You can divide 65 by rationals and irrationals:
Divided by a rational: 65 ÷ 13/2 = 10
Divided by an irrational: 65 ÷ √13 = 5√13
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The answer depends on the variable a. 67/a is rational if a is non-zero rational, and it is irrational is a is irrational.
Rational.
Irrational.
Yes. Any irrational number can be divided by itself to produce 1, which is a rational number.
No. sqrt(2)/pi is not rational.
It is irrational - unless the divisor is 0 in which case the division is not defined.