Here is a line segment on a grid. How can we determine the length of this line segment?

By drawing some circles, we can tell that it’s longer than 2 units, but shorter than 3 units.

Two circles that have the same center are drawn on a square grid with radii 2 and 3. A line segment slanted upward and to the left is drawn such that the bottom endpoint is the center of the two circles and is 1 unit down and 2 units right of the top endpoint of the line segment.

To find an exact value for the length of the segment, we can build a square on it, using the segment as one of the sides of the square.

The area of this square is 5 square units. That means the exact value of the length of its side is $\sqrt5$ units.

Notice that 5 is greater than 4, but less than 9. That means that $\sqrt5$ is greater than 2, but less than 3. This makes sense because we already saw that the length of the segment is in between 2 and 3.

With some arithmetic, we can get an even more precise idea of where $\sqrt5$ is on the number line. The image with the circles shows that $\sqrt5$ is closer to 2 than 3, so let’s find the value of 2.1² and 2.2² and see how close they are to 5. It turns out that $2.1^2=4.41$ and $2.2^2=4.84$, so we need to try a larger number. If we increase our search by a tenth, we find that $2.3^2=5.29$. This means that $\sqrt5$ is greater than 2.2, but less than 2.3. If we wanted to keep going, we could try $2.25^2$ and eventually narrow the value of $\sqrt5$ to the hundredths place. Calculators do this same process to many decimal places, giving an approximation like $\sqrt5 \approx 2.2360679775$. Even though this is a lot of decimal places, it is still not exact because $\sqrt5$ is irrational.