e, √8, pi, √10
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Irrational numbers are infinitely dense. Between any two numbers, there are infinitely many irrational numbers. So if it was claimed that some irrational, x, was the closest irrational to 6, it is possible to find an infinite number of irrationals between 6 and x. Each one of these infinite number of irrationals would be closer to 6 than x. So the search for the nearest irrational must fail.
The given four numbers are all rational numbers
Some square roots of whole numbers are integers, some of them are irrational numbers. The square root of four for instance is a rational number, 2. The square root of two however is an irrational number, approximately 1.414211356.
The square root of four is a real number because it is a non-negative number that can be expressed on the number line. It is also a rational number because it can be expressed as the fraction 2/1. In fact, it is both a real number and a rational number, as all rational numbers are also real numbers.
It is {sqrt(2), sqrt(3.7), pi, and e}.