REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way—this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2, 3
and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say
(0)
p
q
q
, is terminating and when it is non-
terminating repeating. We do so by looking at the prime factorisation of the denominator
q of
p
q
. You will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
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2 MATHEMATICS
1.2 The Fundamental Theorem of Arithmetic
In your earlier classes, you have seen that any natural number can be written as a
product of its prime factors. For instance, 2 = 2, 4 = 2 × 2, 253 = 11 × 23, and so on.
Now, let us try and look at natural numbers from the other direction. That is, can any
natural number be obtained by multiplying prime numbers? Let us see.
Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply
some or all of these numbers, allowing them to repeat as many times as we wish,
we can produce a large collection of positive integers (In fact, infinitely many).
Let us list a few :
7 × 11 × 23 = 1771 3 × 7 × 11 × 23 = 5313
2 × 3 × 7 × 11 × 23 = 10626 2
3
× 3 × 7
3
= 8232
2
2
× 3 × 7 × 11 × 23 = 21252
and so on.
Now, let us suppose your collection of primes includes all the possible primes.
What is your guess about the size of this collection? Does it contain only a finite
number of integers, or infinitely many? Infact, there are infinitely many primes. So,
if we combine all these primes in all possible ways, we will get an infinite
collection of numbers, all
the primes and all possible
products of primes. The
question is – can we
produce all the composite
numbers this way? What
do you think? Do you
think that there may be a
composite number which
is not the product of
powers of primes?
Before we answer this,
let us factorise positive
integers, that is, do the
opposite of what we have
done so far.
We are going to use
the factor tree with which
you are all familiar. Let us
take some large number,
say, 32760, and factorise
it as shown.
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Carl Friedrich Gauss
(1777 – 1855)
An equivalent version of Theorem 1.2 was probably
first recorded as Proposition 14 of Book IX in Euclid’s
Elements, before it came to be known as the
Fundamental Theorem of Arithmetic. However, the
first correct proof was given by Carl Friedrich Gauss
in his Disquisitiones Arithmeticae.
Carl Friedrich Gauss is often referred to as the ‘Prince
of Mathematicians’ and is considered one of the three
greatest mathematicians of all time, along with
Archimedes and Newton. He has made fundamental
contributions to both mathematics and science.
So we have factorised 32760 as 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 as a product of
primes, i.e., 32760 = 2
3
× 3
2
× 5 × 7 × 13 as a product of powers of primes. Let us try
another number, say, 123456789. This can be written as 3
2
× 3803 × 3607. Of course,
you have to check that 3803 and 3607 are primes! (Try it out for several other natural
numbers yourself.) This leads us to a conjecture that every composite number can be
written as the product of powers of primes. In fact, this statement is true, and is called
the Fundamental Theorem of Arithmetic because of its basic crucial importance
to the study of integers. Let us now formally state this theorem.
Theorem 1.1 (Fundamental Theorem of Arithmetic) : Every composite
number can be expressed (factorised) as a product of primes, and this factorisation
is unique, apart from the order in which the prime factors occur.
The Fundamental Theorem of Arithmetic says that every composite number can
be factorised as a product of primes. Actually it says more. It says that given any
composite number it can be factorised as a product of prime numbers in a ‘unique’
way, except for the order in which the primes occur. That is, given any composite
number there is one and only one way to write it as a product of primes, as long as we
are not particular about the order in which the primes occur. So, for example, we
regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which
these primes are written. This fact is also stated in the following form:
The prime factorisation of a natural number is unique, except for the order
of its factors.
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4MATHEMATICS
In general, given a composite number x, we factorise it as x = p
1
p
2
... p
n
, where
p
1
, p
2
,..., p
n
are primes and written in ascending order, i.e., p
1
p
2
. . . p
n
. If we combine the same primes, we will get powers of primes. For example,
32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2
3
× 3
2
× 5 × 7 × 13
Once we have decided that the order will be ascending, then the way the number
is factorised, is unique.
The Fundamental Theorem of Arithmetic has many applications, both within
mathematics and in other fields. Let us look at some examples.
Example 1 : Consider the numbers 4
n
, where n is a natural number. Check whether
there is any value of n for which 4
n
ends with the digit zero.
