Why is the sum of two rational numbers always rational?

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Let ab and cd represent two rational numbers. This means a, b, c, and d are, and. The sum of the numbers is ad+bcbd, where bd is not 0. Because integers are closed under, the sum is the ratio of two integers, making it a rational number.

Question

Callahan

Mathematics

3 July, 12:25

Why is the sum of two rational numbers always rational?

Select from the drop-down menus to correctly complete the proof.

Let ab and cd represent two rational numbers. This means a, b, c, and d are, and. The sum of the numbers is ad+bcbd, where bd is not 0. Because integers are closed under, the sum is the ratio of two integers, making it a rational number.

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Answers (1)

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Lara Forbes · Answer 1

1) A number is rational if it can be formed as the ratio of two integer numbers:

m = p/q where p and q are integers.

2) then a/b is a rational if a and b are integers, and c/d is rational if c and d are integers.

3) the sum a/b + c/d = [ad + cb] / (cd)

then given that the integers are closed under the product ad, cb and cd are integers, so the sum ad + cb is also an integer.

So, it has been proved that the result is also the ratio of two integer numbers which is a rational number.