Real Numbers and Properties Natural Numbers n Known

Real Numbers and Properties

Natural Numbers…… n Known as “Counting” Numbers n Example: 1, 2, 3, 4, 5, …….

Whole Numbers…… n n You add the number 0 to the natural numbers. Example: 0, 1, 2, 3, 4, 5…….

Integers…… n n Integers are made up of whole numbers and their opposites. Example: …-4, -3, -2, -1, 0, 1, 2, 3, 4….

Rational Numbers…… n The set of rational numbers is made up of all of the following a. Natural Numbers b. Whole Numbers c. Integers d. Plus every repeating and terminating decimal.

Examples of Rational Numbers…… n A. ½ = 0. 5 (Terminating Decimal) n B. 1. 2323 (Repeating Decimal) n C. 0. 256256256 (Repeating Decimal) D. 2. 735 (Terminating Decimal)

Irrational Numbers…. n n Consists of Non-Terminating and Non. Repeating Decimals. Example: 0. 9482137507264

Real Numbers (ℝ) Rational Numbers (ℚ) Integers (ℤ) Whole Numbers Natural Numbers (ℕ) 1, 2, 3, … 0, 1, 2, 3, … …-3, -2, -1, 0, 1, 2, 3, … Decimal form either terminates or repeats All rational and irrational numbers Irrational Numbers Decimal form is non-terminating and non-repeating

The Number Line…… n n A number line consists of positive numbers (right of 0) and negative numbers (left of 0). A real life example of a number line is a temperature thermometer.

For example…. . n n -5 would represent 5 degrees below zero. +4 would represent 4 degrees above zero.

Make the Comparison…… n n 7 degrees below 0 is (warmer/colder) than 4 degrees above 0. 7 degrees below 0 is a (lower/higher) temperature than 4 degrees above 0. n Colder n Lower

Coordinates on a Graph…. n Find the best estimate of the point. a. -2 b. 2 c. -1. 75 d. -1. 5 Answer: -1. 75

Sets and Subsets…… n n n A set is a group of numbers. Example: Set A = {1, 2, 3, 4, 5} A subset is a group of numbers in which every member is in another set. Example: Set B = {1, 2, 3} So, B is a subset of A.

Which of the following would represent a subset of integers? n n States Sales Tax Rate Amount of Gas in a Car Number of Students in Class A Dinner Receipt n n Strategy: Eliminate those that are NOT integers. n 7. 5% - NO 6. 5 Gallons – NO 12 – YES $10. 31 - NO

You Try…Which of the following would represent a subset of integers? n n Costs of a TV # of miles on the odometer of a car A person’s weight Number of residents in South Carolina n No n Yes

Inequalities…. . n n We use inequalities to compare numbers. The following are inequalities:

Examples……. n n “ 4 is less than 7” “ 9 is greater than or equal to 5” -

You Try……Insert the appropriate inequality sign. 1. -5 -2 1. < 2. -7 2 2. < 3. 4 -12 3. >

Least to Greatest…… n n This means to arrange numbers in the order from the smallest to the largest. HINT: If there are fractions it might be easier to convert to decimals first.

Which Number is Smaller?

Which Number is Larger? . . . .

You Try…Compare n -0. 67 > -0. 68 n -0. 8 > -0. 86

Which Set is Ordered from Least to Greatest? 1. {-3/2, -3, 0, 2/3} 2. {-3, -3/2, 0, 2/3} 3. {0, 2/3, -3/2, -3} 3. {0, 2/3, -3/2, -3} 4. {0, -3/2, -3, 2/3} 4. {0, -3/2, -3, 2/3

n What kinds of numbers are used to represent numbers below zero? n Answer: n NEGATIVE Numbers

n Make -8 -4 a true statement. n Answer: n <

Quick Review -400 -200 0 200 1) Coordinate of A: a) -250 b) -300 c) -325 d) -500 2) Coordinate of B: a) -210 b) -350 c) -100 d) -50 3) Coordinate of C: a) 350 b) 425 c) 325 d) 275 400

Quick Review 4) Use , : -8 5) Which is smaller? 5 or 6) Write from smallest to largest: -3, -3. 8, -5, 5. 6, -5. 6

Number Properties

Commutative Property. Changes Order n For Addition n For Multiplication A+B = B+A Ex. 2+3 = 5 3+2 = 5 2+3=3+2 AB = BA Ex. 4(8) = 32 8(4) = 32 4(8) = 8(4) THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!

Associative Property. Changes Grouping n For Addition A + (B + C) = (A + B) + C Ex. 5 + (2 + 4) =5+6 = 11 (5 + 2) + 4 =7+4 = 11 5 + (2 + 4) = (5 + 2) + 4 n For Multiplication A(BC) = (AB)C Ex. 2 (3 5) = 2(15) = 30 (2 3) 5 = (6)5 = 30 2 x (3 x 5) = (2 x 3) x 5 This is not true for subtraction or division!

Which Property? 1) 2) 3) 4) 5) 6) 3 x 4 = 4 3 x 6 y + (7 + 3 z) = (6 y +7) +3 z (5 x + 7) + 8 y = 5 x + (7 + 8 y) (3 x)(2 x + 5) = (2 x + 5)(3 x) 10 x + 4 y = 4 y + 10 x (2 x 5)(10 y) = (2 x)(5 10 y)

Distributive Property A (B + C) = AB + AC A (B – C) = AB – AC (B + C) A = BA + CA (B – C) A = BA – CA Ex. -3 (4 – 2 x) Strategy: Think -3 (4 – 2 x) means -3 (4 + -2 x) = -3(4) + (-3)(-2 x) = -12 + 6 x TRY THESE: A) 4 (6 +2 a) B) -7 (-3 m – 5)

Which Property? 1) -3 x(y + 2) + 4 y = -3 x(y) – 3 x(2) + 4 y 2) -3 y + 4 x(y + 2) = -3 y + 4 x(2) 3) 6 x + (3 y + 1) = (3 y +1) + 6 x

What is an example of the commutative prop. of addition? A) B) C) D) 3 3 + + 5 m 5 m = = 3 + (1 + 4)m 5 m + 3 5 + 3 m 3 m + 5 A) B) C) D) 7 + 4 m = (7 +4)m (5 + 2) + 4 m = 7 + 4 m = 4 + 7 m 7 + 4 m = 4 m + 7

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