Prove that if we are given a line of length 1, then we can construct a line of length for all .

* Proof. * First, given a line of length 1 we can construct a line of length

by taking the hypotenuse of the right triangle with legs of length 1 (which we can construct with our unit length line).

Now, assume we have a line of length 1 and a line of length

for some integer

. Then, we can form a right triangle with legs of length 1 and length

. The hypotenuse of this triangle is

. Hence, if we can construct a line of length

, then we can construct a line of length

. Since we can construct a line of length

in the base case, then we can construct a line of length

for all integers

*Related*

The orignal question in the book mentioned straightedge and compass but the given answer doesn’t use compass though this one does https://math.stackexchange.com/questions/705/compass-and-straightedge-construction-of-the-square-root-of-a-given-line/708

I don’t understand why such a question was in the book.