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August 17, 2012 Section P.1 Real Numbers A real number is any number that can be written as a decimal. They can terminate, repeat, or continue endlessly with no discernible pattern. Subsets of the set of real numbers include: natural numbers whole numbers integers Rational numbers are real numbers that can be written as a ratio of two integers. They will either terminate or repeat. A real number that neither terminate nor repeats is called an irrational number. August 17, 2012 The set of all real numbers are often represented on the real number line. -5 -4 -3 -2 -1 0 1 2 3 π• 4 5 Plot Let a and b be any two real numbers. If a −b is positive then we say a is greater than b and we write a >b and a will be to the right of b on the number line. If a −b is negative then we say a is less than b and we write a <b and b will be to the right of a on the number line. The Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: August 17, 2012 Interval Notation - bounded intervals [a, b] means a ≤ x ≤ b. This is a closed interval because the endpoints are included in the interval. Graph of the interval [−2, 3]. -5 -4 -3 -2 -1 0 1 2 3 5 4 ° ° ° (a, b) means a < x < b. This is an open interval because the endpoints are not included in the interval. Graph of the interval (−4, 1). -5 ° -4 -3 -2 -1 ° 0 ° 1 2 3 4 5 August 17, 2012 An interval may be open on one side and closed on the other such as (0, 4]. Write this as a compound inequality. Graph of the interval (0, 4]. -5 -4 -2 -3 ° -1 1 0 2 3 5 4 ° Interval Notation - unbounded intervals The set of all real numbers greater than π is an unbounded interval since there is no largest real number. The interval notation for this set of numbers is (π, ∞). If we want to include the number π in our interval we write [π, ∞). Since the symbol ∞ does not represent a number it would never be appropriate to use a square bracket next to the infinity sign. Graph (−∞, π] -5 ° -4 -3 -2 -1 0 1 2 π 3 4 5 ° August 17, 2012 How could we represent the set of all real numbers using interval notation? Write inequality and interval notation for each graph. • -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 ° 0 1 2 3 4 5 -5 -4 ° -2 -1 0 2 3 4 5 -3 • 1 Review the properties of exponents found on page 7. August 17, 2012