How estimate the square root of 2?

Answer 1

The answer depends on the extent of your knowledge of mathematics.

If you know derivatives, a simple method is the Newton-Raphson method. You need a differentiable function defined on the set of real numbers which takes the value 0 when your quest is satisfied.

f(x) = x2 - 2 meets that requirement: if x = sqrt(2) then f(x) = 0

Then f'(x) = 2x

Let x0 be a first guess at the square root of 2. Then

x1 = x0 - f(x0)/f('(x0) is a better estimate.

Continue iteratively using

xn+1 = xn - f(xn)/f('(xn) to improve your estimates.

If you don't know derivatives then trial and improvement is one way.

Again, rather than finding a number such that its sqrt is two, it is simpler to find one whose square is 2. The two statements are equivalent but the calculations are simpler with the second.

Pick two numbers such that you think the target, 2, is between them. You know that 12 = 1 (smaller than 2) and 22 = 4 (bigger than 2) so start with them.

So now you know that 1 < x < 2

Since there is no integer between these, find a number between them at an extra decimal place. Go for 1.5 and square it. That is 2.25 which is bigger than 2.

So 1 < x < 1.5.

Find a number between 1 and 1.5 without adding a decimal place, say 1.3.

Its square is 1.69 which is too small.

So 1.3 < x < 1.5

Now try 1.4, whose square is 1.96. Again too small.

So 1.4 < x < 1.5

There is no number between these to without adding another decimal so do that and go for the midway point: 1.45.

1.45 squared is 2.1025 which is too big

so 1.4 < x < 1.45

So, to one decimal place, you now know that the answer is 1.4

If you want more accuracy, keep going.