Square numbers
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Although "perfect square numbers" is more descriptive.
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perfect squares
They are squares of rational numbers. there is no particular name for them.
No. Perfect squares as the squares of the integers, whereas irrational squares as the squares of irrational numbers, but some irrational numbers squared are whole numbers, eg √2 (an irrational number) squared is a whole number.
Integers
They are called irrational numbers; numbers that can be expressed as a ratio of integers are called rational numbers. Some common irrational numbers are pi (3.14159...) and the square root of two.