# i. How Could Something 'Imaginary' Be So Important? **URL:** \ **Category:** Factual Questions\ **Created:** [January 20, 2016, 4:01am UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654 "2016-01-20T04:01:02Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Novelty\_Bobble](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)\ **Post date:** [January 20, 2016, 9:51am UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/21 "2016-01-20T09:51:18Z")
As with imaginary numbers, the concept of time dilation and the implications of applying the theories of relativity are difficult to grasp for our reptilian brains. However, We rely upon such slippery concepts every day in order for our satnavs to function, the maths **works** whether it makes intellectual sense to us or not. And what the hell is anti-matter anyway? Well, it is the stuff that makes a PET scanner work. In short, don’t worry about it. If you can’t visualise “i” in a way that makes sense to you then that puts you firmly alongside the majority of human beings. Even those who use “i” on a daily basis don’t necessarily intuitively grasp the concept but they do know how to use it for practical purposes. ---
**Author:** ![jtur88](https://avatars.discourse-cdn.com/v4/letter/j/e9c0ed/32.png) [@jtur88](https://boards.straightdope.com/u/jtur88)\ **Post date:** [January 20, 2016, 12:15pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/22 "2016-01-20T12:15:32Z")
> [@gazpacho](#): > > The best thing to remember about math is that it is all made up. The rules that are around that you get taught in school are rules that people have found useful. And, you can say the same thing about Language. ---
**Author:** ![GuanoLad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/guanolad/32/42_2.png) [@GuanoLad](https://boards.straightdope.com/u/GuanoLad)\ **Post date:** [January 20, 2016, 1:00pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/23 "2016-01-20T13:00:59Z")
> [@CalMeacham](#): > > I still don’t understand what’s so exciting about factorials! myself. Now I’m imagining a trailer for a 1950s monster movie with “FACTORIALS!” spread across the screen. ---
**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:** [January 20, 2016, 1:21pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/24 "2016-01-20T13:21:37Z")
> [@GuanoLad](#): > > Now I’m imagining a trailer for a 1950s monster movie with “FACTORIALS!” spread across the screen. Even worse, you could have **DOUBLE FACTORIALS!!** > **[Double factorial](https://en.wikipedia.org/wiki/Double_factorial)** > > In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, > > > > > n > ! > ! > = > > ∏ > > k > = > 0 > > > > ⌈ > > > n > 2 > > > ⌉ > > ... ---
**Author:** ![Your\_Great\_Darsh\_Face](https://avatars.discourse-cdn.com/v4/letter/y/ac8455/32.png) [@Your\_Great\_Darsh\_Face](https://boards.straightdope.com/u/Your_Great_Darsh_Face)\ **Post date:** [January 20, 2016, 1:24pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/25 "2016-01-20T13:24:22Z")
> [@friedo](#): > > If you can’t have π apples, you can’t have 1/7 of an orange. But you can have apple π. Also you can touch a live cable carrying 500_i_ volts and I believe you will find nothing imaginary about the resulting shock. 😃 ---
**Author:** ![John\_DiFool](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_difool/32/19543_2.png) [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)\ **Post date:** [January 20, 2016, 1:32pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/26 "2016-01-20T13:32:56Z")
> [@Thudlow\_Boink](#): > > “Imaginary” (and its counterpart “real”) in this sense are technical terms, not a judgment on whether these numbers actually “exist.” Except that it originally was intended as a judgement-when Rene Descartes first heard of the concept, he thought it was so preposterous that he coined the term “imaginary” to mock the entire notion. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [January 20, 2016, 1:44pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/27 "2016-01-20T13:44:18Z")
Piggybacking on **engineer\_comp\_geek** ’s superb post … You may see outfits selling power factor conversion devices for the home. They’re simultaneously valid science, an engineered solution to a engineering non-problem, and pure commercial snake oil. Save your money; don’t fall for the snake oil. ---
**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:** [January 20, 2016, 2:22pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/28 "2016-01-20T14:22:36Z")
> [@friedo](#): > > What does this mean? That calculations involving _i_ are not meaningful? Can you have -6 apples, or π oranges? I thought it was clear that I was saying it IS useful in doing some calculations. Mainly, it comes does to [Euler’s Identity.](https://en.wikipedia.org/wiki/Euler%27s_identity) As noted earlier, this comes in handy when analyzing wave mechanics, and so physicists used this in certain QM formulations. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [January 20, 2016, 9:14pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/29 "2016-01-20T21:14:27Z")
