Calculus – Why Does Hitting the Square Root Button Always Result in 1?

calculatorcalculusconvergence-divergencefixed-point-theorems

I asked myself this question when I was a young boy playing around with the calculator.

Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman playing around with a calculator.

Hence I'm interested in answers suitable for a person that, say, knows what the square root of a square number is, but doesn't know about sequences, functions, convergence and the like.

Best Answer

Here is how I had justified it to myself when I was a kid (I was convinced :-))

If $x > 1$ then $\sqrt x > 1$ and $x > \sqrt x$.

So we keep reducing the number while still being $> 1$.

Also, we cannot end up at a number $>1$ as then taking the square root would reduce it.

If we do end up at $1$, we stay there. Since the calculator has a limited precision, we end up at $1$, and pretty quickly.

Of course, the assumption here is that we do end up somewhere :-) (Which seemed justified by the fact that the first non-zero digit after the decimal point always seemed to reduce)

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