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Prove or disprove that the product of a nonzero rational number and an irrational number is irrational.

User Anton Nakonechnyi
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1 Answer

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9 votes

Answer: The claim is true

The proof is below

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A = some nonzero rational number

B = some irrational number

Because value A is rational, this means A = p/q where p,q are nonzero integers.

Let's do a proof by contradiction. The claim is that A*B is irrational, but let's do the opposite and assume A*B is rational. We should get to a contradiction somewhere based on this assumption.

If A*B is rational, then A*B = r/s for some integers r and s

Then we can have the following steps

A*B = r/s

(p/q)*B = r/s

B = (r/s)*(q/p)

B = (rq)/(sp)

B = (some integer)/(some other nonzero integer)

B = some rational number

Recall we made B irrational. There's no way to have B rational and irrational at the same time. This is where the contradiction happens. This causes the original assumption of "A*B is rational" to be false. Therefore, the opposite must be the case and A*B has been proven to be irrational.

Therefore, the product of a rational number and an irrational number is irrational.

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Example:

5 is rational

sqrt(3) is irrational

5*sqrt(3) is irrational

User Loomer
by
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