Transcript Rational and Irrational Numbers

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Rational and Irrational Numbers
Rational Numbers
a
b
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
Examples
4
5
2
8
2
=
3
3
7
0.7 = 10
1
0.3 = 3
6
6
= 1
0.625
0.27
5
= 8
3
=
11
-3 = -
3
1
2.7
34.56
27
= 10
3456
= 100
1
0.142857 = 7
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
a
b
Show that the terminating decimals below are rational.
0.6
3.8
56.1
3.45
2.157
6
10
38
10
561
10
345
100
2157
1000
Rational
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
a
b
To show that a repeating decimal is rational.
Example 1
Example 2
To show that 0.333… is rational.
To show that 0.4545… is rational.
Let x = 0.333…
Let x = 0.4545…
10x = 3.33…
100x = 45.45…
9x = 3
x = 3/9
x = 1/3
99x = 45
x = 45/99
x = 5/11
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
Question 1
a
b
Question 2
Show that 0.222… is rational.
Show that 0.6363… is rational.
Let x = 0.222…
Let x = 0.6363…
10x = 2.22…
100x = 63.63…
9x = 2
x = 2/9
99x = 63
x = 63/99
x = 7/11
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
Question 3
Show that 0.273is rational.
Let x = 0.273
1000x = 273.273
999x = 273
x = 273/999
x = 91/333
a
b
Question 4
Show that 0.1234 is rational.
Let x = 0.1234
10000x = 1234.1234
9999x = 1234
x = 1234/9999
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
a
b
By looking at the previous examples can you spot a quick method of
determining the rational number for any given repeating decimal.
0.3
0.45
3
9
45
99
0.273
273
999
0.1234
1234
9999
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
a
b
Write the repeating part of the decimal as the numerator and write the
denominator as a sequence of 9’s with the same number of digits as the
numerator then simplify where necessary.
0.3
0.45
3
9
45
99
0.273
273
999
0.1234
1234
9999
Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
All terminating and repeating decimals can be expressed in
this way so they are irrational numbers.
a
b
Write down the rational form for each of the repeating decimals below.
0.32
0.7
0.1543
0.628
32
99
7
9
1543
9999
628
999
Rational and Irrational Numbers
Irrational
Irrational Numbers
An irrational number is any number that cannot be
expressed as the ratio of two integers.
The history of irrational numbers begins with a
discovery by the Pythagorean School in ancient
Greece. A member of the school discovered that
the diagonal of a unit square could not be
expressed as the ratio of any two whole
numbers. The motto of the school was “All is
Number” (by which they meant whole numbers).
Pythagoras believed in the absoluteness of whole
numbers and could not accept the discovery. The
member of the group that made it was Hippasus
and he was sentenced to death by drowning.
a
b
Pythagoras
2
(See slide 19/20 for more history)
1
1
1
1
1
Rational Numbers
1
1
1
11
10
Irrational Numbers
12
1
9
13
8
1
1
14
7
6
15
1
1
5
1
4
1
1
2
3
1
16
1
17
1
18
1
Rational and Irrational Numbers
Irrational Numbers
An irrational number is any number that cannot be
expressed as the ratio of two integers.
Intuition alone may convince you that all points
on the “Real Number” line can be constructed
from just the infinite set of rational numbers,
after all between any two rational numbers we
can always find another. It took
mathematicians hundreds of years to show
that the majority of Real Numbers are in fact
irrational. The rationals and irrationals are
needed together in order to complete the
continuum that is the set of “Real Numbers”.
a
b
Pythagoras
2
1
1
Rational and Irrational Numbers
a
b
Irrational Numbers
An irrational number is any number that cannot be
expressed as the ratio of two integers.
Surds are Irrational Numbers
We can simplify numbers such as
into rational numbers.
1 1
1
1
3
 and

4 2
27 3
However, other numbers involving
3
roots such as those shown cannot
2
,
8
,
12
be reduced to a rational form.
Any number of the form n
which cannot be written
Pythagoras
m
as a rational number is called a surd.
Other irrational numbers include
2
1
 and e, (Euler’s number)
All irrational numbers are non-terminating, non-repeating decimals.
Their decimal expansion form shows no pattern whatsoever.
1
Rational and Irrational Numbers
Multiplication and division of surds.
ab  a x b
For example:
and
also
36  4 x 9  4 x 9  2 x 3  6
50  5 x 10  5 x 10
a
a

b
b
for example
and
4

9
4
2

9 3
6

7
6
7
Rational and Irrational Numbers
Example questions
a
Show that
3 x 12
is rational
3 x 12  3 x 12  36  6
b
Show that
45
rational
is rational
5
45
5

