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.

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. Irrational numbers Proof of the “IF PART”

f =0.d d d ...d ... 1 2 3 k ∈R ⇓ f has a terminating representation, or f has a non-terminating representation with a repeating pattern.

Rational numbers vs. Irrational numbers 1. Euclidean Division Theorem 2. Pigeon Hole Principle

Tools for Proof of the if part of Main Theorem about Rational Numbers

Two tools to prove this result:

Rational numbers vs. Irrational numbers 2. Pigeon Hole Principle

Tools for Proof of the if part of Main Theorem about Rational Numbers

Two tools to prove this result: 1. Euclidean Division Theorem

Rational numbers vs. Irrational numbers Tools for Proof of the if part of Main Theorem about Rational Numbers

Two tools to prove this result: 1. Euclidean Division Theorem 2. Pigeon Hole Principle

Rational numbers vs. Irrational numbers First Tool: Euclidean Division Theorem

M 0 and N 1. Then,≥ there exists≥ a unique pair of integers (d, r), such that: M = d N + r, × or equivalently: M r = d + N N d 0 is the quotient of the division, and r ≥ 0, 1, ..., N 1 is the remainder.

∈{ − } Rational numbers vs. Irrational numbers “Successive Multiplications by 10 and Divisions by n (SMD)”

Application of Euclidean Division Theorem on f, 0 < f < 1 m 1 1 1 1 f = = d ( )+d ( )+d ( )+... + d ( )+... n 1 10 2 100 3 1000 k 10k 10m 1 1 1 = d1+f1 where f1 = d2( )+d3( )+...+dk( k 1 )+... n 10 100 10 − 10m r1 1 10m = d1n + r1 n = d1 + f1 f1 = n = d2( 10 )+...

10r1 r2 1 10r1 = d2n + r2 n = d2 + f2 f2 = n = d3( 10 )+... . . 10rk 1 rk 1 − 10rk 1 = dkn + rk n = dk + fk fk = n = dk+1( 10 )+... − . .

Each of r ,r, ..., r , .. 0 , 1, ..., n 1 1 2 k ∈{ − } # $! "

Rational!"#$ numbers vs. Irrational numbers Application of Euclidean Division Theorem on f, 0 < f < 1 m 1 1 1 1 f = = d ( )+d ( )+d ( )+... + d ( )+... n 1 10 2 100 3 1000 k 10k 10m 1 1 1 = d1+f1 where f1 = d2( )+d3( )+...+dk( k 1 )+... n 10 100 10 − 10m r1 1 10m = d1n + r1 n = d1 + f1 f1 = n = d2( 10 )+...

10r1 r2 1 10r1 = d2n + r2 n = d2 + f2 f2 = n = d3( 10 )+... . . 10rk 1 rk 1 − 10rk 1 = dkn + rk n = dk + fk fk = n = dk+1( 10 )+... − . .

Each of r ,r, ..., r , .. 0 , 1, ..., n 1 1 2 k ∈{ − } # $! "

Rational!"#$ numbers vs. Irrational numbers

“Successive Multiplications by 10 and Divisions by n (SMD)” ! yes, when rk =0.

! If not, d ,r starts repeating. { i i}

The Algorithm of Successive Multiplications by 10 and Divisions by n

! Can this procedure terminate?

Rational numbers vs. Irrational numbers ! If not, d ,r starts repeating. { i i}

The Algorithm of Successive Multiplications by 10 and Divisions by n

! Can this procedure terminate?

! yes, when rk =0.

Rational numbers vs. Irrational numbers The Algorithm of Successive Multiplications by 10 and Divisions by n

! Can this procedure terminate?

! yes, when rk =0.

! If not, d ,r starts repeating. { i i}

Rational numbers vs. Irrational numbers Proof of Terminating Sequences using Successive Multiplications and Divisions 10m 1 1 1 = d1 + d2( )+d3( )+... + dk( k 1 )+... n 10 100 10 −

10m r1 1 10m = d1n + r1 n = d1 + f1 f1 = n = d1 + d2( 10 )+...

10r1 r2 1 10r1 = d2n + r2 n = d2 + f2 f2 = n = d2 + d3( 10 )+... . . 10rk 1 10rk 1 = dkn +0 − = dk + fk fk =0 − n

Algorithm stops at k : rk =0implies:

rk+1 = rk+2 = .... =0and dk+1 = dk+2 = ... =0. m = =0.d d ....d . ⇒ n 1 2 k

Rational numbers vs. Irrational numbers Examples of fractions with terminating decimal representation

m 1 1. n = 4 ,m=1,n=4

10 1 2 10 1=2 4+2 × = 2 + , (d =2,r = 2) × × ⇔ 4 4 1 1 10 2 0 10 2=5 4+0 × = 5 + , (d =5,r = 0) × × ⇔ 4 4 2 2

1 r2 =0implies 4 =0.d1d2 =0.25

Rational numbers vs. Irrational numbers 10 5 2 10 5=6 8+2 × = 6 + , (d =6,r = 2) × × ⇔ 8 8 1 1 10 2 4 10 2=2 8+4 × = 2 + , (d =2,r = 4) × × ⇔ 8 8 2 2 10 4 0 10 4=5 8+0 × = 5 + , (d =5,r = 0) × × ⇔ 8 8 3 3 5 r3 =0implies 8 =0.d1d2d3 =0.625

