# Practical uses for the number i **URL:** \ **Category:** Factual Questions\ **Created:** [December 9, 2013, 8:19pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920 "2013-12-09T20:19:32Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![drewtwo99](https://avatars.discourse-cdn.com/v4/letter/d/d07c76/32.png) [@drewtwo99](https://boards.straightdope.com/u/drewtwo99)\ **Post date:** [December 10, 2013, 4:09pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/41 "2013-12-10T16:09:01Z")
These explanations about rotations are great, but really, you could do all of that with vectors and special operators and you wouldn’t ever need _i_. The only place (so I’ve been told) where i is uniquely needed to describe physical situations and you couldn’t get around it any other way because it’s the only math we have that works, is quantum mechanics. The quantum mechanical states/operators depend on i being its own unique thing and not just a clever way of rotating things. Or so I was told by a number of physics professors. Perhaps \*\*Chronos \*\*could expand on this? ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [December 10, 2013, 4:13pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/42 "2013-12-10T16:13:15Z")
> [@Great\_Antibob](#): > > That’s a troublesome statement. The same reasoning leads one to say that irrational numbers don’t have real-world meaning either, yet I would argue that’s not the case. Nor is it is the case for complex numbers. Claiming that irrational numbers have no real world meaning is very irrational. pi. It’s irrational and represents a very useful real-world ratio. ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [December 10, 2013, 4:16pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/43 "2013-12-10T16:16:10Z")
> [@Indistinguishable](#): > > He’ll never have i fish, or i apples, or i cousins, but then, he’ll never have -1 cousins either. He’ll never come in 0.5th place in a race, and he’ll never enter a lottery with probability 2 of winning. Different number systems for different purposes. One of my cousins had an imaginary friend when she was 7, does that count? ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [December 10, 2013, 4:18pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/44 "2013-12-10T16:18:54Z")
> [@Indistinguishable](#): > > He’ll never have i fish, or i apples, or i cousins, but then, he’ll never have -1 cousins either. He’ll never come in 0.5th place in a race, and he’ll never enter a lottery with probability 2 of winning. Different number systems for different purposes. This is a good point. When you start to think of mathematics as building blocks to build conceptual models of the real world, you see that you don’t need every block for every model even though most of the blocks end up getting used in some model or another. In some models, it makes sense to have half of some amount or a negative amount of it. In others, it doesn’t make sense. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [December 10, 2013, 4:19pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/45 "2013-12-10T16:19:21Z")
> [@robert\_columbia](#): > > Claiming that irrational numbers have no real world meaning is very irrational. > > pi. It’s irrational and represents a very useful real-world ratio. They have as much meaning as complex numbers, which is the point. I was responding to a post claiming that complex numbers “have no real world meaning”. That’s clearly not true, or if it is true, just as true for irrationals. I challenge you to find anything in the real world with a ratio of precisely pi and NOT simply a rational number whose value is close to what we call “pi” up to the limits of our measuring equipment. In other words, we can’t measure truly irrational numbers. No fundamental constants can be physically measured to infinite precision. We only get to infinite precision if we jump into theory. And if we’re jumping into theory anyway, complex numbers are in the same boat. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [December 10, 2013, 4:25pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/46 "2013-12-10T16:25:07Z")
> [@drewtwo99](#): > > These explanations about rotations are great, but really, you could do all of that with vectors and special operators and you wouldn’t ever need _i_. > > The only place (so I’ve been told) where i is uniquely needed to describe physical situations and you couldn’t get around it any other way because it’s the only math we have that works, is quantum mechanics. The quantum mechanical states/operators depend on i being its own unique thing and not just a clever way of rotating things. Or so I was told by a number of physics professors. > > Perhaps \*\*Chronos \*\*could expand on this? You can indeed handle rotations with other math, but in many situations the answer you need falls out naturally and obviously with complex notation, especially when there’s phase shifting or a decay envelope (imaginary terms in your oscillation frequency translate into decay terms). Fourier analysis can be done with separate sine and cosine terms, but it’s much easier and concise to do it with complex expressions. You can describe polarized light using 4 X 4 real-valued Mueller Matrices 9which are non-obvious and a pain to derive), or you can do it with intuitively obvious 2 X 2 Jones matrices that I can write down from memory. It’s notb that you need complex numbers to do this. It’s that using them is more simple and direct. the n roots of x[sup]n\*/sup] = B are obviously equally-distributed points around a circle of radius B[sup]1/2[/sup]plotted in the Argand plane, but only a collection of weird complex terms if you write them out ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [December 10, 2013, 4:27pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/47 "2013-12-10T16:27:58Z")
Likewise, we can get around the use of complex numbers in electrical impedance, too, but the result is unintuitive math and a lot of unnecessary complexity. ---
