A square whose area is 25 square units has a side length of $\sqrt{25}$ units, which means that $\sqrt{25} \boldcdot \sqrt{25} = 25$. Since $5 \boldcdot 5 = 25$, we know that $\sqrt{25}=5$.

$\sqrt{25}$ is an example of a rational number. A rational number is a fraction or its opposite. In an earlier grade we learned that $\frac{a}{b}$ is a point on the number line found by dividing the interval from 0 to 1 into $b$ equal parts and finding the point that is $a$ of them to the right of 0. We can always write a fraction in the form $\frac{a}{b}$, where $a$ and $b$ are integers (and $b$ is not 0), but there are other ways to write them. For example, we can write $\sqrt{25}=\frac51=5$ or $\text-\frac{1}{\sqrt{4}} = \text-\frac{1}{2}$. 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 $\sqrt{2}$ units. This means that$\sqrt{2} \boldcdot \sqrt{2} = 2$.

An irrational number is a number that is not rational, meaning it cannot be expressed as a positive or negative fraction. For example,
$\sqrt{2}$ has a location on the number line (it’s a tiny bit to the right of $\frac75$),
but its location can not be found by dividing the segment from 0 to 1 into $b$ equal parts and going $a$ 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.

$\frac{17}{12}$ is close to $\sqrt{2}$ because $\left( \frac{17}{12} \right)^2=\frac{289}{144}$, which is very close to 2 since $\frac{288}{144}=2$. We could keep looking forever for rational numbers that are solutions to $x^2=2$, and we would not find any since $\sqrt{2}$ is an irrational number.

The square root of any whole number is either a whole number, like $\sqrt{36}=6$ or $\sqrt{64}=8$, or an irrational number. Here are some examples of irrational numbers: $\sqrt{10}, \text{ -}\sqrt3, \text{ }\frac{\sqrt5}{2},\text{ } \pi$.