Equality

The term equality, in mathematics, refers to the relationship between two quantities, or expressions, that says that they have the same value or represent the same object. The symbol used to denote this is the equals sign (=). For example:

2 + 2 = 4

(x - 2)² = x² - 4x + 4

In both equations above, both quantities and expressions on the left and right of the equals sign represent the same value. Equality is used throughout mathematics, and is a basis for solving equations in algebra.

Basic properties of equality

There are many properties of equality in different areas of mathematics. Below are some of the basic ones.

Reflexive property

The reflexive property of equality states that for any quantity, such as a, a is equal to itself:

a = a

12 = 12

-72 = -72

Symmetric property

The symmetric property of equality states that for any quantities, a and b, if a is equal to b, then b must be equal to a.

if a = b, then b = a

If a = 2, then 2 = a

Transitive property

The transitive property of equality states that for any quantities, a, b, and c, if a is equal to b, and b is equal to c, then a must be equal to c.

If a = b and b = c, then a = c

If a = b, a = 5, and b = c, then c = 5

Substitution property

The substitution property of equality states that for any quantities or expressions, if a = b, then substituting a or b for the other in a given expression will yield the same result. There are many examples of this, but we can use basic arithmetic operations to demonstrate this property. Assume that a, b, and c are real numbers for the examples below.

Substitution with addition:

If a = b, then a + c = b + c

Substitution with subtraction:

If a = b, then a - c = b - c

Substitution with multiplication:

If a = b, then ac = bc

Substitution with division:

Assume that c is not equal to 0. If c is equal to 0 then the following expressions are undefined.

If a = b, then