# What is the Square root of minus 1? **URL:** \ **Category:** Factual Questions\ **Created:** [November 25, 2005, 10:25pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736 "2005-11-25T22:25:24Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [November 26, 2005, 3:56pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/21 "2005-11-26T15:56:28Z")
> [@Bricker](#): > > Hmmm. > > Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. > > And Lord knows the integers are more “real” still… right? Some of the complex numbers are rational also. No numbers are more real than others, they are all abstract concepts. Some of them happen to correspond more easily to how we percieve reality. As for integers, what’s real about 0 or negative numbers. Some mathematician said (memory quote here), “God invented the natural numbers, man the rest.” ---
**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:** [November 26, 2005, 4:10pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/22 "2005-11-26T16:10:56Z")
> [@muttrox](#): > > Some mathematician said (memory quote here), “God invented the natural numbers, man the rest.” > [@Leopold Kronecker](#): > > Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. Variously translated, as, for example: “God made the integers, all else is the work of man.” Although, “whole numbers” or “natural numbers” would be a more accurate translation than “integers.” ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 26, 2005, 4:21pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/23 "2005-11-26T16:21:33Z")
> [@muttrox](#): > > Some of the complex numbers are rational also. Yeah, the rationals themselves. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 26, 2005, 5:23pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/24 "2005-11-26T17:23:27Z")
Most of the responses are pretty good, but one thing that’s been left out is that i is just _a_ square root of -1. Rigorously, one says “pick a square root of -1 and call it i”. There’s nothing to stop you from adjoining as many square roots of -1 as you want. Whether it’s useful or not is another matter… ---
**Author:** ![JThunder](https://avatars.discourse-cdn.com/v4/letter/j/e95f7d/32.png) [@JThunder](https://boards.straightdope.com/u/JThunder)\ **Post date:** [November 26, 2005, 5:29pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/25 "2005-11-26T17:29:14Z")
> [@Bricker](#): > > Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. A better explanation would be that imaginary numbers are purely a mathematical convenience. That is, they make many types of computations a great deal simpler. Real numbers, on the other hand, can correspond to real-world objects. You can have two apples, or three apples, or even 3.426 apples. However, you can’t have _i_ apples. That’s why these numbers are called “imaginary.” ---
**Author:** ![JThunder](https://avatars.discourse-cdn.com/v4/letter/j/e95f7d/32.png) [@JThunder](https://boards.straightdope.com/u/JThunder)\ **Post date:** [November 26, 2005, 5:30pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/26 "2005-11-26T17:30:44Z")
> [@John Mace](#): > > Yep. It’s called a vector. More specifically, a 3D vector or a triple. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 26, 2005, 5:32pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/27 "2005-11-26T17:32:16Z")
> [@JThunder](#): > > A better explanation would be that imaginary numbers are purely a mathematical convenience. That is, they make many types of computations a great deal simpler. > > Real numbers, on the other hand, can correspond to real-world objects. You can have two apples, or three apples, or even 3.426 apples. However, you can’t have _i_ apples. That’s why these numbers are called “imaginary.” Tell that to Ahronov and Bohm. ---
**Author:** ![JThunder](https://avatars.discourse-cdn.com/v4/letter/j/e95f7d/32.png) [@JThunder](https://boards.straightdope.com/u/JThunder)\ **Post date:** [November 26, 2005, 5:35pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/28 "2005-11-26T17:35:45Z")
Can you be more specific? As far as I can tell, they use these numbers as mathematical conveniences as well. I doubt that Bohm would every declare, “There are 2_i_ particles here!” ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [November 26, 2005, 5:42pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/29 "2005-11-26T17:42:41Z")
> [@Topologist](#): > > In an appropriately defined sense, the reals, complex numbers, quaternions, and octonions are the only cases with nicely behaved multiplications. What if you wanted to program a 16-nion matrix? I’d figure that would provide some interesting insights into M-Theory. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 26, 2005, 5:43pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/30 "2005-11-26T17:43:16Z")
