# Real Number Properties

Real Numbers have properties!

### Example: Multiplying by zero

When we multiply a real number by zero we get zero:

- 5 × 0 = 0
- −7 × 0 = 0
- 0 × 0.0001 = 0
- etc!

It is called the "Zero Product Property", and is listed below.

## Properties

Here are the main properties of the Real Numbers

Real Numbers are Commutative, Associative and Distributive:

Commutative | Example |
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a + b = b + a |
2 + 6 = 6 + 2 | |||

ab = ba |
4 × 2 = 2 × 4 | |||

Associative | Example |
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(a + b) + c = a + ( b + c ) |
(1 + 6) + 3 = 1 + (6 + 3) | |||

(ab)c = a(bc) |
(4 × 2) × 5 = 4 × (2 × 5) | |||

Distributive | Example |
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a × (b + c) = ab + ac |
3 × (6+2) = 3 × 6 + 3 × 2 | |||

(b+c) × a = ba + ca |
(6+2) × 3 = 6 × 3 + 2 × 3 |

Real Numbers are closed (the result is also a real number) under addition and multiplication:

Closure | Example |
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a+b is real |
2 + 3 = 5 is real | |||

a×b is real |
6 × 2 = 12 is real | |||

Adding zero leaves the real number unchanged, likewise for multiplying by 1:

Identity | Example |
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a + 0 = a |
6 + 0 = 6 | |||

a × 1 = a |
6 × 1 = 6 |

For addition the inverse of a real number is its negative, and for multiplication the inverse is its reciprocal:

Additive Inverse | Example |
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a + (−a ) = 0 |
6 + (−6) = 0 | |||

Multiplicative Inverse | Example |
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a × (1/a) = 1 |
6 × (1/6) = 1 | |||

But not for 0 as 1/0 is undefined |

Multiplying by zero gives zero (the Zero Product Property):

Zero Product | Example |
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If ab = 0 then a=0 or b=0, or both |
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a × 0 = 0 × a = 0 |
5 × 0 = 0 × 5 = 0 |

Multiplying two negatives make a positive, and multiplying a negative and a positive makes a negative:

Negation | Example |
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−1 × (−a) = −(−a) = a |
−1 × (−5) = −(−5) = 5 | |||

(−a)(−b) = ab |
(−3)(−6) = 3 × 6 = 18 | |||

(−a)(b) = (a)(−b) = −(ab) |
−3 × 6 = 3 × −6 = −18 |