1. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 1 Introduction to Algebraic Expressions

2. 1-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Fraction Notation Factors and Prime Factorizations Multiplication, Division, and Simplification Addition and Subtraction 1.3

3. 1-3 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factors and Prime Factorizations Natural Numbers can be thought of as the counting numbers: 1, 2, 3, 4, 5… (The dots indicated that the established pattern continues without ending.) To factor a number, we simply express it as a product of two or more numbers.

4. 1-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Write several factorizations of 18. Then list all the factors of 18. Solution The factors of 18 are: 1, 2, 3, 6, 9, and 18.

5. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Prime Number A prime number is a natural number that has exactly two different factors: the number itself and 1. The first several primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23.

6. 1-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Definitions If a natural number, other than 1, is not prime, we call it composite. Every composite number can be factored into a product of prime numbers. Such a factorization is called the prime factorization of that composite number.

7. 1-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find the prime factorization of 48. Solution The prime factorization of 48 is

8. 1-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Fraction Notation The top number is called the numerator and the bottom number is called the denominator.

9. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Multiplication of Fractions For any two fractions a/b and c/d, (The numerator of the product is the product of the two numerators. The denominator of the product is the product of the two denominators.)

10. 1-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Multiply: a)b) Solution a) b)

11. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Fraction Notation for 1 For any number a, except 0, (Any nonzero number divided by itself is 1.)

12. 1-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Reciprocals Two numbers whose product is 1 are reciprocals, or multiplicative inverses. The reciprocal of is because

13. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Division of Fractions To divide two fractions, multiply by the reciprocal of the divisor:

14. 1-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Divide: Solution

15. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Identity Property of 1 For any number a, a ● 1 = 1 ● a = a. (Multiplying a number by 1 gives the same number.) The number 1 is called the multiplicative identity.

16. 1-16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: Solution Factoring the numerator and the denominator using a common factor of 5. Rewriting as a product of two fractions Using the identity property of 1

17. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Addition and Subtraction of Fractions For any two fractions a/d and b/d,

18. 1-18 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Add and simplify: Solution Using 60 as the common denominator Perform the multiplication Adding fractions & simplifying

19. 1-19 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Perform the indicated operation and, if possible, simplify. Solution Removing a factor equal to 1

20. 1-20 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Perform the indicated operation and, if possible, simplify. Solution Removing a factor equal to 1