`
To prove 2=1
Let a  b
a 2  ab Multiply across by a
a 2  b 2  ab  b 2 Subtract the same thing from both sides
(a-b)(a+b)=b(a-b) Factorize.
a+b=b Divide across by the same quantity.
Now let a  b  1
So 1+1=1
2=1.
Obviously there is something wrong. I have done some operation that is completely
Incorrect. Can anyone spot it?
A short history of numbers
Ancient man started with natural numbers1,2,3,etc -the positive integers and these arose
from idea of counting items such as people ,animals etc. The next numbers that arose
1 17 25 19 7 101
naturally were the positive rationals or positive fractions eg , , , ,
.These
2 34 50 22 8 19
arose from commerce from measuring quantities i.e. amount of land a farmer had and
measuring out foodstuffs. The next numbers were the negative integers and negative
201
567
and rationals e.g.  12,76,
that arose from the idea of dept. Then we had one
97
43
of the great discoveries in mathematics namely 0(zero) by the indians in India about 500
BC. The idea that you could represent nothing by a symbol 0 that obeyed the laws of
arthimetic like other symbols for numbers say 3,5 and 8 was a great achievemen if you
think that you would not be able to do much mathematics without 0. ( The first
representation of zero by 0 first appeared in a temple in central India).
Now all these numbers can be represented pictorially on a straight line as follows
________________________________________________________________________
-10
-3
-5/2 -2
-1 -1/2 0 1/2 1
2 5/2 3
4
21/5
7
However there was a great crises in mathematics when the ancient Greeks discovered
irrational numbers about 250 BC. The ancient Greeks were great mathematicians.
The outstanding contribution of Greek mathematics was the introduction, for the first
time, of deductive reasoning and the concept of proof in mathematics. Euclid’s
geometry was the supreme example of that and it was a major influence in mathematics
for 2000 years. No other civilization including the Indian civilization, achieved this in
their mathematics. The concept of proof, nowadays an essential part of mathematics, is
what makes the theorems and results of mathematics true for all time.
Remember a rational number is a ratio of two integers
m
where m and n are integers
n
41 9
12
, ,12 
.
71 7
1
Note integers are considered rational numbers as they ca be put over 1.
e.g.
Now Pythagoras proved by one of the great proofs in mathematics, namely the proof by
contradiction, that 2 was an irrational number that is, could not be expressed as a
m
rational number, that is 2  where m and n are integers (which were the only
n
numbers you could do calculations with).Now this led to a crisis in mathematics and
because of this the Greeks did not trust numbers and consequently preferred to do their
mathematics using geometry instead of numbers. This some people felt that it held back
the development of mathematics by 500 years. Not only was 2 irrational but there was
an infinite number of irrational numbers i.e. all numbers that were not perfect squares i.e.
1 1 1
3 , 5 , 7 ,  and their reciprocals
,
,
, .This is a serious from a computing
2 3 5
point of view because who can we calculate irrational numbers?. This was answered by
mathematicians who showed that although we could not calculate irrational numbers
exactly we could approximate them as closely as we wished ,which is sufficient for
practical purposes
The major contribution of Indian mathematics was their number system, which is
used throughout the world to-day. The Indian mathematicians were the first to introduce
the symbol for zero, namely 0 in their decimal place value number system. This enabled
them(unlike their Greek counterparts) to represent very large and very small numbers
with a very small number of symbols i.e using just the ten symbols 0,1,2,3,4,5,6,7,8,9.
Not only that, but the Indian mathematician Brahmagupta [589-668 AD] for the first time
introduced negative numbers and gave definite rules for manipulating negative , positive
numbers and 0 arithmetically. Indeed he regarded numbers in a very modern way,
thinking of them as abstract symbols that obeyed certain rules. This was a major
development in the history of mathematics as it led to greater ease in computation and
rapid progress in mathematics compared with other number systems such as the Roman
number system which it replaced in Europe only in the 14 th century (brought into Europe
by the Muslims who saved a lot of the Greek mathematics for posterity).
Now the rational numbers and the irrational numbers together make up was known as the
real number system. The real numbers satisfy the usual arithmetical operations of
addition, subtraction, multiplication and division and can be ordered and represented on
a straight line called the real line thus
________________________________________________________________________
-10   -3
-5/2 -2
-1 -1/2 0 1/2 1 2 2
3 
4
7
There is a one-to-one correspondence between points on the line and real numbers.
For every real number there is a point on the line and for every point on the line there is a
real number.
Now in mathematics since certain sets of numbers are co important they are given special
symbols to represent them
N  1,2,3,--the set of all natural numbers.
Z    3,2,1,0,1,2,3,--the set of all integers


