In a book they describe finding the square root of 2 manually like so:
$$1^2 = 1 < 2 < 2^2 = 4$$ $$(1.4)^2 = 1.96 < 2 < (1.5)^2 = 2.25$$ $$(1.41)^2 = 1.9881 < 2 < (1.42)^2 = 2.0264$$ $$\vdots$$
And so on. First, I estimate the result will be between 1 and 4. But where does the 1.4 then come from?
I don't undestand what are the steps of this procedure?
On
It's a bit of guess work. Rewrite the inequalities as
$$1^2=1<2<4=2^2\\14^2=196<200<225=15^2\\141^2=19881<20000<20264=142^2\\\vdots$$
As you can see, they were simply finding the nearest perfect squares to $2\times100^n$, then dividing everything by $100^n$.
At least, this is one interpretation...
One way is just guesswork. You know $1<\sqrt{2}<2$ just from familiarity with integers. (Note that the upper bound here is $2$, not $4$.) You try $1.1,1.2,\dots,1.9$ in succession and find that $1.4<\sqrt{2}<1.5$. You try $1.41,1.42,\dots,1.49$ in succession and find $1.41<\sqrt{2}<1.42$ and so forth. In practice you stop when you have found a "bracket" (so for instance you shouldn't bother to square $1.6,\dots,1.9$ once you already found $1.4<\sqrt{2}<1.5$)
More generally there are numerical methods that require much less guesswork, such as Newton's method, the bisection method, etc.