Ex 1.4 , 3 (Method 1)
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , π/π what can you say about the prime factors of q?
43.123456789
43.123456789 is terminating
So, it would be a rational number
43.123456789 = 43123456789/1000000000
= 43123456789/(10)9
= 43123456789/(2 Γ5)9
= 43123456789/(29 Γ59)
Hence 43.123456789 is now in the form of π/π
And the prime factors of q are in terms of 2 and 5
Ex 1.4 , 3 (Method 2)
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , π/π what can you say about the prime factors of q?
43.123456789
43.123456789 is terminating
So, it would be a rational number
In a terminating expansion of π/π,
q is of the form 2n 5m
So, prime factors of q will be 2 or 5 or both only.
Ex 1.4 , 3
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , π/π what can you say about the prime factors of q?
(ii) 0.120120012000120000β¦
0.120120012000120000β¦ is non terminating and non repeating
So, it is not a rational number
Ex 1.4 , 3
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , π/π what can you say about the prime factors of q?
(iii) 43. ("123456789" ) Μ
Here 43. ("123456789" ) Μ is non terminating but repeating.
So, it would be a rational number
In a non- terminating , repeating expansion of π/π,
q will have factors other than 2 or 5.

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.