Square Roots on the Number Line – Construction and Approximation
MathematicsNumbers & ArithmeticAges 12–13
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Numbers & Arithmetic
Construct √2 as the diagonal of a unit square, then chain right triangles with leg 1 to build √3, √4, … √n (the Theodorus spiral), and swing an arc to mark that length on the number line. Below, a zoomed number line shows each decimal digit: drag the test point x to compare x² with n, then click 'Add a digit' to squeeze √n ever tighter and see why √n is irrational when n is not a perfect square.
Lesson: Arithmetic square roots; irrational numbers; representing real numbers on the number line
What it shows
By Pythagoras' theorem, the diagonal of a square with side 1 has length √2. Attaching a right triangle with one leg 1 to a hypotenuse of length √k gives a new hypotenuse of length √(k + 1), so repeating the step builds the spiral of Theodorus and constructs √n for any whole number n. An arc then transfers each length onto the number line. The zoomed number line locates √n digit by digit: a decimal x lies to the left of √n when x² < n and to the right when x² > n. When n is not a perfect square, the digits never end or repeat, the mark of an irrational number.
How to use
Set Number n to find √n to 2, then try a perfect square such as 16. With Show the construction (right-triangle spiral) on, follow how the arc carries the length down to the axis. Next, drag the test point x on the zoomed number line and have students decide from x² whether x lies left or right of √n. Click Add a digit to tighten the bounds and compare the rows of the step-by-step bounding table.
Parameters you can change
- Number n to find √n 2–30
- Number of decimal places to approximate 1–6
- Show the construction (right-triangle spiral)
- Show the step-by-step bounding table
Questions to explore
- Why is the diagonal of a square with side 1 exactly √2 long?
- How does the bounding table end for n = 16, and why is n = 15 different?
- Is 1.41 the exact value of √2? How do the squares of 1.41 and 1.42 show this?
