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
Additive Inverse The additive inverse of a is -a
Double Negative -(-a) = a
Absolute Value Absolute value is distance from 0 on the number line.
Adding two numbers
with the same sign
Add their absolute values and
put the common sign in front.

Adding two numbers
with opposite signs
Subtract their absolute values.

Answer "+" if positive number
has larger absolute value.

Answer "-" if negative number
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:

Property Addition Multiplication
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

0 is the additive
identity element

a x 1 = a
1 x a = a

1 is the multiplicative
identity element

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

0 is the additive inverse
or opposite of a

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

Distributive
(of multiplication over addition)
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