## Properties of Real Numbers

Sets, Rings, and Groups of Numbers

For all real numbers A, B and for when C does NOT equal 0, the equations:

A = B and AC = BC are equivalent to one another.

Each side can be multiplied by the same NONZERO number without changing the solution set.

For all real numbers A, B and C, the equations:

A = B and A + C = B + C are equivalent to one another.

The same number may be added to each side of an equation without altering the solution set.

The Associative Property states that for any real numbers a, b and c:

a + (b + c) = (a + b) + c

a(bc) = (ab)c

With the Associative Property, parentheses amongst 3 terms/factors change, but the order of the terms stays the same.

The Commutative Property states:

For any real numbers a and b:

a + b = b + c

ab = ba

The Commutative Property is when the ORDER CHANGES but the result remains the same.

For any real number a,

a + 0 = 0 + a = a

a * 1 = 1 * a = a

Notes: An easy way to remember this is that the Identity Property leaves the IDENTITY of a real number unchanged. Adding 0 to any number or multiplying any number times a does not change the value of the number in any way.

EXAMPLES:

12m + m

= 12m + 1m

= (12 + 1)m

= 13m

For any real number \( a \neq 0 :\)

\( a + (-a) = 0 \ \) and,

\( a \left( \frac{1}{a} \right) = 1.\)

That is, any number times its reciprocal equals 1.

The reciprocal of zero is not defined.

Very simply stated, the Distributive Property is easy to understand when seen like this:

a(b+c) = ab + ac

OR

(b+c)a = ba + ca

In more complex examples, the Distributive Property can be applied to equations like this one, though the same principles still apply:

EXAMPLES:

**3(x + y)**

= 3x + 3y

**-2(5 + k)**

= -2(5) + (-2)(k)

= -10 - 2k

**4x + 8x**

= (4 + 8)x

= 12x