Sullivan Algebra and Trigonometry: Section R.1 Real Numbers

1. Sullivan Algebra and Trigonometry: Section R.1Real Numbers • Objectives of this Section • Classify Numbers • Evaluate Numerical Expressions • Work with Properties of Real Numbers

2. Describing Sets of Numbers The Roster Method The roster method is used to list the elements in a set. For example, we can describe the set of even digits as follows: E = {0, 2, 4, 6, 8}

3. Describing Sets of Numbers Set Builder Notation Set Builder notation is used to describe a set of numbers by defining a property that the numbers share. For example, we can describe the set of odd digits as follows: O = {x | x is an odd digit}

4. Subsets of the Real Numbers The Rational Numbers A rational number is a number that can be expressed as a quotient a/b . The integer a is called the numerator, and the integer b, which cannot be 0, is called the denominator. All rational numbers can be written as a decimal that either terminates or repeats. For example: 1/3 = 0.3333…

5. Subsets of the Rational Numbers Subsets of the Rational Numbers can be found by letting the denominator equal 1. • The Natural Numbers: {1, 2, 3, 4, …} • The Whole Numbers: {0, 1, 2, 3, 4, …} • The Integers: {… -3, -2, -1, 0, 1, 2, 3, …}

6. Subsets of the Real Numbers The Irrational Numbers Numbers in which the decimal neither terminates nor repeats are called irrational numbers. Examples of irrational numbers include: The set of all rational and irrational numbers form the set of real numbers.

7. Approximations In practice, the decimal representation of an irrational number is given as an approximation. Truncation: Drop all of the digits that follow the specified final digit in the decimal. Rounding: Identify the specified digit in the decimal. If the next digit is 5 or more, add one to the final digit. If the next digit is 4 or less, leave the final digit as is. Then, truncate following the final digit.

8. Order of Operations 1. Begin with the innermost parenthesis and work outward. Remember that in dividing two expressions the numerator and denominator are treated as if they were in parenthesis. For example:

9. Order of Operations 2. Perform multiplication and divisions, working left to right. For example: 3. Perform additions and subtractions, working from left to right.

10. Properties of Real Numbers Commutative Properties Addition: a + b = b + a Multiplication: ab = ba Associative Properties Addition: a + (b + c) = (a + b) + c Multiplication: a(bc) = (ab)c

11. Properties of Real Numbers Distributive Property a(b + c) = ab + ac (a+b)c = ac+bc Example 4(x + 2) = 4(x) + 4(2) = 4x + 8

12. Properties of Real Numbers Identity Properties 0 + a = a + 0 = aa(1) = 1(a) = a Additive Inverse Property a + (- a) = - a + a = 0 Multiplicative Inverse Property

13. Properties of Real Numbers Multiplication by Zero a (0) = 0 Division Properties

14. Properties of Real Numbers Rules of Signs a(-b) = -(ab) = (-a)b (-a)(-b) = ab - ( -a) = a

15. Properties of Real Numbers Cancellation Properties Zero – Product Property If ab= 0, then a= 0 or b= 0 or, both.

16. Properties of Real Numbers Arithmetic of Quotients

17. Arithmetic of Quotients: Example 1

18. Arithmetic of Quotients: Example 2