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THE REAL NUMBER SYSTEM Numbers rule the universe. - Pythagoras
Learning Objectives At the end of the lesson, you should be able to identify subsets of the set of real numbers recognize the various forms of rational numbers distinguish rational numbers from irrational numbers locate numbers on the real number line 3
NUMBER!!!??? evolved over time by expanding the notion of what we mean by the word number. Question: What is the difference between numbers and numerals? at first, number meant something you could count, like how many girlfriends I have how many legs an insect has These are called 4
SET OF NATURAL NUMBERS All natural numbers are truly natural. We find them in nature. 5
The set of NATURAL NUMBERS, (also called COUNTING NUMBERS) is denoted by N = {1, 2, 3, 4, 5, 6, 7, } 6
Special Subsets of N P = the set of prime numbers (divisible only by 1 and itself) What is the smallest prime number? Can an even number be prime? Name some more prime numbers. 7
C = the set of composite numbers What is the smallest composite number? Are all odd numbers composite? Name some more composite numbers. Are P and C disjoint? 8
The Number Zero Is zero a number? How can the number of nothing be a number? 0 is a special number because it does not quite obey the same laws as other numbers (e.g. We can t divide by zero) Zero is also used as a place-holder 9
SET OF WHOLE NUMBERS W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } What is the only difference between this set and the set of natural (or counting) numbers? 10
What is N W? N W? Are N and W disjoint? What is W N? N W? 11
NEGATIVE NUMBERS Negative numbers used to represent losses, debt, depth, etc. These are the natural numbers with the negative (minus) sign 12
NEGATIVE NUMBERS Opposites of the positive numbers When a positive number and its negative are added the result is 0. These pairs of numbers are called additive inverses of one another. 13
NEGATIVE NUMBERS Time to think: What is the additive inverse of 5? 5 What is the additive inverse of 23? 23 14
SET OF INTEGERS Z = {,-4,-3,-2,-1,0,1,2,3,4, } Special Subsets of the Set of Integers 1. Set of Negative integers: N = Z = { -4, -3, -2, -1} 2. The set consisting of Zero alone: {0} 3. Set of Positive integers: Z + = N = {1, 2, 3, 4, } What is the notation for the set of nonpositive integers? nonnegative integers? 15
Other Special Subsets of Z E = the set of even integers = {x x = 2k where k є Z} Which of the following numbers is even? Why? a) 146 b) 2313 c) 1887640 16
O = the set of odd integers = {x x = 2k + 1 where k є Z} = {x x = 2k 1 where k є Z} Which of the following is odd? Why? a) 12345 b) 24670 c) 32146987 17
Time to think: Is E = O? Why? Is E O? Why? Are E and O disjoint? 18
SET OF RATIONAL NUMBERS A rational number is a number that can be expressed as the ratio or quotient of two integers p and q where q 0. The set of rational numbers is denoted as Q p p, q Z, q 0 q 19
Examples of Rational Numbers 1 a) 0.25 4 1 b) 0.5 2 11 c) 5.5 2 20 d) 4 5 2 e) 0.666... 3 20
About Rational Numbers Integers are rational numbers Fractions are rational numbers (similar to the word fracture suggesting breaking something up) proper fraction improper fraction mixed numbers 21
About Rational Numbers Furthermore, rational numbers are numbers with decimals that are terminating non-terminating but repeating Example: (Can you convert the following to fractions?) 0.25 0.1111 0.125125 0.645 0.00222 0.547123 22
POSITIVE, NEGATIVE RATIONAL NUMBERS What are the notations? Positive rational numbers Negative rational numbers Nonpositive rational numbers Nonnegative rational numbers (Zero is neither positive nor negative) 23
Positive rational numbers can be written in any of three forms, any of three notations as fractions as decimals as percents Each form has its own special characteristics The forms are interchangeable. But the ways of doing standard operations with the three notations are very different. 24
25 FRACTIONS are used to name part of a whole object or part of a whole collection of objects or to compare two quantities. Note that (for mixed numbers): 2 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3
The fraction can be thought of as 2 divided by 3 so that when you divide a numerator by a denominator you can express that fraction as a DECIMAL. 26
A PERCENT is a fraction with 100 in the denominator. The word percent comes from the Latin word per centum. per meaning FOR and centum meaning ONE HUNDRED. 60% means 60 out of 100. A percent represents a portion of something and that something is the whole thing, or 100 %. 27
Multiple Representations Fractions and mixed numbers appear in recipes Decimals occur in scientific measurements Percentages used in commerce 28
SET OF IRRATIONAL NUMBERS are those real numbers that can not be expressed as the ratio of two integers denote the set of irrational numbers as Q c (the complement of Q) if U=R can also be described as numbers with decimals that are nonterminating and nonrepeating 29
Examples of Irrational Numbers Non-terminating, non-repeating decimals a) 1.01001000100001 b) 1.414213562 2 c) 3.141592653589 p d) 2.7182818284590 e 30
WARNING!!! p 22 7 31
Examples of Irrational Numbers The square roots of all positive numbers which are not perfect squares are irrationals. (Actually, if an integer is not an exact k th power of another integer then its k th root is irrational.) 3 16 3 8 Determine if rational or irrational: 5 7 32
Examples of Irrational Numbers log 2 3 is also irrational The golden ratio (also called as golden mean, golden section, divine proportion, golden number) and its reciprocal are irrational. 33
The Golden Ratio 1 2 5 1.61803... 34
The Reciprocal of the Golden Ratio 1 1 0.61803... 35
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FYI about RADICALS Some rational and irrational numbers are of the form n p where n p : principal nth root of p. sometimes called radicals p n radicand index p : principal square root of p, p 0. 40
QUESTION: 2 We know that 2 4 so 4 2. 2 Also, 2 4 so is 4 2? 41
Principal Root Definition. If n is even and p is non-negative, we define n p as the positive nth root of p. Therefore, 4 2 and 4 2 42
NOTE: n If p is negative, p is undefined. (Since we assume U=R) Suppose p is positive. If n is odd, n p n p 43
Example Determine the value of the following radicals. 1. 9 3 3 2. 8 2 3 3. 27 3 27 3 4 4. 625 5 5. 4 is undefined 44
x n =p n n If n is odd then p is the solution to x p. 3 3 8 is the solution to x 8 3 3 2 8 so 8 2 n n If n is even then p and p n are the solutions to x p. ( p 0) 2 4 and 4 are the solutions to x 4 2 2 2 4 so 4 2 ; ( 2) 4 so 4 2 45
Example Find the solution(s) to the following. 2 1. x 16 3 2. x 125 4, 4 5 5 3. x 4 5 4 2 4. x 9 no solutions 2 5. x 3 3, 3 46
Set of Real Numbers, R R is the union of the set of rational numbers and set of irrational numbers 47
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Real Number Line One-dimensional coordinate system 1 1 correspondence between the set of points on a line and the set of real numbers 49
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Locating numbers in the number line: Find the following numbers in the number line shown below: 0 a) 1 c) b) -2 d) 2 3 51
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SUMMARY A real number is either rational or irrational. If it is a rational number, it is either an integer or a non-integer fraction. If it is an integer, it is is either a whole number or a negative integer. If it is a whole number, it is either a counting number or zero. There is a 1-1 correspondence between the set of real numbers and the set of points on the line. 53
FYI: The cardinality of N, W, Z, Z, E, O, Q is אּ 0 (countable) The cardinality of Q c and R is אּ 1 (uncountable) 54
FYI: Example of a set outside R is the set of imaginary numbers (I). The union of R and I is the set of complex numbers. 55