In
english, the
square root of
two is
2.
Proof:
take two lines of equal length placed end to end at right angles to each other. Let's pretend that these are two streets you need to walk down to get to, say, the market from your house. Each street is 1 mile long, so you walk a total of two miles.






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Now, one day, you decide to take a road halfway down the first one, .5 miles from your starting point, and after walking half a mile down that road you decide to turn again so that you will meet the second leg of your original path 1/2 way along that second path, so that your walking route now looks like:


____


____
There is no difference in the length of this path  you are just travelling four 1/2 mile lengths instead of two 1 mile lengths.
One day you decide to halve this again, making you walk eight 1/4 mile lengths:

__

__

__
Once again, Same length.
If you continue to follow this process of going shorter and shorter distances while taking more and more turns, your path, as you approach an infinite number of turns, will look like this:
\
\
\
\
\
\
(more or less). In other words, the hypotenuse of the right triangle formed with the two legs of your original path. AND, using the pythagorean theorum, we know the length of that hypotenuse to be (1^2 + 1^2)^(.5) or, the square root of two. AND, seeing as each iteration of our walk did not change the length of the path walked, our path should be equal to the length of the two original sides, i.e., 2. SO, 2 = 2^(1/2).