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?

15.764

Answer 1

The number 15.764 is a terminating decimal number, so it is a rational number.

To express it as a fraction in the form `p/q` where p and q are co-prime integers and q ≠ 0:

`15.764 = 15764/1000`

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD):

`15764/1000 = 3941/250`

So, `15.764 = 3941/250`

Here, p = 3941 and q = 250.

Factorise the denominator q = 250:

250 = 2 × 5³

Thus, the denominator q in simplest form has prime factors only 2 and 5.

In general, a terminating decimal expressed as a fraction `p/q` will have a denominator q whose prime factors are only 2 and/or 5.

Therefore,

15.764 is rational.

Its simplest fractional form is `3941/250`.

The prime factors of the denominator are only 2 and 5.