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 Multiplication does 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.