Recall that the numbers that we use regularly for counting and measuring are called the **real numbers**. They include zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in between.

The real numbers are so familiar to us that we hardly notice that they have special properties that involve the addition and multiplication operations.

For any real numbers a,\,b,\, and c, the following properties are always true.

Addition | Multiplication | |
---|---|---|

Commutative property | \ a+b = b+a | \ a \cdot b = b \cdot a |

Associative property | \ \left(a + b\right) + c = a + \left(b + c\right) | \ a \cdot \left(b \cdot c\right) = \left(a \cdot b\right) \cdot c |

Inverse property | \ a + \left(-a\right) = 0 | \ a \cdot \dfrac{1}{a} = 1 |

Identity property | \ a + 0 = a | \ a \cdot 1 = a |

Distributive property | a\left(b + c\right) = a \cdot b + a \cdot c |

Notice the distributive property only applies to distributing the operation of multiplication. If the positions of the addition and multiplication signs are swapped so that we have a + \left(b \cdot c\right), a similar distributive rule is *not* true. This is one situation in which we must observe the correct order of operations when simplifying expressions.

Verify the distributive property a\left(b + c\right) = a \cdot b + a \cdot c for the values a = 3,\, b = 13,\, and c = -1.

Worked Solution

Using the properties of real numbers, rewrite 3\left(x+4\right) = 9 in another way.

Worked Solution

If 5 \cdot \left(\dfrac{1}{x}\right) = 1, use the properties of real numbers to solve for x.

Worked Solution

Idea summary

The properties of real numbers can be used to simplify expressions and solve equations.

Addition | Multiplication | |
---|---|---|

Commutative property | \ a+b = b+a | \ a \cdot b = b \cdot a |

Associative property | \ \left(a + b\right) + c = a + \left(b + c\right) | \ a \cdot \left(b \cdot c\right) = \left(a \cdot b\right) \cdot c |

Inverse property | \ a + \left(-a\right) = 0 | \ a \cdot \dfrac{1}{a} = 1 |

Identity property | \ a + 0 = a | \ a \cdot 1 = a |

Distributive property | a\left(b + c\right) = a \cdot b + a \cdot c |