The whole numbers are defined to be 0 and the so-called counting numbers, or natural numbers 1, 2, 3, and so forth. Negative integers  and  are not included in this set.

By definition, a complex number is a number with an imaginary term denoted as i.

A complex number is in the form,

where  represents the real part of the number and  represents the imaginary portion of the complex number.

Therefore, the complex number which is the solution is .

By the Quotient of Powers Property,

.

Therefore, each element, the square root of a fraction, can be seen to be the fraction of the square roots of the individual parts. Each numerator and denominator is a perfect square, so each square root is a fraction of integers:

By definition, any integer fraction, being a quotient of integers, is a rational number, so all elements in the set are rational.

To raise  to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Every element in the set is equal to  raised to an even-numbered power, so when each exponent is divided by 4, the remainder will be either 0 or 2. Therefore, each element is equal to either 1 or . Consequently, the set includes no imaginary numbers.

An irrational number is any number that cannot be written as a fraction of whole numbers.  The number pi and square roots of non-perfect squares are examples of irrational numbers.  

 can be written as the fraction .  The term  is a whole number.  The square root of  is , also a rational number. , however, is not a perfect square, and its square root, therefore, is irrational.

A rational number is any number that can be expressed as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to this definition is that the denominator cannot be equal to .

Using the above definition, we see  ,  and   (which is ) cannot be expressed as fractions. These are non-terminating numbers that are not repeating, meaning the decimal has no pattern and constantly changes. When a decimal is non-terminating and constantly changes, it cannot be expressed as a fraction.

  is the correct answer because , which can be expressed as , fullfilling our above defintion of a rational number. 

The definition of an irrational number is a number which cannot be expressed in a simple fraction, or a number that is not rational.

Using the above definition, we see that  is already expressed as a simple fraction.

  any number  and

. All of these options can be expressed as simple fractions, making them all rational numbers, and the incorrect answers.

  cannot be expressed as a simple fraction and is equal to a non-terminating, non-repeating (ever-changing) decimal, begining with 

This is an irrational number and our correct answer.

An irrational number  is one that cannot be written as a fraction. All integers are rational numberes.

Repeating decimals are never irrational,  can be eliminated because

.

  and  are perfect squares making them both integers.

Therefore, the only remaining answer is .

Irrational numbers are numbers that can't be expressed as a fracton. This elminates statement III automatically as it's a fraction.

Statement I's fraction is  so this statement is false.

Statement IV. may not be easy to spot but if you let that decimal be  and multiply that by  you will get . This becomes . Subtract it from  and you get an equation of . 

 becomes  which is a fraction.

Statement II can't be expressed as a fraction which makes this the correct answer.

Irrational numbers can't be expressed as a fraction with integer values in the numerator and denominator of the fraction.

Irrational numbers don't have repeating decimals.

Because of that, there is no definite value of irrational numbers.

Therefore,  is irrational because it can't be expressed as a fraction.