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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