Examples of fractions with terminating decimal representation

m 5 2. n = 8 ,m=5,n=8

Rational numbers vs. Irrational numbers Examples of fractions with terminating decimal representation

m 5 2. n = 8 ,m=5,n=8

10 5 2 10 5=6 8+2 × = 6 + , (d =6,r = 2) × × ⇔ 8 8 1 1 10 2 4 10 2=2 8+4 × = 2 + , (d =2,r = 4) × × ⇔ 8 8 2 2 10 4 0 10 4=5 8+0 × = 5 + , (d =5,r = 0) × × ⇔ 8 8 3 3 5 r3 =0implies 8 =0.d1d2d3 =0.625

Rational numbers vs. Irrational numbers Successive Multiplications and Divisions: Non Terminating Representations

10m 1 1 1 = d1 + d2( )+d3( )+... + dk( k 1 )+... n 10 100 10 −

10m r1 1 10m = d1n + r1 n = d1 + n = d1 + d2( 10 )+... ⇔ 10r1 r2 1 10r1 = d2n + r2 n = d2 + n = d2 + d3( 10 )+... ⇔. . 10rk 1 rk 1 − 10rk 1 = dkn + rk n = dk + n = dk + dk+1( 10 )+... − ⇔. .

Each of r ,r, ..., r , .. 1, ..., n 1 and r =0for all i. 1 2 k ∈{ − } i $ # $! " Rational numbers vs. Irrational numbers Second tool: Use of Pigeon hole Principle in m proving that Inﬁnite representations for n have repeating patterns

Statement: If you have n pigeons

# ...... $! " to occupy n 1 holes: −

# $! "

Then at least 2 pigeons must occupy the same hole.

Rational numbers vs. Irrational numbers Example 10 pigeons and 9 pigeon holes

Rational numbers vs. Irrational numbers Example of 3 pigeons and 2 pigeon holes

Rational numbers vs. Irrational numbers Solution of example of 3 pigeons and 2 pigeon holes

Rational numbers vs. Irrational numbers Application of Pigeonhole Principle for non-terminating sequences

10m r1 10m = d1n + r1 n = d1 + n ⇔ 10r1 r2 10r1 = d2n + r2 n = d2 + n ⇔. . 10rk 1 rk − 10rk 1 = dkn + rk n = dk + n − ⇔. .

r1 = r2 = ...... rn 1 = rn = −

By Pigeonhole principle: At least 2 remainders rj, rk, 1 j < k n: r = r . ≤ ≤ j k Rational numbers vs. Irrational numbers Applying the Pigeon hole Principle to obtain repeating sequences Let j, k be the ﬁrst pair, such that: 1 {j <} k n and r = r then: ≤ ≤ j k 10rj = dj+1n + rj+1 and 10rk = dk+1n + rk+1

⇓ dj+1 = dk+1 and rj+1 = rk+1... More generally, d = d and r = r , 1 l k j. j+l k+l j+l k+l ≤ ≤ − and therefore by recurrence: m =0.d d ...d d ....d n 1 2 j j+1 k Length of pattern: 1 k Rationalj numbersn vs.1 Irrational. numbers ≤ − ≤ − Example m 6 f = = n 7

10 6 = 8 7+4 d1 =8r1 =4 10 × 4 = 5 × 7+5 d =5r =5 × × 2 2 10 5 = 7 7+1 d3 =7r3 =1 10 × 1 = 1 × 7+3 d =1r =3 × × 4 4 10 3 = 4 7+2 d5 =4r5 =2 10 × 2 = 2 × 7+6 d =2r =6 × × 6 6 10 6 = 8 7+4 d7 =8r7 =4 × × . .

Each of r ,r,r,r,r, ... 1, 2, 3, 4, 5, 6 . 1 2 3 4 5 ∈{ } Rational numbers# vs. Irrational$! numbers" m 6 Example f = n = 7 r1 =4 r2 =5 r3 =1 r4 =3 r5 =2 r6 =6 r7 =4

1, 7 is the ﬁrst pair, such that r = r then: { } 1 7 6 =0.d d d d d d d =0.8571428 7 1 2 3 4 5 6 7 Length of pattern is 6.

Rational numbers vs. Irrational numbers Exercise

Find the decimal representation of m 2 f = = n 3 using Successive Multiplications and Divisions

Rational numbers vs. Irrational numbers m 2 Solution of the exercise f = n = 3

10 2 = 6 3+2 d1 =6r1 =2 10 × 2 = 6 × 3+2 d =6r =2 × × 2 2 10 2 = 6 3+2 d3 =6r3 =2 × × . .

1, 2 is the ﬁrst pair, such that r = r and therefore: { } 1 2 2 =0.d d =0.66 3 1 2 Length of pattern is 1

Rational numbers vs. Irrational numbers Answer to the Main question of Module

= Rational Numbers f, 0 < f < 1 R { } = Irrational Numbers f, 0 < f < 1 I { } = with =Φ empty set. S R∪I R∩I

Question: If we pick at random a number f between 0 and 1, what is the probability that this number be rational: f ? ∈R Rational numbers vs. Irrational numbers ! f =0.d1d2..dk or ! f =0.d1d2..dl 1dl...dk. −

! = and = |R| ∞1 |I| ∞2 ! Which one of these two inﬁnities is bigger? ! If f : ∈R

! While if f : f =0.d1d2..dk.... (inﬁnite representation∈I with no speciﬁc pattern).