1. Describing Data with Sets of Numbers
Section 1.1
Describing Data with Sets of Numbers

2. Objectives Natural and Whole Numbers Integers and Rational Numbers
Real Numbers
Properties of Real Numbers

3. Types of Numbers
Natural Numbers: The set of counting numbers. N = {1, 2, 3, 4, 5, 6, …} Set braces, { }, are used to enclose the elements of a set. Whole Numbers: W = {0, 1, 2, 3, 4, 5, …} Integers: I = {…, 3, 2, 1, 0, 1, 2, 3, …} Rational Number: any number that can be expressed as the ratio of two integers; p/q, where q is not equal to 0 because we cannot divide by 0.

4. Example
Classify each number as one or more of the following: natural number, whole number, integer, or rational number. a. b. 8 c. 0 Solution a. natural number, whole number, integer, rational number b. integer, rational number c. whole number, integer, rational number

5. Real Numbers: Can be represented by decimal numbers
Real Numbers: Can be represented by decimal numbers. Every fraction has a decimal form, so real numbers include rational numbers. Irrational Numbers: A number that cannot be expressed by a fraction, or a decimal number that does not repeat or terminate. Examples:

6. Example
Classify each real number as one or more of the following: a natural number, an integer, a rational number, or an irrational number. a. 8 b. 1.6 c. Solution a. natural number, integer, rational number b. rational number c. irrational number

7. Example
A student obtains the following test scores: 91, 96, 89, and 84. a. Find the student’s average test score. b. Is this average a natural, rational, or a real number? Solution a. To find the average, we find the sum of the four test scores and divide by 4: b. rational and real numbers

8. Properties of Real Numbers--Summary
IDENTITY PROPERTIES
For any real number a,
a + 0 = 0 + a = a
and
a ·1 = 1 · a = a.

9. For any real numbers a and b, a + b = b + a and a ·b = b · a.
COMMUTATIVE PROPERTIES
For any real numbers a and b,
a + b = b + a
and
a ·b = b · a.

10. For any real numbers a, b, and c, (a + b) + c = a + (b + c) and
ASSOCIATIVE PROPERTIES
For any real numbers a, b, and c,
(a + b) + c = a + (b + c)
and
(a ·b) · c = a · (b · c).

11. Example
State the property of real numbers that justifies each statement. a. 5 · (2x) = (5 · 2)x b. (1 · 3) · 6 = 3 · 6 c. 7 + xy = xy + 7
Associative property for multiplication
Identity property of 1.
Commutative property for addition

12. For any real numbers a, b, and c, a(b + c) = ab + ac and
DISTRIBUTIVE PROPERTIES
For any real numbers a, b, and c,
a(b + c) = ab + ac
and
a(b  c) = ab  ac.

13. Example
Apply a distributive property to each expression. a. 5(2 + y) b. 8 – (2 + w) c. 5x – 2x d. 3y + 4y – y Solution a. b. c. d.