# How many roots are there in a square root? **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [December 19, 2025, 8:22pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864 "2025-12-19T20:22:31Z")\ **Posts on this page:** 13\ **Page:** 2
**Author:** ![Ponderoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ponderoid/32/19221_2.png) [@Ponderoid](https://boards.straightdope.com/u/Ponderoid)\ **Post date:** [December 21, 2025, 1:39am UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/21 "2025-12-21T01:39:58Z")
> [@Exapno\_Mapcase](#): > > Nevertheless, entering principal square root into Google Ngrams yields an [almost identical graph](https://books.google.com/ngrams/graph?content=principal+square+root&year_start=1800&year_end=2022&corpus=en&smoothing=3) [Not quite](https://books.google.com/ngrams/graph?content=principal+square+root%2Cprinciple+square+root&year_start=1800&year_end=2022&corpus=en&smoothing=2&case_insensitive=false). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [December 21, 2025, 1:44am UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/22 "2025-12-21T01:44:54Z")
Complicating the discussion is that $$\sqrt{-1}=i$$. I recently wrote a paper in which I had identified a certain field as the complex numbers and I then said “Let i denote a square root of -1.” There were two and they are indistinguishable so you choose one of them and call it i. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [December 21, 2025, 2:23am UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/23 "2025-12-21T02:23:30Z")
What the three graphs say is that a very consistent percentage of people confuse principle for principal, so that while principle is used much less often it tracks principal almost exactly. That’s the kind of information linguists live for. Does anyone have a linguist on speed dial? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 22, 2025, 7:11pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/24 "2025-12-22T19:11:16Z")
> [@Chronos](#): > > The existence of multiple contexts makes it nigh impossible to make any sweeping, absolute statements. If a is, let us say, a complex number, then a has two square roots. Unless a=0. However, a positive real number has a unique positive square root. If A is, e.g, a matrix then it might have a whole bunch of square roots, or none at all. On the other hand, a positive (semi)definite Hermitian matrix has a unique positive square root. Etc. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [December 23, 2025, 2:45am UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/25 "2025-12-23T02:45:35Z")
Actually even for complex numbers there is a principle square root. Every complex number can be written in polar form $$c=re^{i\theta}$$ where $$r\ge0$ is a real number and $$0\le\theta\<2\pi$$. Then the principal $$\sqrt{c}=\sqrt{r}e^{i\theta/2}$$. Why is this not formatting properly? ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [December 23, 2025, 2:57am UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/26 "2025-12-23T02:57:21Z")
> [@Hari\_Seldon](#): > > Why is this not formatting properly? Use just one dollar sign at the beginning and one at the end, like this: c=re^{i\theta} where r\ge0 is a real number and 0\le\theta\<2\pi. Then the principal \sqrt{c}=\sqrt{r}e^{i\theta/2}. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 23, 2025, 2:42pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/27 "2025-12-23T14:42:20Z")
Although, this definition would make the principle cube root of -1 be \frac{1}{2}+\frac{\sqrt{3}}{2}i, instead of the more usual convention that \root{3}\of{-1} = -1. Unless you use different definitions for the principle root of a pure-real number and for a partially-imaginary number, but then you’re introducing weird discontinuities (in addition to the discontinuities you get at 2\pi) ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [December 23, 2025, 3:36pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/28 "2025-12-23T15:36:07Z")
You make a good point. Not sure the best way to deal with this. Maybe just accept it. Of course we will always that -1 is the real cube root of -1, but perhaps that \sqrt^3{-1} = (1+\sqrt{3}i/2. ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [December 23, 2025, 5:36pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/29 "2025-12-23T17:36:53Z")
Just to add that what @Hari_Seldon meant to type was > … but perhaps that \sqrt[3]{-1} = (1+\sqrt{3}i)/2. Not much to add mathematically. The main issue is that, because there are multiple roots and because there is no way to define a continuous single-valued n^\text{th} root function on the whole complex plane (n\geq 2 for those nitpickers out there), any attempt at defining a “principal” root is going to involve some arbitrary choices. Any such choice is just a convention, not a set-in-stone mathematical fact. If you were using such in a math paper, you would have to say explicitly what choices you were making. Or, you could do what Riemann did, and invent manifolds. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 23, 2025, 6:02pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/30 "2025-12-23T18:02:15Z")
It is certainly plausible that, if asked for an _n_th root of 1, most posters here would come up with e^{2\pi i/n} rather than… 1. > [@Topologist](#): > > Or, you could do what Riemann did, and invent manifolds. People still talk about “Riemann surfaces”! (Like you can consider \{\,(z,w) \mid w^2 = z \,\} ) ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [December 23, 2025, 6:10pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/31 "2025-12-23T18:10:37Z")
> [@DPRK](#): > > It is certainly plausible that, if asked for an _n_th root of 1, most posters here would come up with e^{2\pi i/n} rather than… 1. That’s so much more interesting. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [December 23, 2025, 7:31pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/32 "2025-12-23T19:31:10Z")
1 = e\*{0\pi i} and no matter how finely you divide 0, you always get 0. Note that the angle is required to be \< 2\pi. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 23, 2025, 7:50pm UTC](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864/33 "2025-12-23T19:50:19Z")
> [@DPRK](#): > > It is certainly plausible that, if asked for an _n_th root of 1, most posters here would come up with e^{2\pi i/n} rather than… 1. e^{2\pi m i/n}, please, where m is any integer 0 \le m \lt n. [Previous page](https://boards.straightdope.com/t/how-many-roots-are-there-in-a-square-root/1025864.md?page=1)