An irrational number is defined as any number that cannot be expressed as a simple fraction or does not have terminating or repeating decimals. Of the answer choices given, the only number that cannot be expressed as a simple fraction or with repeating or terminating decimals is .

An irrational number is any number that can not be expressed as a ratio of integers, i.e. a fraction. Therefore, the only irrational number listed is .

The square root of an integer is either an irrational number or an integer. The latter is the case if and only if there is an integer which, when multiplied by itself, or squared, yields the number inside the symbol (the radicand) as the product. Of , only 81 is the square of an integer (9).

Since , 

.

We can determine which is closer by evaluating .

Since , 9 is the closer integer, and it is the correct choice.

Pi is the only irrational number listed. Irrational numbers are in the form of infinite non-repeating decimals. 

A root of an integer is one of two things, an integer or an irrational number. By testing all five on a calculator, only  comes up an exact integer - 5. This is the correct choice.

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 .

A rational number can be expressed as a fraction of integers, while an irrational number cannot.  

can be written as .   

 is simply , which is a rational number.  

The number  can be rewritten as a fraction of whole numbers, , which makes it a rational number.  

is also a rational number because it is a ratio of whole numbers.  

The number, , on the other hand, is irrational, since it has an irregular sequence of numbers (...) that cannot be written as a fraction.

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.