Question

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

`18.bar(32)`

Answer 1

The number given is `18.bar(32)`, which means the decimal part “32” repeats indefinitely: 18.323232...

This is a recurring decimal, so the number is rational.

Let’s express it as a fraction `p/q` where p and q are co-prime integers and q ≠ 0:

Let `x = 18.323232... = 18.bar(32)`

Since the repeating block is 2 digits long, multiply x by 100:

100x = 1832.323232...

Subtract the original x:

100x – x = 1832.323232... – 18.323232...

100x – x = 1814

99x = 1814

`x = 1814/99`

Now, simplify `1814/99` if possible.

Prime factorise the numerator and denominator:

1814 = 2 × 907

907 is prime

99 = 9 × 11

99 = 3² × 11

No common prime factors, so `1814/99` is in simplest form.

Therefore, `18.bar(32) = 1814/99`.

About the prime factors of q = 99: 99 = 3² × 11.

So the prime factors of the denominator are 3 and 11.