Question

Which of the following is always irrational?

the sum of two fractions
the product of a fraction and a repeating decimal
the sum of a terminating decimal and the square root of a perfect square
the product of a repeating decimal and the square root of a non-perfect square

Answer 1

The product of a repeating decimal (non-zero) and the square root of a non-perfect square would always be irrational.

Explanation:

The sum and product between rational numbers are rational.

Fractions, finite decimals, and infinite repeating decimals are all rational numbers.

The square root of a perfect square is a rational number. However, the square root of a non-perfect square is not rational.

Let
`x` denote a repeating (non-zero) decimal (
`x \\e 0`). Let and
`y` denote the square root of a non-perfect square. Note that
`x\!` would be a rational number, but
`y` would not be rational.

Assume the product
`x\, y` to be rational by contradiction. Since
`x \\e 0` and
`x` is a rational number, the multiplicative inverse
`(1/x)` of
`x\!` would be a rational number.

Left-multiply
`x\, y` by this multiplicative inverse to obtain:
`(1/x) \, (x\, y) = ((1/x)\, x)\, y = y`.

Since
`(1/x)` is a rational number, and
`x\, y` is assumed to be rational, the product
`(1/x) \, (x\, y) = y` should also be rational. This observation is a contradiction with the assumption that
`y` is not rational.