The weird thing is, I’ve come to accept how i can be used to model rotation, but I still have very little intuitive concept of what this has to do with the square root of -1. The best I can come up with (and it’s probably wrong) is that if you’re trying to solve for the roots of a parabola that doesn’t intersect the x axis, you’ll get imaginary or complex answers, which correspond to rotating the parabola 90 degrees and finding where **that** parabola intersects the axis. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 20, 2016, 9:18pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/30 "2016-01-20T21:18:54Z")
> [@Jragon](#): > > The weird thing is, I’ve come to accept how i can be used to model rotation, but I still have very little intuitive concept of what this has to do with the square root of -1. Did **Chronos** ’s comment from Post #3 help? > [@Chronos](#): > > Really, all i means is “rotate by 90 degrees”, and all that i^2 = -1 means is that if you rotate by 90 degrees and then do it again, you’re now facing the opposite direction from where you started. Or my [link](http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/) from Post #2 that says much the same thing but in more detail and with visual aids? ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [January 20, 2016, 9:31pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/31 "2016-01-20T21:31:26Z")
> [@Jragon](#): > > The weird thing is, I’ve come to accept how i can be used to model rotation, but I still have very little intuitive concept of what this has to do with the square root of -1. The best I can come up with (and it’s probably wrong) is that if you’re trying to solve for the roots of a parabola that doesn’t intersect the x axis, you’ll get imaginary or complex answers, which correspond to rotating the parabola 90 degrees and finding where **that** parabola intersects the axis. You’re not so much rotating the parabola as you are adding another dimension to the graph. When you plot a function on the cartesian plane, you’re only seeing the real-number outputs of the function on the Y axis. But that’s only part of the story. The function may have complex outputs as well, but you can’t see them unless you plot the function in more dimensions. Take the parabola y = x[sup]2[/sup] + 2, which has a y-intercept of 2 and therefore does not touch the x-axis. So we might conclude from the graph that it has no roots, because the curve never reaches zero. But it actually does have two roots at _i_√2 and -_i_√2. If you substitute those for x, you get y = 0. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 20, 2016, 10:00pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/32 "2016-01-20T22:00:17Z")
> [@standingwave](#): > > A recent Numberphile video titled [Fantastic Quaternions](https://youtu.be/3BR8tK-LuB0). Cool video. I had somehow never heard of quaternions before (or if I had, I just never bothered to look up what they were.) That Numberphile guy just makes everything in math seem so fun and interesting. Love his enthusiasm and just pure joy at explaining mathematical concepts. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 20, 2016, 10:51pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/33 "2016-01-20T22:51:49Z")
> [@Jragon](#): > > The weird thing is, I’ve come to accept how i can be used to model rotation, but I still have very little intuitive concept of what this has to do with the square root of -1. The best I can come up with (and it’s probably wrong) is that if you’re trying to solve for the roots of a parabola that doesn’t intersect the x axis, you’ll get imaginary or complex answers, which correspond to rotating the parabola 90 degrees and finding where **that** parabola intersects the axis. The Better Explained blog does an excellent job of [explaining imaginary exponentials](http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/). It’s post #5 of an eight-part series which starts [here](http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/) with imaginary numbers if you want a refresher. Enjoy. I think the author does a great job. As a sparky (EE) I’m cool with the math but I like to _grok_ things on a deeper level as well. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [January 20, 2016, 11:18pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/34 "2016-01-20T23:18:13Z")
> [@Giles](#): > > It makes sense to say that you own -6 apples if you possess 0 apples, but have promised to give 6 apples to a friend. By the same token, it makes sense to say that you own 6_i_ apples if you possess 6 apples each of which has been rotated exactly 90 degrees around its stem from whatever position it was in when you received it. ---
