**Distributive, Identity and Inverse Axioms**

**Distributive, Identity and Inverse Axioms**

An Axiom is a mathematical statement that is assumed to be true. There are five basic axioms of algebra. Axioms are generally statements made about real numbers. Sometimes they are called algebraic postulates.

The **Distributive Axioms** are that x(y + z) = xy + xz and (y + z)x = yx + zx. This is the only property which combines both addition and multiplication.

These equations are true for all numbers x, y and z.

The **Additive Identity Axiom** states that a number plus zero equals that number. A number plus zero equals that number. (The number keeps its identity!)

x + 0 = x or 0 + x = x

The **Multiplicative Identity Axiom** states that a number multiplied by 1 is that number. A number times 1 equals that number. (The number keeps its identity!)

x*1 = x or 1*x = x

The **Additive Inverse Axiom** states that the sum of a number and the Additive Inverse of that number is zero. Every real number has a unique additive inverse. Zero is its own additive inverse. The sum of a number and the Additive Inverse of that number is zero. Every real number has a unique additive inverse. Zero is its own additive inverse.

x + (-x) = 0

The **Multiplicative Inverse Axiom** states that the product of a real number and its multiplicative inverse is 1. Every real number has a unique multiplicative inverse. The reciprocal of a nonzero number is the multiplicative inverse of that number. Reciprocal of x is 1/x.

x * 1/x = 1

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