷ 2 = 10

And here are a few examples of more-complicated arithmetic equations:

1,000 – 1 – 1 – 1 = 997

(1 · 1) + (2 · 2) = 5

## What is an expression?

An expression is any string of mathematical symbols that can be placed on one side of an equation. Here are a few examples of simple expressions:

2 + 2

–17 + (–1)

14 ÷ 7

And here are a few examples of more-complicated expressions:

(88 – 23) ÷ 13

100 + 2 – 3 · 17

## How to evaluate expressions

At the root of the word *evaluation* is the word *value*. In other words, when you evaluate something, you find its value. Evaluating an expression is also referred to as *simplifying*, *solving*, or *finding the value of *an expression. The words may change, but the idea is the same — boiling a string of numbers and math symbols down to a single number.

When you evaluate an arithmetic expression, you simplify it to a single numerical value — that is, you find the number that it’s equal to. For example, evaluate the following arithmetic expression:

7 · 5

How? Simplify it to a single number:

35

## How do you connect equality, expressions, and evaluation?

The Three E’s — equality, expressions, and evaluation — are all connected. Evaluation allows you to take an expression containing more than one number and reduce it down to a single number. Then, you can make an equation, using an equal sign to connect the expression and the number. For example, here’s an expression containing four numbers:

1 + 2 + 3 + 4

When you evaluate it, you reduce it down to a single number:

10

And now, you can make an equation by connecting the expression and the number with an equal sign:

1 + 2 + 3 + 4 = 10