Concepts and Objectives1‐1 Number Theory Unit 1 Functions and Relations Number Theory (Obj. #1) Identify subsets of real numbers Simplify expressions using order of operations Identify real number axioms Rational Numbers (Obj. #2) Convert between fractions and decimals Number Systems What we currently know as the set of real numbers was only formulated around 1879. We usually present this as sets of numbers. Number Systems The set of natural numbers ( ) and the set of integers ( ) have been around since ancient times, probably prompted by the need to maintain trade accounts. Ancient civilizations, such as the Babylonians, also used ratios to compare quantities. One of the greatest mathematical advances was the introduction of the number 0. Properties of Real Numbers For all real numbers a, b, and c: Closure Property a + b ∈ ab ∈ Properties of Real Numbers The properties are also called axioms. 0 is called the additive identity and 1 is called the multiplicative identity. Notice the relationships between the identities and the inverses (called the additive inverse and the multiplicative inverse). Saying that a set is “closed” under an operation (such as multiplication) means that performing that operation on numbers in the set will always produce an answer that is also in the set – there are no answers outside the set. Identity Property a + 0 = a a i 1 = a Commutative Property a + b = b + a ab = ba Inverse Property a + (–a) = 0 1 a i =1 a Associative Property (a + b) + c = a + (b + c) (ab)c = a(bc) Distributive Property a(b + c) = ab + ac Properties of Real Numbers Examples The set of natural numbers ( ) is not closed under the operation of subtraction. Why? Order of Operations Parentheses (or other grouping symbols, such as square brackets or fraction bars) – start with the innermost set, following the sequence below, and work outward. Exponents Multiplication working from left to right Division Addition working from left to right Subtraction –20 ÷ 5 = –4. Does this show that the set of integers is closed under division? Order of Operations Use order of operations to explain why Order of Operations Work the following examples without using your calculator. 1. 2. 3. ( −3) ≠ −32 2 −2 i 5 + 12 ÷ 3 −4 ( 9 − 8 ) + ( −7 )( 2) 3 −8 + ( −4 )( −6 ) ÷ 12 4 − ( −3) Absolute Value The absolute value of a real number a, denoted by |a|, is the distance from 0 to a on the number line. This distance is always taken to be nonnegative. ⎧ x if x ≥ 0 x =⎨ ⎩− x if x < 0 Absolute Value Properties For all real numbers a and b: 1. a ≥ 0 2. 3. 4. 5. −a = a a i b = ab a b = a (b ≠ 0) b a+b ≤ a + b Absolute Value Example: Rewrite each expression without absolute value bars. 1. 2. 3. Rational Numbers The Greeks, specifically Pythagoras of Samos, originally believed that the lengths of all segments in geometric objects could be expressed as ratios of positive integers. A number is a rational number ( ) if and only if it can be expressed as the ratio (or quotient) of two integers. Rational numbers include decimals as well as fractions. The definition does not require that a rational number must be written as a quotient of two integers, only that it can be. 3 −1 2− π x x , if x < 0 Examples Example: Prove that the following numbers are rational numbers by expressing them as ratios of integers. Irrational Numbers Unfortunately, the Pythagoreans themselves later discovered that the side of a square and its diagonal could not be expressed as a ratio of integers. Prove is irrational. 2 2 Proof (by contradiction): Assume is rational. This means that there exist relatively prime integers a and b such that a a2 2 = ⇒2= 2 b b 2b2 = a2 , therefore, a is even 1. 2‐4 2. 64‐½ 3. 4π π 4. 0.9 6.3 5. 20.3 6. –5.4322986 Irrational Numbers This means there is an integer j such that 2j=a. 2 2b2 = ( 2 j ) 2b2 = 4 j 2 b2 = 2 j 2 ⇒ b is even If a and b are both even, then they are not relatively 2 prime. This is a contradiction. Therefore, is irrational. n Theorem: Let n be a positive integer. Then is either an integer or it is irrational. Real Numbers The number line is a geometric model of the system of real numbers. Rational numbers are thus fairly easy to represent: What about irrational numbers? Consider the following: (1,1) • 2 Real Numbers In this way, if an irrational number can be identified with a length, we can find a point on the number line corresponding to it. What this emphasizes is that the number line is continuous—there are no gaps. Intervals Name of Interval finite, open finite, closed finite, half open Notation Inequality Description (a, b) [a, b] (a, b] [a, b) a < x < b a≤x≤b a < x ≤ b a ≤ x < b Number Line Representation a b a a a a a a a a a a b b b b b b b b b b b infinite, open (a, ∞) a < x < ∞ (‐∞, b) ‐∞ < x < b [a, ∞) a ≤ x < ∞ (‐∞, b] ‐∞< x ≤ b infinite, closed Finite and Repeating Decimals If a nonnegative real number x can be expressed as a finite sum of of the form d1 d2 dt + + ... + t 10 102 10 where D and each dn are nonnegative integers and 0 ≤ dn ≤ 9 for n = 1, 2, …, t, then D.d1d2…dt is the finite decimal representing x. x = D+ Finite and Repeating Decimals If the decimal representation of a rational number does not terminate, then the decimal is periodic (or repeating). The repeating string of numbers is called the period of the decimal. a It turns out that for a rational number where b > 0, b the period is at most b – 1. Finite and Repeating Decimals Example: Use long division (yes, long division) to find 462 the decimal representation of and find its period. 13 Finite and Repeating Decimals The repeating portion of a decimal does not necessarily start right after the decimal point. A decimal which starts repeating after the decimal point is called a simpleperiodic decimal; one which starts later is called a delayedperiodic decimal. Type of Decimal Examples General Form What is the period of this decimal? terminating simpleperiodic delayedperiodic 0.5, 0.25, 0.2, 0.125, 0.0625 0.d1 d2 d3 ...dt ( dt ≠ 0) 0.3, 0.142857, 0.1, 0.09, 0.076923 0.16, 0.083, 0.0714285, 0.06 0.d1 d2 d3 ...dp 0.d1 d2 d3 ...dt dt +1 dt + 2 dt + 3 ...dt + p Decimal Representation If we know the fraction, it’s fairly straightforward (although sometimes tedious) to find its decimal representation. What about going the other direction? How do we find the fraction from the decimal, especially if it repeats? We’ve already seen how to represent a terminating decimal as the sum of powers of ten. More generally, we can state that the decimal 0.d1d2d3…dt can be written as M , where M is the integer d1d2d3…dt. 10t Decimal Representation For simple‐periodic decimals, the “trick” is to turn them into fractions with the same number of 9s in the denominator as there are repeating digits and simplify: 0.3 = 3 1 = 9 3 0.09 = 9 1 = 99 11 0.153846 = 153846 2 = 999999 13 To put this more generally, the decimal 0.d1d2d3 ...dp M can be written as the fraction , where M is the p 10 − 1 integer d1d2d3…dp. Decimal Representation For delayed‐periodic decimals, the process is a little more complicated. Consider the following: What is the decimal representation of ? 1 12 0.083 Decimal Representation It turns out you can break a delayed‐periodic decimal into a product of terminating and simple‐periodic decimals, so the general form is also a product of the general forms: 0.d1d2d3 ...dt dt +1dt +2dt +3 ...dt + p The decimal can be written N as the fraction , where N is the integer 10t 10p − 1 1 1 1 is the product of what two fractions? i 12 4 3 Notice that the decimal representation has characteristics of each factor. ( ) d1d2d3…dtdt+1dt+2dt+3…dt+p – d1d2d3…dt . Decimal Representation Example: Convert the decimal to a 0.467988654 fraction.