1. Slide 5 - 1 Copyright © 2009 Pearson Education, Inc. Welcome to MM 150 Survey of Mathematics

2. Slide 5 - 2 Copyright © 2009 Pearson Education, Inc. Weekly Requirements Seminar – Live or Option 2 Quiz My Math Lab Homework – 20 Questions  View An Example  Help Me Solve This  Ask My Instructor Discussion Board Question  Initial Response + Replies to at least 2 others

3. Slide 5 - 3 Copyright © 2009 Pearson Education, Inc. 1.2 The Integers

4. Slide 5 - 4 Copyright © 2009 Pearson Education, Inc. Whole Numbers The set of whole numbers contains the set of natural numbers and the number 0. Whole numbers = {0,1,2,3,4,…}

5. Slide 5 - 5 Copyright © 2009 Pearson Education, Inc. Integers The set of integers consists of 0, the natural numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…} On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

6. Slide 5 - 6 Copyright © 2009 Pearson Education, Inc. Writing an Inequality Insert either > or < in the box between the paired numbers to make the statement correct. a)  3  1 b)  9  7  3 <  1  9 <  7 c) 0  4d) 6 8 0 >  4 6 < 8

7. Slide 5 - 7 Copyright © 2009 Pearson Education, Inc. Subtraction of Integers a – b = a + (  b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

8. Slide 5 - 8 Copyright © 2009 Pearson Education, Inc. Properties Multiplication Property of Zero Division For any a, b, and c where b  0, means that c b = a.

9. Slide 5 - 9 Copyright © 2009 Pearson Education, Inc. Rules for Multiplication The product of two numbers with like signs (positive  positive or negative  negative) is a positive number. The product of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

10. Slide 5 - 10 Copyright © 2009 Pearson Education, Inc. Examples Evaluate: a) (3)(  4)b) (  7)(  5) c) 8 7d) (  5)(8) Solution: a) (3)(  4) =  12b) (  7)(  5) = 35 c) 8 7 = 56d) (  5)(8) =  40

11. Slide 5 - 11 Copyright © 2009 Pearson Education, Inc. Rules for Division The quotient of two numbers with like signs (positive  positive or negative  negative) is a positive number. The quotient of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

12. Slide 5 - 12 Copyright © 2009 Pearson Education, Inc. Example Evaluate: a) b) c) d) Solution: a) b) c) d)

13. Slide 5 - 13 Copyright © 2009 Pearson Education, Inc. 1.3 The Rational Numbers

14. Slide 5 - 14 Copyright © 2009 Pearson Education, Inc. The Rational Numbers 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  0. The following are examples of rational numbers:

15. Slide 5 - 15 Copyright © 2009 Pearson Education, Inc. Fractions Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line.

16. Slide 5 - 16 Copyright © 2009 Pearson Education, Inc. Reducing Fractions In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor. Example: Reduce to its lowest terms. Solution:

17. Slide 5 - 17 Copyright © 2009 Pearson Education, Inc. Mixed Numbers A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. 3 ½ is read “three and one half” and means “3 + ½”.

18. Slide 5 - 18 Copyright © 2009 Pearson Education, Inc. Improper Fractions Rational numbers greater than 1 or less than –1 that are not integers may be written as mixed numbers, or as improper fractions. An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is.

19. Slide 5 - 19 Copyright © 2009 Pearson Education, Inc. Converting a Positive Mixed Number to an Improper Fraction Multiply the denominator of the fraction in the mixed number by the integer preceding it. Add the product obtained in step 1 to the numerator of the fraction in the mixed number. This sum is the numerator of the improper fraction we are seeking. The denominator of the improper fraction we are seeking is the same as the denominator of the fraction in the mixed number.

20. Slide 5 - 20 Copyright © 2009 Pearson Education, Inc. Example Convert to an improper fraction.

21. Slide 5 - 21 Copyright © 2009 Pearson Education, Inc. Converting a Positive Improper Fraction to a Mixed Number Divide the numerator by the denominator. Identify the quotient and the remainder. The quotient obtained in step 1 is the integer part of the mixed number. The remainder is the numerator of the fraction in the mixed number. The denominator in the fraction of the mixed number will be the same as the denominator in the original fraction.

22. Slide 5 - 22 Copyright © 2009 Pearson Education, Inc. Convert to a mixed number. The mixed number is Example

23. Slide 5 - 23 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

24. Slide 5 - 24 Copyright © 2009 Pearson Education, Inc. Division of Fractions Multiplication of Fractions

25. Slide 5 - 25 Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

26. Slide 5 - 26 Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

27. Slide 5 - 27 Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Fractions

28. Slide 5 - 28 Copyright © 2009 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

29. Slide 5 - 29 Copyright © 2009 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b  0, c  0, then

30. Slide 5 - 30 Copyright © 2009 Pearson Education, Inc. Example: Evaluate: Solution:

31. Slide 5 - 31 Copyright © 2009 Pearson Education, Inc. Commutative Property Addition a + b = b + a for any real numbers a and b. Multiplication a b = b a for any real numbers a and b.

32. Slide 5 - 32 Copyright © 2009 Pearson Education, Inc. Example 8 + 12 = 12 + 8 is a true statement. 5  9 = 9  5 is a true statement. Note: The commutative property does not hold true for subtraction or division.

33. Slide 5 - 33 Copyright © 2009 Pearson Education, Inc. Q & A ??