**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 10, 2013, 6:56pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/48 "2013-12-10T18:56:08Z")
And for quantum mechanics as well. As **Indistinguishable** says, really all that complex numbers are is just two-dimensional rotations. Anything that uses complex numbers, you can replace them with any other means of describing two-dimensional rotations. The reason we use complex numbers specifically, in electrical impedance and quantum mechanics and Fourier analysis and all the other places we use them, is that they’re a nice simple way to do it, that follows in a natural way from other things we do on a regular basis. ---
**Author:** ![Larry\_Borgia](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/larry_borgia/32/156_2.png) [@Larry\_Borgia](https://boards.straightdope.com/u/Larry_Borgia)\ **Post date:** [December 10, 2013, 7:10pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/49 "2013-12-10T19:10:38Z")
Is there a system that allows you to describe three dimensional rotations? ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [December 10, 2013, 7:19pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/50 "2013-12-10T19:19:42Z")
> [@Larry\_Borgia](#): > > Is there a system that allows you to describe three dimensional rotations? [Sure](http://en.wikipedia.org/wiki/Quaternion). ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 10, 2013, 7:26pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/51 "2013-12-10T19:26:47Z")
> [@drewtwo99](#): > > These explanations about rotations are great, but really, you could do all of that with vectors and special operators and you wouldn’t ever need _i_. > > The only place (so I’ve been told) where i is uniquely needed to describe physical situations and you couldn’t get around it any other way because it’s the only math we have that works, is quantum mechanics. The quantum mechanical states/operators depend on i being its own unique thing and not just a clever way of rotating things. Or so I was told by a number of physics professors. > > Perhaps \*\*Chronos \*\*could expand on this? You can actually do quantum mechanics using only real numbers, but you have to impose a special rule (a so-called ‘superselection rule’) according to which all operators must commute with a special operator which in a simple system (a qubit) takes the form of the matrix ```auto ( 0 1 0 0) J = (-1 0 0 0) ( 0 0 0 1) ( 0 0-1 0) ``` (The state space of a qubit is two-complex dimensional, that is, two complex numbers are required to describe its wave function; since a complex number can be capture using two real numbers, its state space is four dimensional in real number quantum mechanics.) This requirement was first formulated by Stückelberg in the '60s, I think. More recently, William Wootters has worked on a theory in which Stückelberg’s rule would emerge dynamically (by adding a ‘universal re(al)-bit’ that essentially accounts for the global phase degree of freedom, for the experts), and which thus is also a theory equivalent to QM done with real numbers. One might think, however, that this is a kind of cheat, because in a way, using the operator J reintroduces complex numbers ‘through the backdoor’ in a sense, since the square of J is equal to ```auto (-1 0 0 0) J²= ( 0-1 0 0) ( 0 0-1 0) ( 0 0 0-1) ``` Which is basically -1 for 4x4 matrices. But still, one can make a case that this theory can teach us something about the world: first of all, time reversal in quantum mechanics is related to complex conjugation (changing i to -i everywhere, basically). In the real theory, this is related to the superselection rule: any such rule effectively ‘decomposes’ your theory into certain fragments, which for instance can’t enter into superposition, etc. So the real theory can incorporate time reversal symmetry in a more natural way than complex QM. Secondly, the operator J is related to interference, which is in many ways the central phenomenon of QM. Nevertheless, it’s something that’s not widely used (or known), and perhaps more of a peculiarity than anything else. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 10, 2013, 7:26pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/52 "2013-12-10T19:26:47Z")
Fourier analysis. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [December 10, 2013, 7:34pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/53 "2013-12-10T19:34:24Z")
> [@Asympotically\_fat](#): > > Fourier analysis. Mentioned in post #46, where I point out that it’s not strictly necessary, but is an awful lot easier if you simply have complex exponentials rather than separate sine and cosine functions. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 10, 2013, 7:35pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/54 "2013-12-10T19:35:52Z")
> [@Half\_Man\_Half\_Wit](#): > > Which is basically -1 for 4x4 matrices. But still, one can make a case that this theory can teach us something about the world: first of all, time reversal in quantum mechanics is related to complex conjugation (changing i to -i everywhere, basically). In the real theory, this is related to the superselection rule: any such rule effectively ‘decomposes’ your theory into certain fragments, which for instance can’t enter into superposition, etc. So the real theory can incorporate time reversal symmetry in a more natural way than complex QM. Secondly, the operator J is related to interference, which is in many ways the central phenomenon of QM. > > Nevertheless, it’s something that’s not widely used (or known), and perhaps more of a peculiarity than anything else. I know I’m being massively oversimplistic here, but if your using objects with the algebraic properties of the complex/imaginary numbers it’s difficult to say that you have avoided using them (especially when you are selecting said objects for their algebraic properties). ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 10, 2013, 7:38pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/55 "2013-12-10T19:38:55Z")
