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.

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, …

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

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)

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

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

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

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

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

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

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