Question

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, p/q what can you say about the prime factors of q? (i) 43.123456789 (ii) 0.120120012000120000. . .​

Answer 1

Step-by-step explanation:

i) 43.123456789 is terminating or rational

43.123456789=

1000000000

43123456789

=

10

9

43123456789

=

(2×5)

9

43123456789

=

2

9

×5

9

43123456789

This is in the form of

q

p

, and the prime factors of q are in terms of 2 and 5.

ii) 0.120120012000120000.... is non-terminating and non-repeating, it is irrational. Hence cannot be expressed in the form of

q

p

iii) 43.

123456789

is non-terminating but repeating. So, it would be rational.

Let x=43.

123456789

…(1)

1000000000x=43123456789,

123456789

…(2)

(2)−(1)⇒999999999x=43123456746⇒x=

999999999

43123456746

=

3

4

×37

1

×333667

1

43123456746

In a non-termination repeating expansion of

q

p

, q will have factors 3,37,333667.