Math 101 Intermediate Algebra    Properties of and Operations with Real Numbers
Chapter 1, Section 3

For real numbers a and b:

Operation/Property Rule Note
Double Negative -(-a) = a
Absolute Value Absolute value is distance from 0 on the number line.
with the same sign

with opposite signs
Subtract their absolute values.

has larger absolute value.

has larger absolute value.

Sum of 2 positive numbers is positive.

Sum of 2 negative numbers is negative.

In a + b = c, a, b, and c are terms.

Subtraction a - b = a + (-b)
Multiplication Product of two numbers with
like signs is positive.

Product of two numbers with
opposite signs is negative.

Product is the result from multiplying.

In a x b = c, a and b are factors and c is the product.

Multiplicative
property of zero
a x 0 = 0
0 x a = 0

Division Quotient of two numbers with
like signs is positive.

Quotient of two numbers with
opposite signs is negative.

Quotient is the result from dividing.

For real numbers a, b, and c:

Commutative a + b = b + a ab = ba
Associative (a + b) + c = a + (b + c) (ab)c = a(bc)
Identity a + 0 = a
0 + a = 0

identity element

a x 1 = a
1 x a = a

1 is the multiplicative
identity element

Inverse a + (-a) = 0
(-a) + a = 0

or opposite of a  1 is the multiplicative inverse
or reciprocal of a, a nonzero

Distributive
a(b + c) = ab + ac
a(b + c + d + ... + n) = ab + ac + ad + ... + an

Some Useful Rules

For real numbers a, b, c, and d:
Note  Invert

and

Multiply  