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Rational Numbers Vs. Irrational Numbers Handout Rational numbers vs. Irrational numbers by Nabil Nassif, PhD in cooperation with Sophie Moufawad, MS and the assistance of Ghina El Jannoun, MS and Dania Sheaib, MS American University of Beirut, Lebanon An MIT BLOSSOMS Module August, 2012 Rational numbers vs. Irrational numbers “The ultimate Nature of Reality is Numbers” A quote from Pythagoras (570-495 BC) Rational numbers vs. Irrational numbers “Wherever there is number, there is beauty” A quote from Proclus (412-485 AD) Rational numbers vs. Irrational numbers Traditional Clock plus Circumference 1 1 min = of 1 hour 60 Rational numbers vs. Irrational numbers An Electronic Clock plus a Calendar Hour : Minutes : Seconds dd/mm/yyyy 1 1 month = of 1year 12 1 1 day = of 1 year (normally) 365 1 1 hour = of 1 day 24 1 1 min = of 1 hour 60 1 1 sec = of 1 min 60 Rational numbers vs. Irrational numbers TSquares: Use of Pythagoras Theorem Rational numbers vs. Irrational numbers Golden number ϕ and Golden rectangle 1+√5 1 1 √5 Roots of x2 x 1=0are ϕ = and = − − − 2 − ϕ 2 Rational numbers vs. Irrational numbers Golden number ϕ and Inner Golden spiral Drawn with up to 10 golden rectangles Rational numbers vs. Irrational numbers Outer Golden spiral and L. Fibonacci (1175-1250) sequence = 1 , 1 , 2, 3, 5, 8, 13..., fn, ... : fn = fn 1+fn 2,n 3 F { } − − ≥ f1 f2 1 n n 1 1 !"#$ !"#$ fn = (ϕ +( 1) − ) √5 − ϕn Rational numbers vs. Irrational numbers Euler’s Number e 1 1 1 s = 1 + + + =2.6666....66.... 3 1! 2 3! 1 1 1 s = 1 + + + =2.70833333...333.... 4 2 3! 4! 1 1 1 1 s = 1 + + + + =2.7166666666...66.... 5 2 3! 4! 5! ............................. 1 1 1 1 1 lim 1+ + + + + .... + = e =2.718281828459........ n →∞ { 2 3! 4! 5! n!} e is an irrational number discovered by L. Euler (1707-1783), a limit of a sequence of rational numbers. Rational numbers vs. Irrational numbers Definition of Rational and Irrational numbers ! A Rational number r is defined as: m r = n where m and n are integers with n =0. $ ! Otherwise, if a number cannot be put in the form of a ratio of 2 integers, it is said to be an Irrational number. Rational numbers vs. Irrational numbers I is its integral part; • 0 f<1 is its fractional part. • ≤ Distinguishing between rational and irrational numbers Any number x, (rational or irrational) can be written as: x = I + f Rational numbers vs. Irrational numbers 0 f<1 is its fractional part. • ≤ Distinguishing between rational and irrational numbers Any number x, (rational or irrational) can be written as: x = I + f I is its integral part; • Rational numbers vs. Irrational numbers Distinguishing between rational and irrational numbers Any number x, (rational or irrational) can be written as: x = I + f I is its integral part; • 0 f<1 is its fractional part. • ≤ Rational numbers vs. Irrational numbers Examples 48 = 1 + 0.92 • 25 8 = • 3 17 = • 7 √2= • π = • ϕ = 1+√5 = • 2 Rational numbers vs. Irrational numbers Answers to Examples 48 = 1 + 0.92 • 25 8 = 2 + 0.6666666..... • 3 17 = 2 + 0.4285714285714..... • 7 √2 = 1 + 0.4142135623731..... • π = 3 + 0.14159265358979..... • ϕ = 1 + 0.6180339887499... • Rational numbers vs. Irrational numbers 1. As x = I + f, I: Integer; 0 < f < 1: Fractional. 2. = Distinction between rational and irrational can⇒ be restricted to fraction numbers f between 0 < f < 1. Distinguishing between rational and irrational numbers Rational numbers vs. Irrational numbers 2. = Distinction between rational and irrational can⇒ be restricted to fraction numbers f between 0 < f < 1. Distinguishing between rational and irrational numbers 1. As x = I + f, I: Integer; 0 < f < 1: Fractional. Rational numbers vs. Irrational numbers Distinguishing between rational and irrational numbers 1. As x = I + f, I: Integer; 0 < f < 1: Fractional. 2. = Distinction between rational and irrational can⇒ be restricted to fraction numbers f between 0 < f < 1. Rational numbers vs. Irrational numbers If we pick a number f at random between 0 • and 1, what is the probability that this number be rational: f ? ∈R Position of the Problem = Rational Numbers f, 0 < f < 1 R { } = Irrational Numbers f, 0 < f < 1 I { } The segment following segment represents all numbers between 0 and 1: S = with =Φ empty set. S R∪I R∩I Basic Question: • Rational numbers vs. Irrational numbers Position of the Problem = Rational Numbers f, 0 < f < 1 R { } = Irrational Numbers f, 0 < f < 1 I { } The segment following segment represents all numbers between 0 and 1: S = with =Φ empty set. S R∪I