3. Write the following rationals in decimal form using Theorem 1.4.
(i)
14/
25
(ii)
15
/
16
(iii) 3 2
23
/
2 .5
(iv) 2 2
7218/
3 .5
(v)
143
110
4. Express the following decimals in the form of
p
/
q
, and write the prime factors of q. What
do you observe?
(i) 43.123 (ii) 0.1201201 (iii) 43.12 (iv) 0.63
Irrational numbers
class IX, you were introduced to irrational numbers and some 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
√2, √3, √5 and √p 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 real number is called irrational if it cannot be written in the form
p
/
q
,
where p and q are integers and q ¹ 0. Some examples of irrational numbers, with which
you are already familiar, are :
√2, √3, √15, π 0.10110111011110…, etc.
Before we prove that √2 is irrational, we will look at a theorem, the proof of
which is based on the Fundamental Theorem of Arithimetic.
Theorem-1.6
Let p be a prime number. If p divides a
2
, (where a is a positive integer),
then p divides a.
Proof : Let the prime factorization of a be as follows :
a = p1 p2
… pn
, wher ep1 p2
, …., pn
are primes, not necessarily distinct.
Therefore a
2
= (p1 p2
… pn
) (p1 p2
… pn
) = p
2
1
p
2
2
… p2
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
p1, p2
,… pn