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A rational number is a number that can be written as the quotient of two integers where the divisor is not zero.
An irrational number is a real number that cannot be written as the quotient of two integers where the divisor is not zero.
Any given real number either can or cannot be written as the quotient of two integers. If it can, it is rational. If it cannot, it is irrational. You can't be both at the same time.
The square root of -1 is not a real number and it cannot be written as the quotient of two integers, so it is neither rational nor irrational.
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A number exists that is both a rational and irrational?
No. Irrational means "not rational". A number either is rational, or it is not rational - tercium non datur.
Is 10.01 a rational or irrational number?
10.01 is a rational number
Is 37.5 a rational or irrational number?
Rational
Is the product of a rational number and an irrational number rational or irrational?
Such a product is always irrational - unless the rational number happens to be zero.
The sum of a rational number and an irrational number is?
The sum of a rational and irrational number must be an irrational number.