> [@JThunder](#): > > Can you be more specific? As far as I can tell, they use these numbers as mathematical conveniences as well. I doubt that Bohm would every declare, “There are 2_i_ particles here!” The Ahronov-Bohm effect. The electromagnetic 4-potential is described by a complex 4-vector. The A-B effect shows that the phase has a real physical significance. Basically, if you run the double-slit experiment and put a solenoid between the slits so that there’s a nonvanishing (complex) 4-potential outside the solenoid, but the e/m fields vanish there, the interference pattern moves to the side. ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 26, 2005, 5:59pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/31 "2005-11-26T17:59:20Z")
> [@JThunder](#): > > More specifically, a 3D vector or a triple. True. I was simply trying to make my answer apply to n dimensions, though, and to avoid the inevitable: But what about 4 dimensions? ---
**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:** [November 26, 2005, 6:41pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/32 "2005-11-26T18:41:16Z")
> [@Bricker](#): > > Hmmm. > > Simple mind that I have, I’d argue that the real numbers are more “real” than the imaginary numbers, because some of the set of real numbers are rational, and rational numbers are more “real” than the irrational numbers. > > And Lord knows the integers are more “real” still… right? The counter to this argument is that you’re being limited by human “common sense” perspective. It’s like believing that the earth is flat or that we are at the center of the universe with the sun revolving around us. That may be everyday “naive” experience but a more sophisticated view of the universe refutes it. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 26, 2005, 6:49pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/33 "2005-11-26T18:49:37Z")
> [@Enola Straight](#): > > What if you wanted to program a 16-nion matrix? You should see a doctor immediately. ---
**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:** [November 26, 2005, 7:14pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/34 "2005-11-26T19:14:40Z")
A note: It’s easy to construct an n-dimensional space with a well-defined definition of addition. The complex numbers are an example in two dimensions, and the ordinary 3-d vector spaces one frequently encounters are a 3-d example. Multiplication, however, is not so straightforward. In general, there will not be a natural multiplication operation to take two n-dimensional vectors and multiply them to get another n-dimensional vector. There is such an operation on the complex numbers (as well as the quaternions), but it’s not completely natural, since it renders the space inherently anisotropic (I presume this is also the case for the octonions, but I never could make heads nor tails of them… It doesn’t seem like there’s enough structure left to figure anything out). For three dimensions, there’s no natural multiplcation at all which gives you back a three-vector (yes, I know about the cross product, but a cross product doesn’t actually live in the same vector space as the original vectors). So a vector space is not quite really a generalization of complex numbers to higher dimension. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 26, 2005, 7:23pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/35 "2005-11-26T19:23:48Z")
> [@Chronos](#): > > I presume this is also the case for the octonions, but I never could make heads nor tails of them… The octionions are constructed from the quaternions in exactly the same [way](http://en.wikipedia.org/wiki/Cayley-Dickson_construction) that the quaternions are constructed from **C**. In particular, multiplication is defined in the same manner. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 26, 2005, 7:34pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/36 "2005-11-26T19:34:39Z")
To expand on what **Chronos** said, it’s actually very easy to come up with multiplication over an arbitrary vector space. For **R** [sup]3[/sup], define (a[sub]0[/sub], a[sub]1[/sub], a[sub]2[/sub]) \* (b[sub]0[/sub], b[sub]1[/sub], b[sub]2[/sub]) as (a[sub]0[/sub]b[sub]0[/sub], a[sub]1[/sub]b[sub]1[/sub], a[sub]2[/sub]b[sub]2[/sub]) and you’re good to go. The trick is to come up with multiplications that have nice algebraic properties. That one’s associative and commutative, but it’s got a slight flaw, which I’ll leave to the reader to discover. Hint: what’s the product of (1, 2, 0), (1, 0, 3) and (0, 2, 3)? ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [November 26, 2005, 10:07pm UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/37 "2005-11-26T22:07:23Z")