R    4, ,3,2, 2 ,1,0,1, 2 ,3,4.56, --the set of all real numbers.
The decimal system The system enables us to represent all numbers both large and small
by only ten digits 0,1,2,3,4,5,6,7,8,9.
Definition A decimal is an expression of the form
 a0 .a1a 2 a3 ,
where a 0 is a non-negative integer and a n is a digit for each n  N (for n belonging to
N)
e.g. -25.87654.
If only a finite number of digits a1 , a2 ,, are non-zero, then the decimal is a terminating
or finite decimal, and we usually omit the tail of zeros.
Terminating decimals are used to represent rational numbers as follows
a
a
a
 a0 .a1a2 a3 an  (a0  1  22    nn
10 10
10
2
3
4
5
7
9
e.g.-56.234579=  56   2  3  4  5  6 .
10 10
10 10
10 10
However if we apply long division to other rationales, then the process of long division
never terminates, and we obtain a non-terminating or infinite decimal.
1
19
For example gives 0.33333 and
gives 0.86363
3
22
The infinite decimals which we obtain by applying the long division process have a
common property. All of them are recurring decimals that is they have e repeating block
of decimals so they can be written in shorthand as follows;
1
 0. 3 or 0.3
 0.3333
3
0.142857142857  0. 1 4 2 8 5 7 or 0.1 42857
The reason why we get a decimal recurring when we apply the long division process to a
p
fraction
is that there are only q possible remainders at each stage of the division, so
q
one of these remainders must eventually repeat. . When this happens, the block of digits
obtained after the first occurrence of this remainder will be repeated infinitely often.
as shown below.
.86363
22 19.00000
176
___
140
132
_____
80
66
___
140
132
____
80
66
___
140
If the remainder 0 occurs, then the resulting decimal is finite that is ,it ends in recurring
0s.Infinite recurring decimals which arise from the long division of fractions are used to
represent the corresponding ration numbers.
1
This representation is not quite so straightforward as for finite decimals(  0.25 ).For
4
example the statement
1
3
3
3
 0. 3 
 2  3   is justified by using the concept of a convergent series.
3
10 10
10
3
1
[1  ( ) n ]
1
3
3
3
3
3
3
3
10 GP


    2  3    n  10
(  0. 3 
1
3
10 10 2 10 3
10 10 10
10
1
10
3
1
1
1
1
 [1  ( ) n ] then as n   ( ) n  0 so  [1  0]  .)
9
10
10
3
3
The decimal representation of rational numbers has the advantage it enables us to decide
immediately which of two distinct positive rationals is the greater.(We could do
7 19 7  11  19  4 77  76 1
7 19




so 
) We only examine their decimal
8 22
88
88
88
8 22
7
19
0.86363 Then 0.875>0.86363.
representations  0.875 and
8
22
Irrational numbers in their decimal representation. All irrational numbers are
represented by non-recurring decimals and it is this that distinguishes them from the
rational numbers.
Thus
2  1.41421356 and   3.14159265 
They will always have an infinite number of decimal points and so cannot be even
stored on a computer exactly since it would take an infinite number of storage places
Together, the rational numbers (recurring decimals) and the irrational numbers(nonrecurring decimals) form the set of real numbers, denoted by R .Now the reals (both
rational and irrational numbers obey the same rules of arithmetic as given below
Decimals which end in recurring 9s sometimes arise as alternative representations for
terminating decimals. For example
1  0. 9  0.9999
and 1.35  1.34 9  1.3499999 
This may seem strange , but it is important to realize this representation is a matter of
definition. We wish to allow the decimal 0.999 to represent a number x so x must
be less than or equal to 1 and greater than each of the numbers 0.9,0.99,0.999,0.9999, .
The only rational with these properties is 1.
Laws of Arithmetic in R
Addition
Multiplication
A1 If a, b  R, then
a bR
M1 If a, b  R,
ab R
closure
A2 If a  R, then
a0  0a  a
M2 If a  R, then
a  1  1 a  a
identity
A3 If a  R , then there is a
M3 If a  R  0, then there is a number
1
a 1   R such that inverses
a
a  a 1  a 1  a  1 .
number  a  R such that
a   a   a  a  0
A4 If a, b, c  R ,then
(a+b )+c= a+(b+c)
M4 If a, b, c  R , then associativity
(a  b)  c  a  (b  c)
A5 If a, b  R , then
ab  ba
M5 If a, b  R , then
ab  b a
D. If a, b, c  R ,then a  (b  c)  a  b  a  c
commutativity
distributivity
Note1 R  0means all the real numbers except zero.
Note 2 Why is (1)  (1)  1? Answer because of the distributive law.
Now If a, b, c  R ,then a  (b  c)  a  b  a  c This holds for all real numbers
including the integers. In particular if a=-1,b=1 and c=-1 then by the distributive law
 1  (1  (1))  (( 1)  1)  (( 1)  (1))
 (1)  (( 1)  (1)) .
0
Add one to each side and we get
1  1  (1)  (( 1)  (1))
1  0  (( 1)  (1))
Hence (1)  (1)  1 must be true for the distributive law to hold and this is why
(1)  (1)  1 .
Note 3 It can be proved that real numbers satisfy the laws of arithmetic that is
he terminating and recurring decimals(the rationals )and the non-recurring decimals (the
irrationals both satisfy the laws of arithmetic that is 2   makes sense(even though
they both need an infinite number of decimals to represent them.
Exercise Order 17 / 20 and 45/53 by (i) taking a common denominator and
(ii) by finding their decimal representation.
Summary
In summary you should remember that the real number system consists of rational
301
numbers e.g.
(which are the only numbers we compute with) and irrational
199
numbers e.g. 2 which cannot be computed exactly since they require an infinite
number of decimal points. However mathematicians proved that you could approximate
an irrational number to any degree of accuracy that you wished and so from a practical
point could be compute with them just like rational numbers. Thus the fears of the
Ancient Greeks of using irrational numbers were unfounded.
When the real numbers are expressed in decimals the difference between rational and
irrational numbers are clearly shown as follows
Rational numbers will either be finite that is a finite number of decimal points
17
19
 0.86363636363
e,g,  2.125 or have a recurring number of decimal points e.g.
8
22
Irrational numbers will always have an infinite number of decimal places with no
recurring pattern and so cannot be stored exactly on a computer.
e.g. 2  1.41421356 infinite no. of decimals with no pattern and
  3.14159265  infinite no. of decimals with no pattern.
The advantage of expressing real numbers in decimal form is that you can see
immediately which of two numbers is the greater.