> [@CalMeacham](#): > > Mentioned in post #46, where I point out that it’s not strictly necessary, but is an awful lot easier if you simply have complex exponentials rather than separate sine and cosine functions. Yes, but still a very definite practical application (in as much as that Fourier analysis has lots of practical applications). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 10, 2013, 7:47pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/56 "2013-12-10T19:47:02Z")
> [@drewtwo99](#): > > These explanations about rotations are great, but really, you could do all of that with vectors and special operators and you wouldn’t ever need _i_. There’s no _difference_ between doing it “with vectors and special operators” and doing it "with complex numbers. Whether you call something “the 90 degree turn operator” or “i” or “j” or what have you, it has exactly the same properties: applying it twice to a value is the same as negating that value. Call it a complex number, or a matrix, or a linear operator, or whatever you like… the label makes no difference, except pyschologically. In the same way, one might say “Why do we need fractions? What can you do with fractions that you can’t do with just a pair of whole numbers, and suitable rules for manipulating these as ratios?”… Well, yes. That’s what fractions are. They’re a pair of whole numbers representing a ratio. For that matter, we don’t “need” any numbers… Instead of saying “I have 3 apples”, you could always just say “I have an apple, and another apple, and yet another apple”. But there isn’t really a difference between these! The former is just another way of saying the latter (one which in its terseness is convenient if you’ll be saying that sort of thing a lot). ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [December 10, 2013, 10:06pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/57 "2013-12-10T22:06:51Z")
> [@Malacandra](#): > > One thing that I like about complex numbers is that you can use them to prove interesting facts about real numbers - and by the time you reach the end of the proof, all the imaginary terms have disappeared. > > For instance, if you take two numbers that are sums of squares and multiply them together, the product is also a sum of squares: that is, 5 = 1[sup]2[/sup] + 2[sup]2[/sup], 13 = 2[sup]2[/sup] + 3[sup]2[/sup], and 13 x 5 = 65 = 1[sup]2[/sup] + 8[sup]2[/sup], and so on. You can prove this algebraically, and the proof begins with the complex factorisation x[sup]2[/sup] +y[sup]2[/sup] = (x + iy)(x - iy), and ends with all imaginary terms disappearing. Nice. The proof took about three lines, thanks to that pointer. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [December 11, 2013, 12:55am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/58 "2013-12-11T00:55:25Z")
> [@Andy\_L](#): > > Nice. The proof took about three lines, thanks to that pointer. Yeah, imaginary numbers weren’t invented to make things difficult. They exist to make things easy. Not unlike [Laplace transforms](http://en.wikipedia.org/wiki/Laplace_transform). ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [December 11, 2013, 1:38am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/59 "2013-12-11T01:38:46Z")
> [@friedo](#): > > > [@Larry Borgia](#): > > > > Is there a system that allows you to describe three dimensional rotations? > > [Sure](http://en.wikipedia.org/wiki/Quaternion). Or maybe not: 😉 > [@friedo's link](#): > > “Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.” — Lord Kelvin, 1892. > > “I came later to see that, as far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work.” Oliver Heaviside, Electromagnetic Theory, Volume I, pp. 134–135 (The Electrician Printing and Publishing Company, London, 1893). > > “… quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.” — Simon L. Altmann, 1986. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 11, 2013, 2:00am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/60 "2013-12-11T02:00:02Z")
Quaternions don’t directly describe scaling-and-rotation in 3d the way complex numbers directly describe scaling-and-rotation in 2d; rather, quaternions directly describe particular kinds of scaling-and-rotation in 4d (specifically, for rotations on 4d space which move every vector by the same angle [aka, “isoclinic rotations”], there is a natural notion of “orientation”, of which there are two possibilities; the quaternions, then, will describe the combinations of scaling and rotation of a particular orientation). It just so happens that, if you have a distinguished way to decompose a 4d space into perpendicular 3d and 1d subspaces, then you can represent every rotation of the 3d subspace as a composition of two “mirror-image” isoclinic rotations of the 4d space, and thus represent 3d rotations by quaternions as well. But you’ll find that the additive properties of quaternions don’t match those of rotations on 3d space thought of as linear operators, because quaternions aren’t fundamentally for 3d space, but for 4d space. Another way to understand what’s going on here is that, in some sense, the reason the complex numbers are so nice is because, for any two directions in a plane, there’s a unique rotation of the plane which takes the one to the other, so that we can think of complex numbers as “ratios” of 2d vectors. The same is no longer true for two directions in 3d space (for example, there are infinitely many ways to rotate the Earth so that the North pole is sent to the North pole). However, in 4d space, we can make it true again IF we add in the demands that A) the rotation must move all directions by the same angle [as automatically happens in the 2d case], and B) the rotation must have a particular “orientation”. And thus, we get the quaternions as ratios of vectors in 4d space the same way the complex numbers are ratios of vectors in 2d space and the real numbers are ratios of vectors in 1d space, but there is no analogous 3d structure. [Previous page](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920.md?page=2) [Next page](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920.md?page=4)