Short answer:
you basically need to divide the paper up into equal parts of either '1', '√2', or '1+√2'. Then be able to measure '1' and '√2' from those 'equal parts' on the opposite side.
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Detailed (possibly confusing) answer:
First off making sure we understand how to fold nths (like dividing paper into thirds, fifths, thirteenths, etc.):
(This one shows how to get fifths using fourths(four 'equal parts' of '1's))
We basically
make use of Similar Triangles.
For all of the shapes in the pic above, we just need to
make two triangles created by two straight lines such that the ratio of the two triangles' size is the ratio we're looking for.
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I will use 4√2:3√2 - 1 as an example. (prev. asked by Mir Numaan)
For this example, I divided the paper up into 'equal parts' of √2 (not 'equal parts' of 1)
-> for #4, I divided the paper up into 4 'equal parts' of √2
____-> Then, I measured 1 'part' of √2 + 1 'part' of 1 on the other side (Note: 4√2 - (3√2 -1) = 1+√2)
-> for #5, I divided the paper up into 3 'equal parts' of √2
____-> Then, I measured 1 'part' of 1 from the other side.
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I'm sorry if this was all confusing. (It means I'm bad at explaining
) But if you can understand my first picture (dividing into 5), you're on the right track.