# Real Numbers, Exponents and Radicals

TED LAI

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

integers.

**Definition.** A real number is an irrational number if it is not a rational
number.

Some examples of irrational numbers are:
(the golden

ratio).

**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.

**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.

TED LAI

**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 |

R Q Z R ^{2} |
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) |

CHAPTER 1 NOTES

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

is

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.

**Radicals.**

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

b such that b^{2} = 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 2^{2} = 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