Real numbers have certain properties when it comes to operations, identities, and equalities.
When performing operations on real numbers five different properties can be used.
| Name | Property |
|---|---|
| Distributive Property | a⋅(a+c)=a⋅b+a⋅c |
| Commutative Property of Addition | a+b=b+a |
| Commutative Property of Multiplication | a⋅b=b⋅a |
| Associative Property of Addition | (a+b)+c=a+(b+c) |
| Associative Property of Multiplication | (a⋅b)⋅c=a⋅(b⋅c) |
These properties can be used when solving equations or simplifying expressions to easier find the correct solution.
Real numbers have two important identities (equations that always hold true).
| Additive Identity | a+0=0 |
| Multiplicative Identity | a⋅1=a |
The equalities for real numbers indicates that different operations can be performed on equations and still yield the same equation.
| Name | Property |
|---|---|
| Addition Property of Equality | If a=b, then a+c=b+c |
| Subtraction Property of Equality | If a=b, then a−c=b−c |
| Multiplication Property of Equality | If a=b, then a⋅c=b⋅c |
| Division Property of Equality | If a=b, then ca=cb |
Finally, there are two special equalities; Symmetric Property of Equality and Transitive Property of Equality.
Real numbers can be written in different ways. For example 0.75=43 and 2.5=221. The Symmetric Property of Equality then implies that the order does not matter. 0.75=432.5=221⇔43=0.75⇔2.5
Given three real numbers, a,b, and c, the transitive property of equality refers to:
Real numbers are said to be closed under addition and multiplication. That is, adding or multiplying two real numbers results in a real number.