Lesson 4: Square Roots on the Number Line

Let’s explore square roots. 

4.1: Diagonals


  1. What is the exact length of the line segment?
  2. Find a decimal approximation of the length.

4.2: Squaring Lines

  1. Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).
  2. Find the exact length of the segment.

4.3: Square Root of 3

Diego said that he thinks that $\sqrt{3}\approx 2.5$.

  1. Use the square to explain why 2.5 is not a very good approximation for $\sqrt{3}$. Find a point on the number line that is closer to $\sqrt{3}$. Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for $\sqrt{3}$.
  2. Use the fact that $\sqrt{3}$ is a solution to the equation $x^2 = 3$ to find a decimal approximation of $\sqrt{3}$ whose square is between 2.9 and 3.1.


Here is a line segment on a grid. What is the length of this line segment?

A line segment slanted down from left to right. The right endpoint is 1 unit down and 2 units right from the left endpoint.

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.

A square on a grid with side lengths equal to the hypotenuse of triangle with side lengths of 1 and 2 units. The square has an area of 5 square units.

The area of this square is 5 square units. (Can you see why?) 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.

Practice Problems ▶