R∩I Basic Question: • If we pick a number f at random between 0 • and 1, what is the probability that this number be rational: f ? ∈R Rational numbers vs. Irrational numbers The Decimal Representation of a number Any number f :0< f < 1 has the following decimal representation: Notation f =0.d1d2d3...dk... #$!" d 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 i ∈{ } 1 1 1 1 f = d ( )+d ( )+d ( )+... + d ( )+... (Equality) 1 10 2 100 3 1000 k 10k with at least one of the di’s =0. Rational$ numbers vs. Irrational numbers f is either Terminating: d =0for i > l 1 i ≥ or f is Non-Terminating with a repeating pattern. Main Theorem about Rational Numbers The number 0 < f < 1 is rational, that is m f = n , m < n, if and only if its decimal representation: f =0.d1d2d3...dk... 1 1 1 1 = d ( )+d ( )+d ( )+... + d ( )+... 1 10 2 102 3 103 k 10k takes one of the following forms: Rational numbers vs. Irrational numbers or f is Non-Terminating with a repeating pattern. Main Theorem about Rational Numbers The number 0 < f < 1 is rational, that is m f = n , m < n, if and only if its decimal representation: f =0.d1d2d3...dk... 1 1 1 1 = d ( )+d ( )+d ( )+... + d ( )+... 1 10 2 102 3 103 k 10k takes one of the following forms: f is either Terminating: d =0for i > l 1 i ≥ Rational numbers vs. Irrational numbers Main Theorem about Rational Numbers The number 0 < f < 1 is rational, that is m f = n , m < n, if and only if its decimal representation: f =0.d1d2d3...dk... 1 1 1 1 = d ( )+d ( )+d ( )+... + d ( )+... 1 10 2 102 3 103 k 10k takes one of the following forms: f is either Terminating: d =0for i > l 1 i ≥ or f is Non-Terminating with a repeating pattern. Rational numbers vs. Irrational numbers Proof of the Main Theorem about Rational Numbers Theorem The number 0 < f < 1 is rational, that is m f = n , m < n, if and only if its decimal representation: f =0.d1d2d3...dk... is either Terminating (di =0for i > l 1) or is Non-Terminating with a repeating pattern.≥ Rational numbers vs. Irrational numbers 1. If f has a terminating decimal representation, then f is rational. 2. If f has a non-terminating decimal representation with a repeating pattern, then f is rational. Proof of the only if part of Main Theorem about Rational Numbers Proof. Rational numbers vs. Irrational numbers 2. If f has a non-terminating decimal representation with a repeating pattern, then f is rational. Proof of the only if part of Main Theorem about Rational Numbers Proof. 1. If f has a terminating decimal representation, then f is rational. Rational numbers vs. Irrational numbers Proof of the only if part of Main Theorem about Rational Numbers Proof. 1. If f has a terminating decimal representation, then f is rational. 2. If f has a non-terminating decimal representation with a repeating pattern, then f is rational. Rational numbers vs. Irrational numbers Proof of the first Statement of only if part Statement 1: If f has a terminating decimal representation, then f is rational. Consider: 1 1 1 1 f = d ( )+d ( )+d ( )+... + d ( ) 1 10 2 100 3 1000 k 10k then: k k 1 k 2 10 f = d110 − + d210 − + ... + dk. implying: m k 1 k 2 f = with m = d 10 − + d 10 − + ... + d 10k 1 2 k Rational numbers vs. Irrational numbers 625 125 5 0.625 = = × 1, 000 125 8 × 5 0.625 = after simplification: 8 Example Rational numbers vs. Irrational numbers 5 0.625 = after simplification: 8 Example 625 125 5 0.625 = = × 1, 000 125 8 × Rational numbers vs. Irrational numbers Example 625 125 5 0.625 = = × 1, 000 125 8 × 5 0.625 = after simplification: 8 Rational numbers vs. Irrational numbers Proof of the second Statement of only if part Statement 2: If f has a non terminating decimal representation with repeating pattern, then f is rational. Without loss of generality, consider: f =0.d1d2d3...dk =0.d1d2d3...dkd1d2d3...dkd1d2d3...dk... 1 1 1 1 f = d ( )+d ( )+d ( )+... + d ( )+ 1 10 2 100 3 1000 k 10k 1 1 1 1 1 1 [d ( )+d ( )+d ( )+... + d ( )] + [....] 10k 1 10 2 100 3 1000 k 10k 102k then: k k 1 k 2 10 f = d110 − + d210 − + ... + dk +f. m: Integer implying: ! "# m $ (10k 1) f = m f = − ⇐⇒ n n: Integer ! "# $ Rational numbers vs. Irrational numbers Example on Proof of the second Statement f =0.428571 = 0.428571428571428571... 1 1 1 1 1 1 1 f = 4( )+2( )+8( )+5( )+7( )+1 + (f) 10 100 103 104 105 106 106 106 f =4 105 +2 104 +8 103 +5 102 +7 10+ 1+f × × × × × × (106 1) f = 428, 571 − × 428, 571 428, 571 f = = 106 1 999, 999 − After simplification: 428, 571 3 142, 857 3 f = = × = 999, 999 7 142, 857 7 × Rational numbers vs.
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