45
 9 3
5
rational
Rational and Irrational Numbers
Questions
State whether each of the following are rational or irrational.
a
6x 7
b
irrational
e
32
8
rational
20 x 5
c
f
11
rational
d
g
18
2
rational
4x 3
irrational
rational
rational
44
27 x 3
h
25
5
irrational
Rational and Irrational Numbers
Combining Rationals and Irrationals
Addition and subtraction of an integer to an irrational number gives
another irrational number, as does multiplication and division.
Examples of irrationals
2
7
3 8  2 11
3
10
5 6
3
5
5  4
3 5
8 3
3  17  1
( 3  5)( 3  5)
( 6  2)( 6  7)
3  10 3  25
6  9 6  14
 28  10 3
20  9 6
Rational and Irrational Numbers
Combining Rationals and Irrationals
Multiplication and division of an irrational number by another irrational
can often lead to a rational number.
Examples of Rationals
1
2
3( 7 ) 2
5( 2 ) 2  6
9 5
21
26
8
( 2  1)( 2  1)
1
( 3  4)( 3  4)
-13
Rational and Irrational Numbers
Combining Rationals and Irrationals
Determine whether the following are rational or irrational.
(a) 0.73
rational
(f)
(b)
irrational
(g) 4 
7
irrational
(c) 0.666….
2
5
irrational
(j) ( 3  1)( 3  1)
irrational
(d) 3.142
(h) ( 2 )  1
3
rational
(k) ( 6  1)( 6  1)
rational
12.25
irrational
rational
rational
3
(e)
(i) 16
1
2
rational
(j) (3 2) 2
irrational
(l) (1  2 )(1  2 )
rational
The Pythagoreans
Pythagoras was a semi-mystical figure who was born on the Island
of Samos in the Eastern Aegean in about 570 B.C. He travelled
extensively throughout Egypt, Mesopotamia and India absorbing
much mathematics and mysticism. He eventually settled in the
Greek town of Crotona in southern Italy.
Spirit
Pythagoras
He founded a secretive and scholarly society there that become
Air
known as the “Pythagorean Brotherhood”. It was a mystical almost
religious society devoted to the study of Philosophy, Science and
Mathematics. Their work was based on the belief that all natural
phenomena could be explained by reference to whole numbers or
ratios of whole numbers. Their motto became “All is Number”.
They were successful in understanding the mathematical
principals behind music. By examining the vibrations of a single
string they discovered that harmonious tones only occurred when
the string was fixed at points along its length that were ratios of
whole numbers. For instance when a string is fixed 1/2 way along
its length and plucked, a tone is produced that is 1 octave higher
and in harmony with the original. Harmonious tones are produced
when the string is fixed at distances such as 1/3, 1/4, 1/5, 2/3
and 3/4 of the way along its length. By fixing the string at points
along its length that were not a simple fraction, a note is
produced that is not in harmony with the other tones.
History
Water
Earth
Pentagram
Fire
Pythagoras and his followers discovered many patterns and relationships between whole numbers.
Triangular Numbers:
Square Numbers:
Pentagonal Numbers:
Hexagonal Numbers:
1 + 2 + 3 + ...+ n
1 + 3 + 5 + ...+ 2n – 1
1 + 4 + 7 + ...+ 3n – 2
1 + 5 + 9 + ...+ 4n – 3
= n(n + 1)/2
= n2
= n(3n –1)/2
= 2n2-n
These figurate numbers were extended into 3 dimensional space and became
polyhedral numbers. They also studied the properties of many other types of
number such as Abundant, Defective, Perfect and Amicable.
In Pythagorean numerology numbers were assigned characteristics or attributes. Odd numbers were regarded as
male 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 sum
of all possible geometric dimensions. 1 point + 2 points (line) + 3 points (surface) + 4 points (plane)
The Square Root of 2 is Irrational
2
This is a “reductio-ad-absurdum” proof.
To prove that 2 is irrational
Assume the contrary: 2 is rational
1
That is, there exist integers p and q with no common factors such that:
p
 2
q
p2
 2 2
q
 p 2  2 q 2  p is even
(Since 2q2 is even, p2 is even so p even) (odd2 = odd) So p = 2k for some k.
p2
p2
2
Also , as 2  2  q 
 q is even.
q
2
2
p is even, q2 is even so q is even)
(Since p is even
So q = 2m for some m.
2

p
2k
p

 have a factor of 2 in common.
q 2m
q
This contradicts the original assumption.
2 is irrational.
QED
Proof
1