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 units.

Notice that 5 is greater than 4, but less than 9. That means that 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 is on the number line. The image with the circles shows that 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 and , so we need to try a larger number. If we increase our search by a tenth, we find that . This means that is greater than 2.2, but less than 2.3. If we wanted to keep going, we could try and eventually narrow the value of to the hundredths place. Calculators do this same process to many decimal places, giving an approximation like . Even though this is a lot of decimal places, it is still not exact because is irrational.