Post on 02-Jun-2018
transcript
8/11/2019 m13 m16 Rational and Irrational Numbers
1/21
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
8/11/2019 m13 m16 Rational and Irrational Numbers
2/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
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
8/11/2019 m13 m16 Rational and Irrational Numbers
3/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
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
8/11/2019 m13 m16 Rational and Irrational Numbers
4/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
All terminating and repeating decimalscan be expressed inthis way so they are rational numbers.
ab
Question 1
Show that 0.222 is rational.
Letx = 0.222
10x = 2.229x = 2
x = 2/9
Question 2
Show that 0.6363 is rational.
Letx = 0.6363
100x
= 63.6399x = 63
x = 63/99
x = 7/11
8/11/2019 m13 m16 Rational and Irrational Numbers
5/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
All terminating and repeating decimalscan be expressed inthis way so they are 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
Show that 0.1234 is rational.
Letx = 0.1234
10000x = 1234.1234
8/11/2019 m13 m16 Rational and Irrational Numbers
6/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
All terminating and repeating decimalscan be expressed inthis way so they are 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
8/11/2019 m13 m16 Rational and Irrational Numbers
7/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
All terminating and repeating decimalscan be expressed inthis way so they are 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.
8/11/2019 m13 m16 Rational and Irrational Numbers
8/21
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as theratio of two integers.
All terminating and repeating decimalscan be expressed inthis way so they are 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.
8/11/2019 m13 m16 Rational and Irrational Numbers
9/21
ab
Rational and Irrational Numbers
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)
8/11/2019 m13 m16 Rational and Irrational Numbers
10/21
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
8/11/2019 m13 m16 Rational and Irrational Numbers
11/21
ab
Rational and Irrational Numbers
Irrational Numbers
An irrational number is any number that cannot beexpressed as the ratio of two integers.
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.
8/11/2019 m13 m16 Rational and Irrational Numbers
12/21
ab
Rational and Irrational Numbers
Irrational Numbers
An irrational number is any number that cannot beexpressed as the ratio of two integers.
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)
8/11/2019 m13 m16 Rational and Irrational Numbers
13/21
Rational and Irrational Numbers
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
8/11/2019 m13 m16 Rational and Irrational Numbers
14/21
Rational and Irrational Numbers
Example questions
Show that 123x is rational
636123123 xx rational
Show that is rational
5
45
395
45
5
45 rational
a
b
8/11/2019 m13 m16 Rational and Irrational Numbers
15/21
Rational and Irrational Numbers
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
8/11/2019 m13 m16 Rational and Irrational Numbers
16/21
Rational and Irrational Numbers
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
8/11/2019 m13 m16 Rational and Irrational Numbers
17/21
Rational and Irrational Numbers
Combining Rationals and Irrationals
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
8/11/2019 m13 m16 Rational and Irrational Numbers
18/21
Rational and Irrational Numbers
Combining Rationals and Irrationals
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
8/11/2019 m13 m16 Rational and Irrational Numbers
19/21
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
8/11/2019 m13 m16 Rational and Irrational Numbers
20/21
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)
8/11/2019 m13 m16 Rational and Irrational Numbers
21/21
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
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