Testwiki:Reference desk/Archives/Mathematics/2016 November 29

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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)

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)

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)

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)

"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)

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)