Chapter 1: Preliminary Information

1. Chapter 1:Preliminary Information Section 1-1: Sets of Numbers

2. Objectives • Given the name of a set of numbers, provide an example. • Given an example, name the sets to which the number belongs.

3. Two main sets of numbers • Real Numbers • Used for “real things” such as: • Measuring • Counting • Real numbers are those that can be plotted on a number line • Imaginary Numbers- square roots of negative numbers

4. The Real Numbers • Rational Numbers-can be expressed exactly as a ratio of two integers. This includes fractions, terminating and repeating decimals. • Integers- whole numbers and their opposites • Natural Numbers- positive integers/counting numbers • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Irrational Numbers-Irrational numbers are those that cannot be expressed exactly as a ratio of two numbers • Square roots, cube roots, etc. of integers • Transcendental numbers-numbers that cannot be expressed as roots of integers

6. Chapter 1:Preliminary Information Section 1-2: The Field Axioms

7. Objective • Given the name of an axiom that applies to addition or multiplication that shows you understand the meaning of the axiom.

8. The Field Axioms • Closure • Commutative Property • Associative Property • Distributive Property • Identity Elements • Inverses

9. Closure • {Real Numbers} is closed under addition and under multiplication. • That is, if x and y are real numbers then: • x + y is a unique real number • xy is a unique real number

10. More on Closure • Closure under addition means that when two numbers are chosen from a set, the sum of those two numbers is also part of that same set of numbers. • For example, consider the digits. • The digits include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. • If the digits are closed under addition, it means you can pick any two digits and their sum is also a digit. • Consider 8 + 9 • The sum is 17 • Since 17 is not part of the digits, the digits are not closed under addition.

11. More on Closure • Closure under multiplication means that when two numbers are chosen from a set, the product of those two numbers is also part of that same set of numbers. • For example, consider the negative numbers. • If we choose -6 and -4 we multiply them and get 24. • Since 24 is not a negative number, the negative numbers are not closed under multiplication.

12. The Commutative Property • Addition and Multiplication of real numbers are commutative operations. That means: • x + y = y + x • xy =yx • Are subtraction and division commutative?

13. Associative Property • Addition and Multiplication of real numbers are associative operations. That means: • (x + y) + z = x + (y + z) • (xy)z = x(yz)

14. Distributive Property • Multiplication distributes over addition. That is, if x, y and z are real numbers, then: x (y + z) = xy + xz • Multiplicationdoes not distribute over multiplication!

15. Identity Elements • The real numbers contain unique identity elements. • For addition, the identity element is 0. • For multiplication, the identity element is 1.

16. Inverses • The real numbers contain unique inverses • The additive inverse of any number x is the number – x. • The multiplicative inverse of any number x is 1/x, provided that x is not 0.