REAL NUMBERS ALGEBRA REVIEW

Sets of numbers
Natural numbers:
Whole numbers:
Integers:
Rational numbers :
{x : x =a/b, where a and b are integers and b ≠ 0}

1. Any integer m can be written as m/1, thus, all integer are
rational numbers.

2. Rational numbers include repeating terminating decimals ,
for example:

Irrational numbers: includes any real numbers that is not a
rational number

Real numbers: The rational and irrational numbers together.

Operations on the set of real numbers

Addition
Multiplication
Properties of onents /adding-and- subtracting -problem.html">addition and multiplication of real numbers
1. Commutativity

2. Associativity

3. Distributivity

4. Identity
a) There exists a unique number 0, called additive identity
such that
a + 0 = 0 + a = a
b) There exists a unique number 1, called multiplicative
identity such that
a · 1 = 1 · a = a

5. Inverse properties
−a is the additive inverse of a
if b ≠ 0 then the multiplicative inverse of b is1/b

Multiplication by 0
a · 0 = 0 · a =
a) if a · b = 0 then either or
b) ifa/b= 0 and b ≠ 0 then
Note that a/0 is UNDEFINED.

Properties of Negatives
i) −(−a) =
ii) (−a) · b =
iii) (−a) · (−b) =
iv) (−1) · a =

Properties of Quotients
if and only if

Exponents

If a is a real number and n is a natural number then
aⁿ =

Examples 2³ =
(−3)⁴ =
−(3)⁴ =
(−2)³ =
−(2)³ =

Special cases
a⁰ =
a¹ =

0⁰ is UNDEFINED

Negative exponents
If a ≠ 0 is a real number and n is a natural number then
a^-n =
2^-1 =
3^-2 =

The Principal n^th root

Let n ≥ 1 be a natural number and let a be a real number.
1) if a = 0 then
2) if a > 0 then is the positive real number b such that
bⁿ = a
3) if a < 0 then
i) if n is odd then is the negative real number b such
that bⁿ = a.
ii) if n is even then is NOT a real number.

Rational exponents

Let m/n be a rational number with n ≥ 1. If a is a real number
such that   IS a real number then

Laws of Exponents

Let m, n be rational numbers and let a and b be real numbers
for which a^m, aⁿ, b^m, and bⁿ ARE REAL NUMBERS. Then
1. a^m · aⁿ =
2. (a^m)ⁿ =
3. (ab)^m =
4. (a/b)^m =

Note that

Simplify the expressions :

Rationalize the denominator

Special Products

(a − b)(a + b) = a² − b²
(a − b)(a² + ab + b²) = a³ − b³
(a + b)(a² − ab + b²) = a³ + b³

Rationalize the denominator: