Question

The $43.12345678$ real numbers have decimal expansions. Decide whether it is rational or not. If it is rational, and of the form, p q what can you say about the prime factors of $\mathrm{q}$.

Answer 1

Rational numbers are the numbers that can be represented in $\frac{\mathrm{p}}{\mathrm{q}}$ form where $\mathrm{p}$ and $\mathrm{q}$ are integers and $\mathrm{q}\ne 0$.

Rational numbers are terminating decimals but irrational numbers are non-terminating and non-recurring.

$43.123456789$ has a terminating decimal expansion so it's a rational number.

$43.123456789$$=\frac{43123456789}{1000000000}$$=\frac{43123456789}{{10}^9}=\frac{43123456789}{{\left(2\times 5\right)}^9}=\frac{43123456789}{2^9\times 5^9}$

Hence, it is a rational number in the form of $\frac{\mathrm{p}}{\mathrm{q}}$, since the prime factors of $\mathrm{q}$ are in terms of $2$ and $5$.