# 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:** 4
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [December 11, 2013, 5:29am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/61 "2013-12-11T05:29:28Z")
I’m surprised that after 60 posts, no one has mentioned that _time_ is multiplied by _i_ in spacetime, i.e. Minkowski space. (I do know how to use Wikipedia, but hope some Doper has a pithy intuitive explanation of this.) ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [December 11, 2013, 7:44am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/62 "2013-12-11T07:44:44Z")
> [@Andy\_L](#): > > Nice. The proof took about three lines, thanks to that pointer. So I gave enough pointer to git 'er done, and not enough to spoil the fun? I’m getting better at this. 🙂 ---
**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 11, 2013, 12:55pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/63 "2013-12-11T12:55:53Z")
I can’t believe I forget to mention _Cauchy Integration_: > **[Cauchy's integral formula](https://en.wikipedia.org/wiki/Cauchy_integral_formula)** > > In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under un... Using contour integration and the Cauchy integration formula, you can evaluate a lot of integrals over the infinite domain that don’t yield to more traditional methods of integration, and for which the final formulae might not involve complex numbers at all. Getting a simple, closed-form solution to a knotty integral can be very practical. Furthermore, this is a case of the use of imaginaries that doesn’t involve either rotation or the square roots of negative numbers at all\*. \*I realize that the contours taken around the poles in the Argand plane are traditionally taken to be circular, and the limit is taken as the radius goes to zero. But they don’t have to be – the result will be the same if I built little squares or other polygons around the poles and then shrank them. It’s just a lot easier to use circles. ---
**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 11, 2013, 2:14pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/64 "2013-12-11T14:14:46Z")
> [@standingwave](#): > > 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). I’m a big fan of Laplace transforms (like several other people on this thread, I was an electrical engineering major). ---
**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 11, 2013, 4:34pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/65 "2013-12-11T16:34:52Z")
I think the most widely-used method for representing rotations and scalings in 3D space is transformation matrices. Which can also be used for 2D; it just so happens that the set of 2D transformation matrices with the appropriate properties corresponds exactly to the set of complex numbers. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [December 11, 2013, 4:38pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/66 "2013-12-11T16:38:31Z")
It depends, quaternions are used extensively in computer rendering, because rotating a vector with a quaternion is fewer operations than doing it with a transformation matrix. Though perhaps tellingly, the rotations themselves are almost invariably described with transformation matrices or euler angles and then converted to quaternions before use. ---
**Author:** ![GargoyleWB](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gargoylewb/32/199_2.png) [@GargoyleWB](https://boards.straightdope.com/u/GargoyleWB)\ **Post date:** [December 11, 2013, 7:34pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/67 "2013-12-11T19:34:53Z")
> [@Larry\_Borgia](#): > > His complaint is that he can have two fish, or half a fish, and he can even see having a close approximation to square root of two fish, but how is he ever going to have anything close to i fish. Engineer-guy here. I prefer ‘i’ when described as a coordinate vector, but that is just a repackaging of the x,y plane concept. Basically, ‘i’ contains the extra dimensional information that is otherwise not being expressed when you are describing a quantity in a dimension that is missing that extra dimension. Less-babbly 🙂 …the x number line seems to contain all of the attributes needed to manipulate the quantities of anything. 1 dog + 1 dog = 2 dogs. No problem. But there are phenomena in the world that don’t behave this way, we discover something extra has been hidden from view. As mentioned, in anything with an oscillating time-phase component, it becomes quite common. So, using dogs, suppose you returned to your arithmetic lab room (hah!) and found that 1 dog + 1 dog = 3 dogs. What happened? Did mathematics fail? Are the very fundamentals of arithmetic unsound? Well, it turns out you were actually counting your dogs over time, and dogs can reproduce. So if you introduce a time-phased vector to represent pregnancy state of the dog, let’s call it “i”, then you get an “imaginary” number that manifests itself on the arithmetic number line in certain cases. It isn’t that your number system isn’t correct, it’s just that you’ve been taught a simplified version that has left out “i” because you normally don’t need to worry about that (and other) attribute that the real world doesn’t make obvious. ---
**Author:** ![Punoqllads](https://avatars.discourse-cdn.com/v4/letter/p/d2c977/32.png) [@Punoqllads](https://boards.straightdope.com/u/Punoqllads)\ **Post date:** [December 11, 2013, 11:07pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/68 "2013-12-11T23:07:50Z")
> [@Jragon](#): > > It depends, quaternions are used extensively in computer rendering, because rotating a vector with a quaternion is fewer operations than doing it with a transformation matrix. Though perhaps tellingly, the rotations themselves are almost invariably described with transformation matrices or euler angles and then converted to quaternions before use. My experience has been the reverse. The model’s skeleton was specified in quaternion form at the application level and was transformed into transformation matrices when sending geometry into the rendering pipe because that’s what the rendering engine accepted. ---
**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:** [December 11, 2013, 11:14pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/69 "2013-12-11T23:14:38Z")
> [@CalMeacham](#): > > Another really good use of the number **i** is to freak out algebra students, by asking them to calculate **i[sup]i[/sup]**. Can you graph the many values? ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [December 13, 2013, 1:36pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/70 "2013-12-13T13:36:28Z")
