Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Topics An introduction to number theory Prime numbers Integers, rational numbers, irrational numbers,

Slide 5 - 1Copyright © 2009 Pearson Education, Inc.

Topics

• An introduction to number theory

• Prime numbers

• Integers, rational numbers, irrational numbers, and real numbers

• Properties of real numbers

Slide 5 - 2Copyright © 2009 Pearson Education, Inc.

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers

Slide 5 - 3Copyright © 2009 Pearson Education, Inc.

Prime Numbers

Casual definition: A prime number is only divisible by itself and 1

More formal definition: A Prime number has exactly two factors

A composite number has more than 2 factors

Is 13 prime or composite? Why? Is 15 prime or composite? Why? A math fact: 1 is neither prime nor composite

Slide 5 - 4Copyright © 2009 Pearson Education, Inc.

Prime Factorization

Breaking a number down into all the prime factors that go into it.

Example: 30 = 2 * 3 * 5

What is the prime factorization of 40?

Slide 5 - 5Copyright © 2009 Pearson Education, Inc.

Addition and Subtraction of Integers

Evaluate:

a) 7 + 3 = 10

b) –7 + (-3) = –10

c) 7 + (-3) = +4 = 4

d) -7 + 3 = -4

To subtract rewrite as addition a – b = a + (-b)

Evaluate:

a) -7 - 3 = –7 + (–3) = –10

b) -7 – (-3) = –7 + 3 = –4

Slide 5 - 6Copyright © 2009 Pearson Education, Inc.

Evaluate [(–11) + 4] + (–8)

a. –23

b. –1

c. 15

d. –15

Slide 5 - 7Copyright © 2009 Pearson Education, Inc.

Evaluate [(–11) + 4] + (–8)

a. –23

b. –1

c. 15

d. –15

Slide 5 - 8Copyright © 2009 Pearson Education, Inc.

The Rational Numbers (i.e. Fractions)

The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q not equal to 0.

The following are examples of rational numbers:

1

3,

3

4,

7

8, 1

2

3, 2, 0,

15

7

Slide 5 - 9Copyright © 2009 Pearson Education, Inc.

Example: Multiplying Fractions

Evaluate the following.

2

3

7

16

2

3

7

16

27316

14

48

7

24

Slide 5 - 10Copyright © 2009 Pearson Education, Inc.

Example: Dividing Fractions

Evaluate the following.

a)

2

3

6

7

2

3

6

7

2

37

6

2736

14

18

7

9

Slide 5 - 11Copyright © 2009 Pearson Education, Inc.

Terminating or Repeating Decimal Numbers

Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number.

Examples of terminating decimal numbers are 0.7, 2.85, 0.000045

Examples of repeating decimal numbers 0.44444… which may be written 0.4,

and 0.2323232323... which may be written 0.23.

Slide 5 - 12Copyright © 2009 Pearson Education, Inc.

Write as a terminating or repeating decimal

number.

a.

b.

c.

d.

1

16

0.0625

1.666

0.1666

0.125

Slide 5 - 13Copyright © 2009 Pearson Education, Inc.

Write as a terminating or repeating decimal

number.

a.

b.

c.

d.

1

16

0.0625

1.666

0.1666

0.125

Slide 5 - 14Copyright © 2009 Pearson Education, Inc.

Write as a terminating or repeating decimal number.

Write as a decimal. Divide on your calculator: top divided by bottom

2 / 11 = 0.1818181818….. Use a repeat bar to indicate the part that

repeats:

2

11

20.18

11

Slide 5 - 15Copyright © 2009 Pearson Education, Inc.

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers

Slide 5 - 16Copyright © 2009 Pearson Education, Inc.

Irrational Numbers

An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number.

Examples of irrational numbers:

5.12639573...

6.1011011101111...

0.525225222...

Slide 5 - 17Copyright © 2009 Pearson Education, Inc.

are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

Any natural number that is not a perfect square is irrational.

A perfect square is a number that has a rational number as its square root.

Do you recall the first few perfect squares? 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 etc

Radicals

2, 17, 53

Slide 5 - 18Copyright © 2009 Pearson Education, Inc.

Product (or Multiplication) Rule for Radicals

Simplify: (Hint: find the largest perfect square)a)

b) Multiply:

ab a b, a 0, b 0

40 410 4 10 2 10 2 10

40

6 54

6 54 654 324 18

Slide 5 - 19Copyright © 2009 Pearson Education, Inc.

Example: Adding or Subtracting Irrational Numbers (Hint: treat the radical as if it were a “like term” in algebra)

Simplify 8 5 125

8 5 125

8 5 25 5

8 5 5 5

(8 5) 5

3 5

Slide 5 - 20Copyright © 2009 Pearson Education, Inc.

Simplify

a.

b.

c.

d. 390

5 6

30

150

270 120.

Slide 5 - 21Copyright © 2009 Pearson Education, Inc.

Simplify

a.

b.

c.

d. 390

5 6

30

150

270 120.

Slide 5 - 22Copyright © 2009 Pearson Education, Inc.

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers

Slide 5 - 23Copyright © 2009 Pearson Education, Inc.

Properties of the Real Number System

Closure Commutative (multiplication and addition only)

a + b = b + a

and

(a)(b) = (b)(a) Associative (multiplication and addition only)

a + (b + c) = (a + b) + c

and

a (bc) = (ab) c

Slide 5 - 24Copyright © 2009 Pearson Education, Inc.

Properties of the Real Number System

Distributive property of multiplication over addition a • (b + c) = a • b + a • c

for any real numbers a, b, and c.

Example: 6 • (r + 12) = 6 • r + 6 • 12

= 6r + 72Example: -3 • (x + y - 3) = -3x -3y + 9