Solution : If the number 4
n
, for any n, were to end with the digit zero, then it would be
divisible by 5. That is, the prime factorisation of 4
n
would contain the prime 5. This is
not possible because 4
n
= (2)
2n
; so the only prime in the factorisation of 4
n
is 2. So, the
uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no
other primes in the factorisation of 4
n
. So, there is no natural number n for which 4
n
ends with the digit zero.
You have already learnt how to find the HCF and LCM of two positive integers
using the Fundamental Theorem of Arithmetic in earlier classes, without realising it!
This method is also called the prime factorisation method. Let us recall this method
through an example.
Example 2 : Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Solution : We have : 6 = 2
1
× 3
1
and 20 = 2 × 2 × 5 = 2
2
× 5
1
.
You can find HCF(6, 20) = 2 and LCM(6, 20) = 2 × 2 × 3 × 5 = 60, as done in your
earlier classes.
Note that HCF(6, 20) = 2
1
= Product of the smallest power of each common
prime factor in the numbers.
LCM (6, 20) = 2
2
× 3
1
× 5
1
=Product of the greatest power of each prime factor,
involved in the numbers.
From the example above, you might have noticed that HCF(6, 20) × LCM(6, 20)
= 6 × 20. In fact, we can verify that for any two positive integers a and b,
HCF (a, b ) × LCM (a, b) = a × b. We can use this result to find the LCM of two
positive integers, if we have already found the HCF of the two positive integers.
Example 3: Find the HCF of 96 and 404 by the prime factorisation method. Hence,
find their LCM.
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Solution : The prime factorisation of 96 and 404 gives :
96 = 2
5
× 3, 404 = 2
2
× 101
Therefore, the HCF of these two integers is 2
2
= 4.
Also, LCM (96, 404) =
96 404 96 404
9696
HCF(96, 404) 4
Example 4 : Find the HCF and LCM of 6, 72 and 120, using the prime factorisation
method.
Solution : We have :
6 = 2 × 3, 72 = 2
3
× 3
2
, 120 = 2
3
× 3 × 5
Here, 2
1
and 3
1
are the smallest powers of the common factors 2 and 3, respectively.
So, HCF (6, 72, 120) = 2
1
× 3
1
= 2 × 3 = 6
2
3
, 3
2
and 5
1
are the greatest powers of the prime factors 2, 3 and 5 respectively
involved in the three numbers.
So, LCM (6, 72, 120) = 2
3
× 3
2
× 5
1
= 360
Remark : Notice, 6 × 72 × 120 HCF (6, 72, 120) × LCM (6, 72, 120). So, the
product of three numbers is not equal to the product of their HCF and LCM.
EXERCISE 1.1
1. Express each number as a product of its prime factors:
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF =
product of the two numbers.
(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
3. Find the LCM and HCF of the following integers by applying the prime factorisation
method.
(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
4. Given that HCF (306, 657) = 9, find LCM (306, 657).
5. Check whether 6
n
can end with the digit 0 for any natural number n.
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round
of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the
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6 MATHEMATICS
* Not from the examination point of view.
same point and at the same time, and go in the same direction. After how many minutes
will they meet again at the starting point?
1.3 Revisiting Irrational Numbers
In Class IX, you were introduced to irrational numbers and many of their properties.
You studied about their existence and how the rationals and the irrationals together
made up the real numbers. You even studied how to locate irrationals on the number
line. However, we did not prove that they were irrationals. In this section, we will
prove that
and, in general,
is irrational, where p is a prime. One of
the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic.
Recall, a number ‘s’ is called irrational if it cannot be written in the form
where p and q are integers and q ¹ 0. Some examples of irrational numbers, with
which you are already familiar, are :
, etc.
Before we prove that
is irrational, we need the following theorem, whose
proof is based on the Fundamental Theorem of Arithmetic.
Theorem 1.2 : Let p be a prime number. If p divides a
2
, then p divides a, where
a is a positive integer.
*Proof : Let the prime factorisation of a be as follows :
a = p
1
p
2
. . . p
n
, where p
1
,p
2
, . . ., p
n
are primes, not necessarily distinct.
Therefore, a
2
= (p
1
p
2
. . . p
n
)(p
1
p
2
. . . p
n
) = p
2
1
p
2
2
. . . p
2
n
.