**Author:** ![Blue\_Blistering\_Barnacle](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/blue_blistering_barnacle/32/3386_2.png) [@Blue\_Blistering\_Barnacle](https://boards.straightdope.com/u/Blue_Blistering_Barnacle)\ **Post date:** [January 20, 2016, 11:25pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/35 "2016-01-20T23:25:13Z")
> [@CalMeacham](#): > > As others note, it’s just terminology. If it bothers you, then stay away from > > – _Transcendental_ numbers, which have nothing to do with Ralph Waldo Emerson, or > > – _Perfect_ numbers, which might not meet your standards of perfection, or > **– _Irrational_ numbers, which nonetheless make perfect sense, or** > > — \*Triangular \* numbers, and a host of other “shaped” numbers – _pentagonal_, etc., and even _Trapezoidal_ numbers, which don’t look like those shapes at all, or > – _Polite_ numbers, which don’t obviously have any more social skills than other numbers, or > > –_Prime_ numbers, which don;'t seem any more choice than other numbers. > There are lots of other types of numbers with equally odd appellations, but they have to call them _something_. It’s all part of the marriage of math, terminology, and typography. I still don’t understand what’s so exciting about factorials! myself. My Bolding. When I was a kid, I actually thought they were called “irrational” because they were these crazy numbers that caused ridiculous run on non-repeating decimals. Now I now (a little bit) better. 🙂 ---
**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:** [January 21, 2016, 5:38pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/36 "2016-01-21T17:38:34Z")
What a lot of these responses are skirting is the ontological status of any number. You can have 3 apples or even square root of three apples, but you cannot have 3 or square root of 3 or square root of -3. So what is a number. One answer is that it is an agreed upon fiction. And along with the agreement comes agreed upon rules for manipulating them. And one of the rules we agree on is that i\*i = -1. Using these rules we solve real-world problems. And has been pointed out the quaternions give a good handle on the group of symmetries of a sphere, the key to rotation in 3 dimensions. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [January 21, 2016, 5:44pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/37 "2016-01-21T17:44:18Z")
> [@Hari\_Seldon](#): > > What a lot of these responses are skirting is the ontological status of any number. You can have 3 apples or even square root of three apples, but you cannot have 3 or square root of 3 or square root of -3. So what is a number. One answer is that it is an agreed upon fiction. And along with the agreement comes agreed upon rules for manipulating them. And one of the rules we agree on is that i\*i = -1. Using these rules we solve real-world problems. And has been pointed out the quaternions give a good handle on the group of symmetries of a sphere, the key to rotation in 3 dimensions. [Pssst](http://boards.straightdope.com/sdmb/showpost.php?p=19030855&postcount=17). Seriously tho - I think this kind of discussion, a bit of minor philosophizing about what a number really _is_, is one of the huge stumbling blocks in math education that trips a lot of people up. It’s a topic that should be addressed early on, while still learning basic arithmetic: _All this stuff is made up. But we make you learn it because it’s useful._ ---
**Author:** ![Chimera](https://avatars.discourse-cdn.com/v4/letter/c/8e8cbc/32.png) [@Chimera](https://boards.straightdope.com/u/Chimera)\ **Post date:** [January 21, 2016, 6:18pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/38 "2016-01-21T18:18:07Z")
_All of the buildings, all of the cars Were once just a dream In somebody’s head_ - Peter Gabriel, _Mercy Street_ ---
**Author:** ![billfish678](https://avatars.discourse-cdn.com/v4/letter/b/7bcc69/32.png) [@billfish678](https://boards.straightdope.com/u/billfish678)\ **Post date:** [January 21, 2016, 6:33pm UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/39 "2016-01-21T18:33:07Z")
> [@Jragon](#): > > And in turn, computer graphics programmers (at least those of us who have written our own math primitive functions) become comfortable with quaternions, because they end up being computationally cheap ways to do 3D rotation. IIRC quaternions also allow you do 3D rotations without any chance of something along the lines of a “divide by zero” problem. Somebody else here can probably explain it better. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [January 23, 2016, 3:39am UTC](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654/40 "2016-01-23T03:39:53Z")
> [@billfish678](#): > > IIRC quaternions also allow you do 3D rotations without any chance of something along the lines of a “divide by zero” problem. > > Somebody else here can probably explain it better. If used correctly, they avoid [Gimbal Lock](http://mathoverflow.net/questions/95902/the-gimbal-lock-shows-up-in-my-quaternions). Quaternions aren’t unique in this, though. The culprit here is actually Euler angles, which are commonly converted to rotation matrices and/or quaternions, thus not avoiding the problem at all. Euler angles are really easy to work in, and tend to be the “default” for representing rotations unless you anticipate gimbal lock being an issue. [Previous page](https://boards.straightdope.com/t/i-how-could-something-imaginary-be-so-important/743654.md?page=1)