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.

Let's take two irrationals like  and multiply them. The answer is  which is rational.

But what if we took the product of  and . We would get  which doesn't have a definite value and can't be expressed as a fraction.

This makes it irrational and therefore, the answer is sometimes irrational, sometimes rational. 

Rational numbers are those which can be written as a ratio of two integers, or simply, as a fraction.

The solution of  is , which can be written as . Each of the other answers would have a solution with an infinite number of decimal points, and therefore cannot be written as a simple ratio. They are irrational numbers. 

An irrational number cannot be represented as the quotient of two integers.

Irrational numbers do not terminate and are not repeat numbers.

Looking at the possible answers,

 can be reduced to , therefore it is an integer.

 by definition is a quotient of two integers and thus it is not an irrational number.

 can be rewritten as  and by definition is a quotient of two integers and thus it is not an irrational number.

 is a terminated decimal and therefore can be written as a fraction. Thus it is not an irrational number.

 is the number for  and does not terminate, therefore it is irrational.

To add the numerator, first multiply the denominator to find the least common denominator.

The common denominator is:  

Rewrite the fractions.

The meaning of irrational states that numbers cannot be rewritten as a ratio of integers. Of the following that could be simplified, the only possible choice of irrational numbers is .

The answer is .

All other options are rational because they can be written as either a fraction of integers or just an integer.

A rational number can be put in the form , it can be a terminating decimal, or it can be a repeating decimal.  is a continual number, therefore it is an irrational number.

An irrational number is any number that cannot be expressed as a ratio of integers.

Therefore,  is considered irrational because it cannot be expressed as a ratio of integers.

An irrational number is a number that cannot be expressed as a ratio of integers and cannot be expressed as terminating or repeating decimals.  

Therefore, the only answer that follows this definition is .

A rational number is any number that can be expressed as a fraction where both the numerator and denominator are integers. The denominator also cannot be equal to 0. In this set, the irrational number is  because the There is no fraction that can be made, it's decimal goes on and on and does not repeat in a pattern. Using the fraction test, we can prove that the following numbers are rational: