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Chapter 1: Tools of Algebra 1-1: Properties of Real Numbers Essential Question: What are the subsets of the real numbers? Give an example of each.

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Presentation on theme: "Chapter 1: Tools of Algebra 1-1: Properties of Real Numbers Essential Question: What are the subsets of the real numbers? Give an example of each."— Presentation transcript:

1 Chapter 1: Tools of Algebra 1-1: Properties of Real Numbers Essential Question: What are the subsets of the real numbers? Give an example of each.

2 1-1: Properties of Real Numbers  Natural Numbers   Whole Numbers   Integers  1, 2, 3, 4, … 0, 1, 2, 3, 4, … …-3, -2, -1, 0, 1, 2, 3, 4, …

3 1-1: Properties of Real Numbers  All real numbers are either rational or irrational  Rational Numbers  Examples:  Can be written as a fraction using integers (Denominator can’t be 0)  Can also be written as either a terminating or repeating decimal  Irrational Numbers  Examples:  Can’t be written as a fraction using only integers  Decimal form neither terminates or repeats

4 1-1: Properties of Real Numbers  Example: Which set of numbers best describes the values for each variable? 1. The cost C in dollars of admission for n people  C:  n: 2. The maximum speed s in meters per second on a roller coaster of height h in meters (use the formula: )  S:  h: 3. The park’s profit (or loss) P in dollars for each week w of the year  P:  w: Cost is a terminating decimal (like $24.95), so it’s a rational number. Since we can talk about 0 people, and never fractions of people, then n is going to be a whole number. Speed is calculated using a square root, so speed will be an irrational number, unless the square root of h is a rational number Height is measured in rational numbers. Profit, as with anything involving money, is a rational number. The week will be a natural number (1 – 52)

5 1-1: Properties of Real Numbers  Real numbers can be graphed as points on a number line  Example: Graph the numbers   Graph the numbers Use a calculator to find

6 1-1: Properties of Real Numbers  Ordering Real Numbers  If a and b are real numbers, then either a = b, a b  There are a number of ways to prove that a < b  Compare a and b on a number line  Determine a positive number that can be added to a to get b  b – a is a positive number  Example: Compare   Because -0.1 – (-0.5) is positive, -0.1 must be greater, so 

7 1-1: Properties of Real Numbers  Finding Inverses  Opposite (additive inverse)  Flip the sign of the number  “Additive inverse” because a + -a = 0  Reciprocal (multiplicative inverse)  Convert the number to an improper fraction and flip the fraction  “Multiplicative inverse” because  Example  Find the opposite and reciprocal of -3.2  Opposite:  Reciprocal: -(-3.2) = 3.2

8 1-1: Properties of Real Numbers  Let a, b, and c be real numbers PropertyAdditionMultiplicationMeaning Closurea + b is a real numberab is a real numberReal numbers produce real numbers Commutativea + b = b + aab = baOrder doesn’t matter Associative(a + b) + c = a + (b + c)(ab)c = a(bc)Grouping doesn’t matter Identitya + 0 = a, 0 + a = aa 1 = a, 1 a = a+0 or x1 produces original number Inversea + (-a) = 0Reciprocals cancel out Distributivea(b + c) = ab + acMultiply to everything on inside

9 1-1: Properties of Real Numbers  Example  Which property is illustrated?  6 + (-6) = 0   (-4 1) – 2 = -4 – 2  Inverse property of addition Identity property of multiplication

10 1-1: Properties of Real Numbers  Finding Absolute Values  The absolute value of a number is it’s distance from 0 on a number line  Distance is always positive  Find |-4|, |0|, |5 (-2)|, and -|5 (-2)|  |-4| =  |0| =  |5 (-2)| =  -|5 (-2)| =  With absolute value signs, treat them like parenthesis and simplify everything inside the absolute value signs first. 4 0 |-10| = 10 -|-10| = -10

11 1-1: Properties of Real Numbers  Assignment  Page 8-9  2 – 54, even problems


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