I'm studying the book: "The Princeton companion to mathematics", I'm in the part of fundamentals: continuity. There is an example I don't get:
How close to $\pi$ do we need $x$ to be for $x^2$ to be within $10^{−100}$ of $\pi ^2$? To answer this, we can use our earlier argument. Let $x = \pi + \delta$ again. Then $$x^2−\pi^2 = 2\delta \pi+\delta^2$$ and an easy calculation shows that this has modulus less than $10^{-100}$ if $\delta$ has modulus less than $10^{-101}$. So we will be all right if we take the first $101$ digits of $\pi$ after the decimal point.
I don't get which was the easy calculation. Can someone help me with that?
And, when it says modulus, is it talking about modular arithmetic?
We have $\delta \pi + \delta^2 < \delta \pi + \delta$ if $\delta < 1$.
We want $2\delta \pi + \delta^2 < \varepsilon$.
So, it is enough that $2\delta \pi + \delta < \varepsilon$, or $\delta < \frac{\varepsilon}{2\pi + 1}$.
Now, $2\pi + 1 < 10$ and so it is enough that $\delta < \frac{\varepsilon}{10}$.
Putting this all together: $$ \delta < \frac{\varepsilon}{10} < 1 \implies x^2−\pi^2 = 2\delta \pi+\delta^2 < 2\delta \pi + \delta = (2\pi + 1)\delta < 10 \delta < \varepsilon $$