How do you work out the square root of 7?

Answer 1

We can do this using Newton's Method, starting by describing the problem as a simple equation:

x2 = 7

In this case, we'll want to find the value of x. Instead of simply solving it though, we'll create a function that gets a zero value when the above statement is true:

f(x) = x2 - 7

The next step is to find the derivative of that expression. Explaining how to do that is beyond the scope of this question, and I would recommend reading some introductory calculus if you are unfamiliar with the concept of derivatives. In this case (or in any case where you're finding the square root of a number), it would be 2x.

Next we'll take an initial guess at the root. Let's go with 3. We'll call that initial guess X0. Our terms should be assembled together in this format:

x1 = x0 - f(x0) / f'(x0)

Where x0 is our initial guess, f() is the initial function we created, and f'() the derivative that we already took. That gives us the following equation:

x1 = 3 - (32 - 7) / 2(3) = 3 - 2/6 ≈ 2.66667

Now we take that new value that we found for x1, and repeat the calculation, replacing x0 with x1, and finding a new value, which we'll call x2.

x2 = x1 - f(x1) / f'(x1)

x2 = 2.66667 - (2.666672 - 7) / (2 * 2.66667) ≈ 2.64583

And we continually repeat this process until we are happy with the answer:

x3 = 2.64583 - (2.645832 - 7) / (2 * 2.64583) ≈ 2.64575

x4 = 2.64575 - (2.645752 - 7) / (2 * 2.64575) ≈ 2.64575

The fact that the number we reached is no longer changing indicates that we have reached an answer which is correct to the number of digits that we have chosen. The square root of seven is approximately 2.64575.