# What are all the properties of multiplication and addition? As in Distributive property of addition? So on so forth. Thanks!

Oct 21, 2017

Here's a list...

#### Explanation:

Here are the properties of addition and multiplication of rational or real numbers...

• Addition has an identity $0$:

$a + 0 = 0 + a = a$ for any number $a$

• Any number $a$ has an inverse under addition (called an opposite) $- a$ such that:

$a + \left(- a\right) = \left(- a\right) + a = 0$

$a + \left(b + c\right) = \left(a + b\right) + c$ for any numbers $a , b , c$

$a + b = b + a$ for any numbers $a , b$

• Multiplication has an identity $1$:

$a \cdot 1 = 1 \cdot a = a$ for any number $a$

• Any non-zero number has an inverse under multiplication (called a reciprocal) $\frac{1}{a}$ such that:

$a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1$

• Multiplication is associative:

$a \cdot \left(b \cdot c\right) = \left(a \cdot b\right) \cdot c$ for any numbers $a , b , c$

• Multiplication is commutative:

$a \cdot b = b \cdot a$ for any numbers $a , b$

• Multiplication is left and right distributive over addition:

$a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot c\right) \text{ }$ left distributive

$\left(a + b\right) \cdot c = \left(a \cdot c\right) + \left(b \cdot c\right) \text{ }$ right distributive

We should also note that the set of rational numbers (or the set of real numbers) is closed under addition and multiplication. That is, if you add two rational numbers or multiply two rational numbers then you get another rational number. Similarly for real numbers.

Remarks

If you have a set of objects (numbers), with addition and multiplication satisfying all of the above properties, then it is called a field. The rational numbers are a field, as are the real numbers. The integers lack multiplicative inverses, so do not form a field. They are a type of ring. Interestingly, integer arithmetic modulo a prime number gives you a field.