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Real Number Properties and Basic Word Problems
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Natural Numbers…… Known as “Counting” Numbers
Example: 1, 2, 3, 4, 5,…….
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Whole Numbers…… You add the number 0 to the natural numbers.
Example: 0, 1, 2, 3, 4, 5…….
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Integers…… Integers are made up of whole numbers and their opposites.
Example: …-4,-3,-2,-1,0,1,2,3,4….
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Rational Numbers…… 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.
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Examples of Rational Numbers……
A. ½ = 0.5 (Terminating Decimal) B (Repeating Decimal) C (Repeating Decimal) D (Terminating Decimal)
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Irrational Numbers…. Consists of Non-Terminating and Non-Repeating Decimals. Example:
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Real Numbers (ℝ) Rational Numbers (ℚ) Irrational Numbers Integers (ℤ)
Decimal form is non-terminating and non-repeating 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
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The Number Line…… 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.
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For example….. -5 would represent 5 degrees below zero.
+4 would represent 4 degrees above zero.
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Make the Comparison…… 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. Colder Lower
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Coordinates on a Graph….
Find the best estimate of the point. a b c d Answer:
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Sets and Subsets…… 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.
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Which of the following would represent a subset of integers?
States Sales Tax Rate Amount of Gas in a Car Number of Students in Class A Dinner Receipt Strategy: Eliminate those that are NOT integers. 7.5% - NO 6.5 Gallons – NO 12 – YES $ NO
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You Try…Which of the following would represent a subset of integers?
Costs of a TV # of miles on the odometer of a car A person’s weight Number of residents in South Carolina No Yes
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Inequalities….. We use inequalities to compare numbers.
The following are inequalities:
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Examples……. “4 is less than 7” - “9 is greater than or equal to 5” -
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You Try……Insert the appropriate inequality sign.
< >
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Least to Greatest…… 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.
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Which Number is Smaller?
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Which Number is Larger?
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You Try…Compare > -0.8 >
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Which Set is Ordered from Least to Greatest?
{-3/2, -3, 0, 2/3} {-3, -3/2, 0, 2/3} {0, 2/3, -3/2, -3} {0, -3/2, -3, 2/3} {-3/2, -3, 0, 2/3} {-3, -3/2, 0, 2/3} 3. {0, 2/3, -3/2, -3} 4. {0, -3/2, -3, 2/3
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What kinds of numbers are used to represent numbers below zero?
Answer: NEGATIVE Numbers
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Make -8 -4 a true statement.
Answer: <
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Number Properties
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Quick Review -400 -200 200 400 Coordinate of A:
Coordinate of A: a) b) c) d) -500 2) Coordinate of B: a) b) c) d) -50 3) Coordinate of C: a) b) c) d) 275
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Quick Review 4) Use , : -8 5 5) Which is smaller? or
6) Write from smallest to largest: -3, -3.8, -5, 5.6, -5.6
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Commutative Property- Changes Order
For Addition A+B = B+A Ex = 5 3+2 = 5 2+3=3+2 For Multiplication AB = BA Ex. 4(8) = 32 8(4) = 32 4(8) = 8(4) THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!
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Associative Property- Changes Grouping
For Multiplication A(BC) = (AB)C Ex. 2 (3 5) (2 3) 5 = 2(15) = (6)5 = = 30 2 x (3 x 5) = (2 x 3) x 5 For Addition A + (B + C) = (A + B) + C Ex. 5 + (2 + 4) (5 + 2) + 4 = = 7 + 4 = = 11 5 + (2 + 4) = (5 + 2) + 4 This is not true for subtraction or division!
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Which Property? 3x 4 = 4 3x 6y + (7 + 3z) = (6y +7) +3z
(5x + 7) + 8y = 5x + (7 + 8y) (3x)(2x + 5) = (2x + 5)(3x) 10x + 4y = 4y + 10x (2x 5)(10y) = (2x)(5 10y)
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Distributive Property
A (B + C) = AB + AC (B + C) A = BA + CA A (B – C) = AB – AC (B – C) A = BA – CA Ex. -3 (4 – 2x) Strategy: Think -3 (4 – 2x) means -3 (4 + -2x) = -3(4) + (-3)(-2x) = x TRY THESE: A) 4 (6 +2a) B) -7 (-3m – 5)
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Which Property? -3x(y + 2) + 4y = -3x(y) – 3x(2) + 4y
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What is an example of the commutative prop. of addition?
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Word Problems
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Words that tell you to add….
Plus Increased By Sum More Than
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Words that tell you to subtract…
Minus Decreased By Difference Less Than
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Multiply and Divide Words…..
Product times Divide – Quotient Ratio of
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Homework Worksheet
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