m13 m16 Rational and Irrational Numbers

8/11/2019 m13 m16 Rational and Irrational Numbers

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Rational and Irrational Numbers

Rational Numbers

A rational number is any number that can be expressed as theratio of two integers.

Examples

All terminating and repeating decimalscan be expressed inthis way so they are irrational numbers.

ab

45

2 23 =

83 6 =

61 2.7 =

2710

0.625 =58 34.56 =

3456100

-3= 31-

0.3= 13 0.27 =311 0.142857 =

17

0.7=710


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Rational Numbers


All terminating and repeating decimalscan be expressed inthis way so they are irrational numbers.

abShow that the terminating decimalsbelow are rational.

0.6 3.8 56.1 3.45 2.157

610

3810

56110

345100

21571000

RATIONAL


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Rational Numbers


All terminating and repeating decimalscan be expressed inthis way so they are rational numbers.

abTo show that a repeating decimalis rational.

Example 1

To show that 0.333 is rational.

Letx = 0.333

10x = 3.339x = 3

x = 3/9

x = 1/3

Example 2

To show that 0.4545 is rational.

Letx = 0.4545

100x

= 45.4599x = 45

x = 45/99

x = 5/11


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Rational Numbers



ab

Question 1

Show that 0.222 is rational.

Letx = 0.222

10x = 2.229x = 2

x = 2/9

Question 2


Letx = 0.6363

100x

= 63.6399x = 63

x = 63/99

x = 7/11


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Rational Numbers



ab

999x = 273

x = 273/999

9999x = 1234

x = 1234/9999

Question 3

Show that 0.273is rational.

Letx = 0.273

1000x = 273.273

x = 91/333

Question 4


Letx = 0.1234

10000x = 1234.1234


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Rational Numbers



abBy looking at the previous examples can you spot a quick method of

determining the rational number for any given repeating decimal.

0.1234

12349999

0.273

273999

0.45

4599

0.3

39


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Rational Numbers



ab

0.1234

12349999

0.273

273999

0.45

4599

0.3

39

Write the repeating part of the decimal as the numerator and write the

denominator as a sequence of 9s with the same number of digitsas thenumerator then simplify where necessary.


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Rational Numbers



ab

15439999

628999

3299

79

0.1543 0.6280.32 0.7Write down the rational form for each of the repeating decimals below.


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ab


Irrational Numbers

An irrational number is any number that cannot beexpressed as the ratio of two integers.

1

1

2

Pythagoras

The history of irrational numbers begins with adiscovery by the Pythagorean School in ancientGreece. A member of the school discovered thatthe diagonal of a unit square could not beexpressed as the ratio of any two wholenumbers. The motto of the school was All isNumber (by which they meant whole numbers).Pythagoras believed in the absoluteness of whole

numbers and could not accept the discovery. Themember of the group that made it was Hippasusand he was sentenced to death by drowning.(See slide 19/20 for more history)


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1

1

11

1

1

1

1

1

11

1

1

1

1

1

1

1

Rational Numbers

Irrational Numbers

2

3

4

5

6

7

8

9

10 11

12

13

14

15

16

17

18


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ab


Irrational Numbers


1

1

2

Pythagoras

Intuition alone may convince you that all pointson the Real Number line can be constructed

from just the infinite set of rational numbers,after all between anytwo rational numbers wecan always find another. It tookmathematicians hundreds of years to showthat the majority of Real Numbers are in factirrational. The rationalsand irrationalsareneeded together in order to complete thecontinuum that is the set of Real Numbers.


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ab


Irrational Numbers


1

1

2

Pythagoras

Surds are Irrational Numbers

3

1

27

1

2

1

4

13 andWe can simplify numbers such as

into rational numbers.However, other numbers involving

roots such as those shown cannot

be reduced to a rational form.

3 1282 ,,

Any number of the form which cannot be written

as a rational number is called a surd.

nm

All irrational numbers are non-terminating, non-repeatingdecimals.

Their decimal expansion form shows no patternwhatsoever.

Other irrational numbers include and e, (Eulers number)


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Multiplication and division of surds.

baab x

632949436 xxxFor example:

10510550 xx and

b

a

b

aalso

3

2

9

4

9

4

for example

and

7

6

7

6


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Example questions

Show that 123x is rational

636123123 xx rational

Show that is rational

5

45

395

45

5

45 rational

a

b


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Questions

8

32

a

e

State whether each of the following are rational or irrational.

