No. There are numbers that are beyond real numbers, such as complex numbers which are neither rational nor irrational.
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Integers are rational. In the set of real numbers, every number is either rational or irrational; a number can't be both or neither.
Next to any rational number is an irrational number, but next to an irrational number can be either a rational number or an irrational number, but it is infinitely more likely to be an irrational number (as between any two rational numbers are an infinity of irrational numbers).
It is usually irrational but it can be rational if the ration number in the pair is zero. So the correct answer is "either".
The square of a rational number can be either rational or irrational. However, the square of an irrational number is always irrational.
No, it is not. And it is not irrational either.