Answer – The square root of 16 belongs to the numbers sets of N, W, Z and Q, while the square root of 17 belongs to the Q’ set. Both numbers also belong to set R.

Explanation:

All numbers fall into the following sets based on their characteristics.

• Natural numbers (N): These are also called ‘counting numbers’, and begin at 1 and go on to infinity. 
• Whole numbers (W): The set of whole numbers includes all natural numbers along with zero.
• Integers (Z): Integers are all whole numbers with the addition of negative numbers (those below zero).
• Rational numbers (Q): This set includes all numbers that can be expressed as fractions where the numerator and the denominator are both integers, or as ratios of two integers. However, the denominator cannot be equal to zero. 
• Irrational numbers (Q’ or “q prime”): These are numbers that cannot be expressed as fractions or ratios. As numerals, they are written in decimals with the digits after the decimal point going on indefinitely. 
• Real numbers (R): This set is the combination of all rational and irrational numbers. Between the two of them, the sets of rational and irrational numbers comprise all possible numbers. Therefore, the set of real numbers is composed of all numbers that can be found in the real world.

Using the list above as a guide, we can determine the number sets that $\sqrt{16}$ and $\sqrt{17}$ belong to.

Since $\sqrt{16}=\pm 4$, it falls into several number sets. +4 is a natural number (N) and a whole number (W). On the other hand both, +4 and -4, are included in sets Z and Q as well.

$\sqrt{17}$=4.1231056256…… As the fractional digits (numbers to the right of the decimal point) go on indefinitely, $\sqrt{17}$ is an irrational number (Q’). 

Both the square roots of 17 and 16 are real numbers.

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