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Yes. The proof is easy. Let x be the irrational number and assume there exists some rational number r = a/b where a and b are integers (that's what it means to be rational).
Now suppose x/r is a rational number. Then x/r = (b/a)x = c/d where c and d are some other integers.
Since (b/a)x=c/d, then
x = bd/ac
which means that x itself is rational, but we assumed it was irrational.
The contradiction proves that the assertion is wrong. An irrational divided by a rational must be irrational.
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Is an irrational number divided by an irrational number always a rational?
No. sqrt(2)/pi is not rational.
When an irrational number is divided by a rational number is the quotient rational or irrational?
Irrational.
Is the product of a rational number and irrational number always rational?
No.A rational times an irrational is never rational. It is always irrational.
Can An irrational number divided by an irrational number equals a rational number?
Yes. Any irrational number can be divided by itself to produce 1, which is a rational number.
Is the sum of a rational and irrational number rational or irrational?
It is always irrational.