76x b 5x20 c 3x27 d 3x4

11

44f

2

18g

5

25h

irrational rational rational irrational

rational rational rational irrational


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Combining Rationals and Irrationals

Addition and subtraction of an integer to an irrational number givesanother irrational number, as does multiplication and division.

Examples of irrationals

11733853651072 3

))(())(( 76265353455

311283

31028

253103

6920

14696


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Multiplication and division of an irrational number by another irrationalcan often lead to a rational number.

Examples of Rationals

))(())(()()( 434312125962573 21

22

21 26 8 1 -13


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Determine whether the following are rational or irrational.

(a) 0.73 (b) (c) 0.666. (d) 3.142 (e) .2512

(f) (g) (h) (i) (j)7 54 13)2(3 16 2

123 2)(

(j) (k) (l)1)31)(( 3 )61)(( 16 ))(( 2121

irrationalrational rational rational irrational

irrational irrational rational rational irrational

irrational rational rational

2


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HISTORY

The Pythagoreans

Pythagoras was a semi-mystical figure who was born on the Islandof Samos in the Eastern Aegean in about 570 B.C. He travelledextensively throughout Egypt, Mesopotamia and India absorbingmuch mathematics and mysticism. He eventually settled in theGreek town of Crotona in southern Italy.

He founded a secretive and scholarly society there that becomeknown as the Pythagorean Brotherhood. It was a mystical almostreligious society devoted to the study of Philosophy, Science andMathematics. Their work was based on the belief that all naturalphenomena could be explained by reference to whole numbers orratios of whole numbers. Their motto became All is Number.

They were successful in understanding the mathematicalprincipals behind music. By examining the vibrations of a singlestring they discovered that harmonious tones onlyoccurred when

the string was fixed at points along its length that were ratios ofwhole numbers. For instance when a string is fixed 1/2 way alongits length and plucked, a tone is produced that is 1 octave higherand in harmonywith the original. Harmonious tones are producedwhen the string is fixed at distances such as 1/3, 1/4, 1/5, 2/3and 3/4 of the way along its length. By fixing the string at pointsalong its length that were nota simple fraction, a note isproduced that is notin harmony with the other tones.

Pentagram

PythagorasSpirit

WaterAir

Earth Fire


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Pythagoras and his followers discovered many patterns and relationships between whole numbers.

Triangular Numbers:

1 + 2 + 3 + ...+ n

= n(n + 1)/2

Square Numbers:

1 + 3 + 5 + ...+ 2n 1

= n2

Pentagonal Numbers:

1 + 4 + 7 + ...+ 3n 2

= n(3n 1)/2

Hexagonal Numbers:

1 + 5 + 9 + ...+ 4n 3

= 2n2-n

These figuratenumbers were extended into 3 dimensional space and becamepolyhedral numbers. They also studied the properties of many other types ofnumber such as Abundant, Defective, Perfect and Amicable.

In Pythagorean numerology numbers were assigned characteristics or attributes. Odd numbers were regarded asmale and even numbers as female.

1. The number of reason (the generator of all numbers)

2. The number of opinion (The first female number)

3. The number of harmony (the first proper male number)

4. The number of justice or retribution, indicating the squaring of accounts (Fair and square)

5. The number of marriage (the union of the first male and female numbers)

6. The number of creation (male + female + 1)

10. The number of the Universe (The tetractys.The most important of all numbers representing the sumof all possible geometric dimensions. 1 point + 2 points (line) + 3 points (surface) + 4 points (plane)


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The Square Root of 2 is Irrational

1

1

This is a reductio-ad-absurdum proof.

To prove that is irrational

Assume the contrary: is rational

That is, there exist integers p and q with no common factorssuch that:

2q

p2

2

2

q

pevenispqp 2 22

(Since 2q2is even, p2is even so p even) So p =2kfor some k.

., evenisqp

qq

pasAlso

22

2

2

2

2

(Since p is even is even, q2is even so q is even)2

2

pSo q =2mfor some m.

p

q

2k

2m

p

qhaveafactorof2 incommon.

This contradicts the original assumption.

is irrational. QED

(odd2= odd)

PROOF

2

2

2

2

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m13 m16 Rational and Irrational Numbers