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Real Numbers, Exponents and Radicals


Section 1.2 Real Numbers

Definition. A real number r is a rational number if
where a and b are integers and b ≠ 0.

Some examples of rational numbers are: Also note
that each integer n is a rational number since , which is a fraction of two

Definition. A real number is an irrational number if it is not a rational number.
Some examples of irrational numbers are: (the golden

Properties of Real Numbers. Let a, b, and c be real numbers. Then they satisfy
the following properties:

Property 1 (The Commutative Property of Addition).

a + b = b + a

Property 2 (The Commutative Property of Multiplication).
a b = b a

Property 3 (The Associative Property of Addition).
(a + b) + c = a + (b + c)

Property 4 (The Associative Property of Multiplication).
(a b) c = a (b c)

Property 5 (The Distributive Property).
a (b + c) = a b + ac and (b + c) a = b a + ca

So, what’s the point of these trivial properties? Let’s illustrate their use by an

Example 1. Let x, y, z, and w be real numbers. Show (x+ y) (z +w) = x z +y z +
x w + y w.

(x + y) (z + w)  
= (x + y) z + (x + y) w by Property 5
= (x z + y z) + (x w + y w) by Property 5
= [(x z + y z) + x w] + y w by Property 1
= x z + y z + x w + y w  

Alternatively, one could verify the property numerically with explicit numbers,
but it would take forever to check all possible combinations of real numbers.


Properties of Fractions. Let a, b, c, d be real numbers with b≠0 and d ≠ 0.
Then the following properties hold:

Property 6.

Property 7.

Property 8.

Property 9.

If c ≠ 0, then

Sets and Intervals. Loosely speaking, a set is a collection of objects. The objects
contained in a set are called elements of the set. A set could contain a finite number
of elements or infinitely many elements.

Notation. Let S be a set. a∈ S denotes that a is an element of S. denotes
that a is not an element of S.

Definition. Let S and A be sets. If every element contained in A is also contained
in S, then A is a subset of S.

Sets frequently seen in class:

Notation Description
the set of all real numbers
the set of all rational numbers
the set of all integers
the set of all ordered pairs of real numbers

Notation. Let S and A be sets. A S denotes that A is a subset of S.

We say two sets A and B are equal if A B and B A.

Notation. denotes the set of all x in S such that . . .

Invervals are subsets of R (see table above), and they correspond to line segments
on the real number line. There are 9 types of intervals. Each one is represented in
interval notation below.

Let a and b be real numbers such that a≤ b.

Notation Set Description
(see pg 15 in textbook for graphs)


Section 1.3 Exponents and Radicals

Integer Exponents.

Notation. If a is a real number and n is a positive integer, then the nth power of a

Observe that for any real number a and any positive integers m and n,

by Property 4 in Section 1.2

Also one could show that   for any real number b, and
for any nonzero real number b.


Definition. Let a be a real number such that a ≥0. A square root of a is a number
b such that b2 = a.

It can be shown that every positive number has exactly two distinct square roots:
a positive one and a negative one. For example, 2 and -2 are both square roots of
4 since 22 = 2 × 2 = 4 and (−2)2 = (−2) × (−2) = 4.

Notation. Let a be a real number such that a  ≥0. If a is positive,
denotes the positive square root of a. If a = 0, then

Properties of Square Roots. Let a, b be real numbers with a≥ 0 and b≥ 0.
Then the following properties hold:

Property 10.

Property 11.If b ≠ 0, then