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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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**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, …

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**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 Can’t be written as a fraction using only integers Decimal form neither terminates or repeats

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**1-1: Properties of Real Numbers**

Example: Which set of numbers best describes the values for each variable? The cost C in dollars of admission for n people C: n: The maximum speed s in meters per second on a roller coaster of height h in meters (use the formula: ) S: h: 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)

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**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

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**1-1: Properties of Real Numbers**

Ordering Real Numbers If a and b are real numbers, then either a = b, a < b, or 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

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**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

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**1-1: Properties of Real Numbers**

Let a, b, and c be real numbers Property Addition Multiplication Meaning Closure a + b is a real number ab is a real number Real numbers produce real numbers Commutative a + b = b + a ab = ba Order doesn’t matter Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Grouping doesn’t matter Identity a + 0 = a, 0 + a = a a • 1 = a, 1 • a = a +0 or x1 produces original number Inverse a + (-a) = 0 Reciprocals cancel out Distributive a(b + c) = ab + ac Multiply to everything on inside

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**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

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**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 |-10| = 10 -|-10| = -10

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**1-1: Properties of Real Numbers**

Assignment Page 8-9 2 – 54, even problems

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Algebra 2. Objectives 1. Know the classifications of numbers 2. Know where to find real numbers on the number line 3. Know the properties and operations.

Algebra 2. Objectives 1. Know the classifications of numbers 2. Know where to find real numbers on the number line 3. Know the properties and operations.

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