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November 29

Construction of higher roots of two

Can ${\sqrt[{4}]{2}}$ or ${\sqrt[{2n}]{2}}$ be constructed? יהודה שמחה ולדמן (talk) 17:28, 29 November 2016 (UTC)[reply]

You can always construct the geometric mean of two given lengths. Which means that if you are given a unit interval, you can construct the square root of any given number. This means you can construct

{\sqrt {2}},{\sqrt[{4}]{2}},{\sqrt[{8}]{2}}

and in general

{\sqrt[{2^{n}}]{2}}

. But I believe you can't construct any other roots of 2. For example,

{\sqrt[{3}]{2}}

is famously non-constructible, so

{\sqrt[{6}]{2}}

can't be constructed either. -- Meni Rosenfeld (talk) 18:11, 29 November 2016 (UTC)[reply]

So how do we construct

{\sqrt[{4}]{2}}

as height in the right triangle with hypotenuse

1+{\sqrt {2}}

? יהודה שמחה ולדמן (talk) 18:54, 29 November 2016 (UTC)[reply]

You can see the details of the construction at http://math.stackexchange.com/a/708/153429. -- Meni Rosenfeld (talk) 19:35, 29 November 2016 (UTC)[reply]

"Equivalent to" vs "Approximately" symbols

What is the difference between these symbols: ≍ ≈ ? I've seen "≈" used many times before (i.e. Pi ≈ 3.14). But "≍" is new to me. It seems they have a similar meaning? --209.203.125.162 (talk) 21:06, 29 November 2016 (UTC)[reply]

The ≍ symbol is sometimes used to denote equivalent "order of magnitude" in asymptotic analysis, for example, in Asymptotic notations. --Mark viking (talk) 21:49, 29 November 2016 (UTC)[reply]