Now, we are given that p divides a
2
. Therefore, from the Fundamental Theorem of
Arithmetic, it follows that p is one of the prime factors of a
2
. However, using the
uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only
prime factors of a
2
are p
1
, p
2
, . . ., p
n
. So p is one of p
1
, p
2
, . . ., p
n
.
Now, since a = p
1
p
2
. . . p
n
, p divides a.
We are now ready to give a proof that
is irrational.
The proof is based on a technique called ‘proof by contradiction’. (This technique is
discussed in some detail in Appendix 1).
Theorem 1.3 :
is irrational.
Proof : Let us assume, to the contrary, that
is rational.
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So, we can find integers r and s (¹ 0) such that
=
.
Suppose r and s have a common factor other than 1. Then, we divide by the common
factor to get
where a and b are coprime.
So, b
= a.
Squaring on both sides and rearranging, we get 2b
2
= a
2
. Therefore, 2 divides a
2
.
Now, by Theorem 1.3, it follows that 2 divides a.
So, we can write a = 2c for some integer c.
Substituting for a, we get 2b
2
= 4c
2
, that is, b
2
= 2c
2
.
This means that 2 divides b
2
, and so 2 divides b (again using Theorem 1.3 with p = 2).
Therefore, a and b have at least 2 as a common factor.
But this contradicts the fact that a and b have no common factors other than 1.
This contradiction has arisen because of our incorrect assumption that
is rational.
So, we conclude that
is irrational.
Example 5 : Prove that
is irrational.
Solution : Let us assume, to the contrary, that
is rational.
That is, we can find integers a and b (¹ 0) such that
=
Suppose a and b have a common factor other than 1, then we can divide by the
common factor, and assume that a and b are coprime.
So,
Squaring on both sides, and rearranging, we get 3b
2
= a
2
.
Therefore, a
2
is divisible by 3, and by Theorem 1.3, it follows that a is also divisible
by 3.
So, we can write a = 3c for some integer c.
Substituting for a, we get 3b
2
= 9c
2
, that is, b
2
= 3c
2
.
This means that b
2
is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3
with p = 3).
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8MATHEMATICS
Therefore, a and b have at least 3 as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction has arisen because of our incorrect assumption that
3
is rational.
So, we conclude that
3
is irrational.
In Class IX, we mentioned that :
the sum or difference of a rational and an irrational number is irrational and
the product and quotient of a non-zero rational and irrational number is
irrational.
We prove some particular cases here.
Example 6 : Show that
5– 3
is irrational.
Solution : Let us assume, to the contrary, that
5– 3
is rational.
That is, we can find coprime a and b (b 0) such that
53
a
b
Therefore,
53
a
b
Rearranging this equation, we get
5
35–
aba
bb
Since a and b are integers, we get
5–
a
b
is rational, and so
3
is rational.
But this contradicts the fact that
3
is irrational.
This contradiction has arisen because of our incorrect assumption that 5 –
3
is
rational.
So, we conclude that
53
is irrational.
Example 7 : Show that
32
is irrational.
Solution : Let us assume, to the contrary, that
32
is rational.
That is, we can find coprime a and b (b 0) such that
32
a
b
Rearranging, we get
2
3
a
b
Since 3, a and b are integers,
3
a
b
is rational, and so
2
is rational.
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REAL NUMBERS 9
But this contradicts the fact that
2
is irrational.
So, we conclude that
32
is irrational.
EXERCISE 1.2
1. Prove that
5
is irrational.
2. Prove that
325
is irrational.
3. Prove that the following are irrationals :
(i)
1
2
(ii)
75
(iii)
62
1.4 Summary
In this chapter, you have studied the following points:
1. The Fundamental Theorem of Arithmetic :
Every composite number can be expressed (factorised) as a product of primes, and this
factorisation is unique, apart from the order in which the prime factors occur.
2. If p is a prime and p divides a
2
, then p divides a, where a is a positive integer.
3. To prove that
2, 3
are irrationals.
A NOTE TO THE READER
You have seen that :
HCF (p, q, r) × LCM (p, q, r) p × q × r, where p, q, r are positive integers
(see Example 8). However, the following results hold good for three numbers
p, q and r :
LCM (p, q, r) =
HCF( , , )
HCF( , ) HCF( , ) HCF( , )
pqr pqr
p q qr pr
HCF (p, q, r) =
LCM( , , )
LCM( , ) LCM( , ) LCM( , )
pqr pqr
pq qr pr
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