A square whose area is 25 square units has a side length of units, which means that . Since , we know that .

is an example of a rational number. A rational number is a fraction or its opposite. In an earlier grade we learned that is a point on the number line found by dividing the interval from 0 to 1 into equal parts and finding the point that is of them to the right of 0. We can always write a fraction in the form , where and are integers (and is not 0), but there are other ways to write them. For example, we can write or . Because fractions and ratios are closely related ideas, fractions and their opposites are called rational numbers.

Now consider a square whose area is 2 square units with a side length of  units. This means that .

An irrational number is a number that is not rational, meaning it cannot be expressed as a positive or negative fraction. For example,
has a location on the number line (it’s a tiny bit to the right of ),
but its location can not be found by dividing the segment from 0 to 1 into  equal parts and going of those parts away from 0.

A number line with 10 evenly spaced tick marks. The first tick mark is labeled 0 and the sixth tick mark is labeled 1. An arrow points to the eighth tick mark and is labeled seven-fifths. A second arrow points to a point slightly to the right of the eighth tick mark and is labeled the square root of 2.

is close to because , which is very close to 2 since . We could keep looking forever for rational numbers that are solutions to , and we would not find any since is an irrational number.

The square root of any whole number is either a whole number, like or , or an irrational number. Here are some examples of irrational numbers: .