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

2. Slide 5 - 2 Copyright © 2009 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

3. Slide 5 - 3 Copyright © 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

4. Slide 5 - 4 Copyright © 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?

5. Slide 5 - 5 Copyright © 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

6. Slide 5 - 6 Copyright © 2009 Pearson Education, Inc. Evaluate [(–11) + 4] + (–8) a.–23 b.–1 c.15 d. –15

7. Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. Evaluate [(–11) + 4] + (–8) a.–23 b.–1 c.15 d. –15

8. Slide 5 - 8 Copyright © 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:

9. Slide 5 - 9 Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following.

10. Slide 5 - 10 Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a)

11. Slide 5 - 11 Copyright © 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

12. Slide 5 - 12 Copyright © 2009 Pearson Education, Inc. Write as a terminating or repeating decimal number. a. b. c. d.

13. Slide 5 - 13 Copyright © 2009 Pearson Education, Inc. Write as a terminating or repeating decimal number. a. b. c. d.

14. Slide 5 - 14 Copyright © 2009 Pearson Education, Inc. Write as a terminating or repeating decimal number. Writeas a decimal. Divide on your calculator: top divided by bottom 2 / 11 = 0.1818181818….. Use a repeat bar to indicate the part that repeats:

15. Slide 5 - 15 Copyright © 2009 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

16. Slide 5 - 16 Copyright © 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:

17. Slide 5 - 17 Copyright © 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

18. Slide 5 - 18 Copyright © 2009 Pearson Education, Inc. Product (or Multiplication) Rule for Radicals Simplify: (Hint: find the largest perfect square) a) b) Multiply:

19. Slide 5 - 19 Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers (Hint: treat the radical as if it were a “like term” in algebra) Simplify

20. Slide 5 - 20 Copyright © 2009 Pearson Education, Inc. Simplify a. b. c. d.

21. Slide 5 - 21 Copyright © 2009 Pearson Education, Inc. Simplify a. b. c. d.

22. Slide 5 - 22 Copyright © 2009 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

23. Slide 5 - 23 Copyright © 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

24. Slide 5 - 24 Copyright © 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 + 72 Example: -3 (x + y - 3) = -3x -3y + 9