> [@Punoqllads](#): > > My experience has been the reverse. The model’s skeleton was specified in quaternion form at the application level and was transformed into transformation matrices when sending geometry into the rendering pipe because that’s what the rendering engine accepted. Quaternions are more common in CPU-side computations. Also keep in mind that a Quaternion can be easily implemented on top of a 4D vector, so as long as your shader language has a vec4 it’s trivial to implement the 3-4 functions you really need to work with Quaternions (hell, most of the time all you need is qTransform which is q_v_qInverse. You also may need interpolation, but that’s usually CPU-side and works exactly the same regardless of whether it’s a quat or vector). ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [December 15, 2013, 5:46am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/71 "2013-12-15T05:46:43Z")
Someone upthread linked to quaternions. It uses complex space and imaginary numbers to elegantly solve rotation vectors not just in engineering and navigation, but even in computer science and software engineering. This sounds esoteric and only of use to a small fraction of people, but as one who entered the graphics, animation and gaming industry, imaginary space comes into play everyday, so much so that I take it for granted, and it’s become almost second nature. In CGI and 3D modeling, it comes in very handy if you ever need to define a vector in space around a local or world origin. To the young or cynical: Despite a 1:1 application between the real numbers and however many fish you have, we build, create, invent, imagine, and live with an abstract mind. Not just mathematics itself, but numbers are an abstract notion; so much more can be done when you begin to think beyond just whole integers or how many fish you have. How about how many fish you _don’t_ have, and how can you find them? That requires some… imagination… ---
**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 15, 2013, 1:08pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/72 "2013-12-15T13:08:18Z")
> [@](#): > > How about how many fish you don’t have, and how can you find them? That requires some… imagination… But still has nothing to do with “imaginary” numbers. ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [December 15, 2013, 6:17pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/73 "2013-12-15T18:17:33Z")
> [@Chronos](#): > > But still has nothing to do with “imaginary” numbers. Not literally, sure, but like zero, there was a time it wasn’t considered a number, or was seen as a useless abstraction. “Pshhh…” They’d laugh, “when will I ever have to count how many fish I _don’t_ have?” Negative, zero and imaginary numbers are having the last laugh. If they had mouths. And could laugh. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 16, 2013, 4:11am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/74 "2013-12-16T04:11:10Z")
> [@CalMeacham](#): > > Furthermore, this is a case of the use of imaginaries that doesn’t involve either rotation or the square roots of negative numbers at all\*. Sort of hard to have a use of imaginaries that doesn’t involve square roots of negative numbers at all… 🙂 [I mean, if it really didn’t make any use of imaginaries qua square roots of negative numbers, then it could just as well be understood as about something other than imaginaries, specifically. And thus probably not provide much ammo for the OP.] ---
**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 16, 2013, 1:03pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/75 "2013-12-16T13:03:05Z")
> [@Indistinguishable](#): > > Sort of hard to have a use of imaginaries that doesn’t involve square roots of negative numbers at all… 🙂 > > [I mean, if it really didn’t make any use of imaginaries qua square roots of negative numbers, then it could just as well be understood as about something other than imaginaries, specifically. And thus probably not provide much ammo for the OP.] The final result of many Cauchy integrations are formulas that only involve real quantities, with real results. They’re derived using an equation that does itself use the imaginary number I, but it disappears from the final result. So you have something that, in its final form, doesn’t seem to use the square roots of negative numbers, but was achieved using them. Supports the OP perfectly. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 16, 2013, 3:31pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/76 "2013-12-16T15:31:25Z")
Gotcha. I was just misunderstanding what you meant by “doesn’t involve either rotation or the square roots of negative numbers at all”. I see now you were speaking only in terms of the ends and not the means. ---
**Author:** ![BlinkingDuck](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@BlinkingDuck](https://boards.straightdope.com/u/BlinkingDuck)\ **Post date:** [December 16, 2013, 4:32pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/77 "2013-12-16T16:32:59Z")
> [@Larry\_Borgia](#): > > Thanks, guys, this is helpful. I printed [This old page](http://www.math.toronto.edu/mathnet/questionCorner/complexinlife.html) which I will inflict on him tomorrow. This is similar to what I used when I taught. I basically said Complex numbers are 2-dimensional and sometimes that extra dimension is useful. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 16, 2013, 5:01pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/78 "2013-12-16T17:01:24Z")
Whoa … we’re talking about a 14-year-old … _i_ times a tensor isn’t what they need to hear. Time to introduce the lad or lassy to the concept of “Faith” … don’t worry what it’s for, just know it shows up occasionally and this is how you deal with it. I can’t imagine them seeing it again until long past the delta/epsilon proofs. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 16, 2013, 7:55pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/79 "2013-12-16T19:55:23Z")
“faith” is the worst way to teach mathematics, and there’s absolutely no reason to delay understanding of complex numbers until after delta/epsilon proof, for a child who quite probably finds the former more interesting and readily graspable than the latter. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 16, 2013, 8:50pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/80 "2013-12-16T20:50:56Z")
Whoa … the question is what complex numbers are good for … understanding them is easy, it’s taught to 14-year-olds. The sciences where it’s used is generally taught after Calculus. At least I don’t remember complex numbers turning up in High School physics, chemistry or biology. [Previous page](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920.md?page=3) [Next page](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920.md?page=5)