The Calculator Puzzler

In this Activity we will investigate how your calculator's [] (squared-root) key works. You might wonder how you would compute a square-root yourself, or at least how you would approximate it. If you answered "I would use my calculator", remember that someone had to program your calculator to approximate the square root of a number, so that doesn't really answer the question of how one might carry out such an approximation.

  • Notice what happens when you repeatedly press the [] key on your calculator. Let's try this with 5 as the initial input number and see what sequence of numbers we get:

  • It looks like the fractional part is roughly halved at each successive stage, that is, each time we take the square-root, the next number we get seems to have a fractional part about half that of the preceding number. But it can't be that that is all our calculator really does!

    What is going on?

    Let's investigate...

  • Follow the steps below, assuming you are dealing with input numbers of the sort which you have experimentally determined would produce the "roughly half the fractional part" phenomenon.

  • Part A: Non-calculus Method

  • Part B: Calculus to the Rescue!

    Copyleft notice: Copyright © 1995, by Harel Barzilai -- "Share and Enjoy".
    Non-profit educational use explicitly allowed and encouraged.
    May not use for any other purpose without written permission.

    Return to Projects Calculus Home Page