You can have i apples. Make an array of apples, i by i. Now add two more. Voila! You have an apple! ---
**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:** [November 27, 2005, 2:42am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/38 "2005-11-27T02:42:46Z")
> [@Cabbage](#): > > Actually, historically it wasn’t quadratic equations that led to the discovery of imaginary numbers, it was cubics. > > Quadratic equations with no real solutions weren’t really considered unusual, or to be “hiding” anything (such as imaginary numbers)–these equations simply didn’t have a solution, and nobody was really troubled by it. > > It wasn’t until the cubic formula was discovered that people began to discover imaginary numbers. The cubic formula was discovered in the 16th century; the strange thing was that while occasionally the steps required you to take the square root of a negative number, the final result would give you an actual (real number) solution. How could this method ultimately work, even when the intermediate steps led you through “fictitious” (imaginary) numbers? (I believe it was Cardano who first began to question what was going on with what he termed as the “fictitious numbers” involved in the steps of the cubic formula). He didn’t manage to figure it out, but this was the genesis of the discovery of imaginary numbers. I suspect that this is right (save for the “ocasionally”). People earlier must have speculated about expanding the number system in order to solve x^2 + 1 = 0, but the solution inevitably involves complex numbers (that is, sums of real and imaginary). A cubic can one real root and two complex roots or three real roots. There can be multiple roots, but I will ignore that case. If it has just one real root, my recollection is that you can solve for that one without involving complex numbers at all. But the interesting case is that of three real roots. In that case complex numbers inevitably arise in the course of the solution and there is no way to avoid them. Even for a polynomial like x^3 - x = 0, whose roots are obvious by inspection, if you use Cardan’s formula (or any other) to solve it, complex numbers must arise at an intermediate step. So yes it is quite reasonable that that must have been a powerful force pushing the complex numbers into the forefront of mathematics. Since Tartaglia’s formula for the fourth degree polynomial involves solving an auxiliary cubic, that will also require complex numbers. ---
**Author:** ![Malodorous](https://avatars.discourse-cdn.com/v4/letter/m/e47774/32.png) [@Malodorous](https://boards.straightdope.com/u/Malodorous)\ **Post date:** [November 27, 2005, 3:48am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/39 "2005-11-27T03:48:13Z")
> [@Chronos](#): > > For three dimensions, there’s no natural multiplcation at all which gives you back a three-vector (yes, I know about the cross product, but a cross product doesn’t actually live in the same vector space as the original vectors). The original vectors are members of **R** [sup]3[/sup], as is their cross product. How is that not part of the same vector space? I see the product isn’t orthogonal to the original two vectors, so it wouldn’t be in the subspace spanned by the original vectors, is that what you mean? I don’t see how that makes in not a “natural multiplication”. ---
**Author:** ![JThunder](https://avatars.discourse-cdn.com/v4/letter/j/e95f7d/32.png) [@JThunder](https://boards.straightdope.com/u/JThunder)\ **Post date:** [November 27, 2005, 5:50am UTC](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736/40 "2005-11-27T05:50:05Z")
> [@Mathochist](#): > > The Ahronov-Bohm effect. The electromagnetic 4-potential is described by a complex 4-vector. The A-B effect shows that the phase has a real physical significance. Yes, it has real physical significance. Nobody denies that. That’s not the issue at hand, though. As I said, imaginary numbers do simplify a great many computations. They are used in phasor multiplication, for example, in which they help establish a “phase” factor. This does not mean that they directly represent any real-world quantities, though. Real-world measurements require real numbers, but the subsequent computations frequently employ imaginary numbers – again, as a matter of simplification and convenience. [Previous page](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736.md?page=1) [Next page](https://boards.straightdope.com/t/what-is-the-square-root-of-minus-1/332736.md?page=3)