# 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:** 1
**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 9, 2013, 8:19pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/1 "2013-12-09T20:19:32Z")
I need to explain to a pretty intelligent but somewhat cynical 9th grader what the use of the number i is. Does it come up in any real world applications? I seem to recall engineers use it for waves? I tried to show him how i (along with the complex plane) allows us to describe any direction using a single number, but I’m looking for something more tangible. 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. As always, thanks in advance for your help. ---
**Author:** ![Measure\_for\_Measure](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/measure_for_measure/32/557_2.png) [@Measure\_for\_Measure](https://boards.straightdope.com/u/Measure_for_Measure)\ **Post date:** [December 9, 2013, 8:27pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/2 "2013-12-09T20:27:25Z")
I haven’t found a satisfactory answer for this question: all presented applications are outside of my purview. That said, here’s one approach. You can’t think of negative fish either. They don’t exist. But negative numbers do. And their applications can be used when plotting positions in space. You can also have 10 fish, _take away 2_, and come up with eight. The negative fish here would be the ones you take away. At any rate, at times it’s helpful to be able to take the square root of a negative number. This happens when solving certain equations. Sometimes the implied answer is, “There is no solution for these assumptions, but there is a solution for those assumptions (or values of your variables).” But I understand in certain physics and engineering applications (which I don’t understand) that complex and imaginary numbers are used meaningfully. ---
**Author:** ![EdwardLost](https://avatars.discourse-cdn.com/v4/letter/e/bbe5ce/32.png) [@EdwardLost](https://boards.straightdope.com/u/EdwardLost)\ **Post date:** [December 9, 2013, 8:33pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/3 "2013-12-09T20:33:26Z")
Yes, in engineering. For instance: calculating currents and voltages in AC circuits. And, in physics: in quantum mechanics, the wave function is described as a complex field. ---
**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 9, 2013, 8:35pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/4 "2013-12-09T20:35:14Z")
It’s extremely useful – damned near essential – in physics, engineering, and especially electrical engineering. It comes into play whenever wave motion gets phase-shifted. This happens al the time in radio, signal processing, and the like. It’s ubiquitous in optics and laser physics. It shows up in vibration analysis in mechanical engineering. Plus, understand mathematicians use it sometimes. ---
**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 9, 2013, 8:49pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/5 "2013-12-09T20:49:36Z")
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. ---
**Author:** ![drewtwo99](https://avatars.discourse-cdn.com/v4/letter/d/d07c76/32.png) [@drewtwo99](https://boards.straightdope.com/u/drewtwo99)\ **Post date:** [December 9, 2013, 9:01pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/6 "2013-12-09T21:01:40Z")
That page is good and electrical impedance is the clearest real world example of a physical quality that involves the use of imaginary numbers. Another good one is quantum mechanics, where it is actually 100% required for the math, and not just a convenient accounting tool like it is in other physical applications. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [December 9, 2013, 9:20pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/7 "2013-12-09T21:20:59Z")
> [@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. From that page: > [@](#): > > Here are some examples of the first kind that spring to mind. In electronics, the state of a circuit element is described by two real numbers (the voltage V across it and the current I flowing through it). A circuit element also may possess a capacitance C and an inductance L that (in simplistic terms) describe its tendency to resist changes in voltage and current respectively. > > These are much better described by complex numbers. Rather than the circuit element’s state having to be described by two different real numbers V and I, it can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can be expressed using complex addition and multiplication. > > Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts. I’m an electrical engineer, I work in electromagnetics, and none of that looks familiar. The part lower down, about sinusoidal variation of voltage and current, where taking the derivative d/dt becomes multiplying by j omega is a very important advantage, and used all the time in circuit analysis and electromagnetics. Maybe someone else has seen the part I quoted in actual use, but to me, it looks like a garbled description of sinusoidal variation. I would avoid using that part unless someone else can tell you who actually uses that. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [December 9, 2013, 9:22pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/8 "2013-12-09T21:22:05Z")
> [@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. I like the population fraction analogy on that page, because it explains what is often missing in these kinds of discussions: the entire history of numbers has consisted of people inventing new kinds of numbers to model new kinds of problems, and then figuring out how to make these numbers behave consistently with the numbers we already know about. What I like to do when explaining complex numbers is to do it by analogy with another kind of number that has two parts, is a superset of the numbers we already know, and has special rules for arithmetic: fractions. ---
**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 9, 2013, 9:25pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/9 "2013-12-09T21:25:12Z")
> [@Measure\_for\_Measure](#): > > …That said, here’s one approach. You can’t think of negative fish either. They don’t exist. But negative numbers do. And their applications can be used when plotting positions in space. You can also have 10 fish, _take away 2_, and come up with eight. The negative fish here would be the ones you take away… Negative numbers can be used to model the reverse of some activity. For example, you could take the concept of “I have two fish” and say that “I have negative two fish” means that I owe an in-kind debt of two fish to someone. Some real life financial statements actually use negative monetary amounts to indicate a debt where one would otherwise expect to see equity, though it’s common to use red lettering and/or parentheses to indicate this rather than to just put a negative sign in front, but this is just a difference in notation rather than a fundamental change in concepts. Likewise, the act of vomiting could be modeled as eating a negative amount of food. If it’s negative 10 miles to grandma’s house, you traveled too far and have to turn around. ---
**Author:** ![barbitu8](https://avatars.discourse-cdn.com/v4/letter/b/839c29/32.png) [@barbitu8](https://boards.straightdope.com/u/barbitu8)\ **Post date:** [December 9, 2013, 9:34pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/10 "2013-12-09T21:34:58Z")
Some of the posts in this thread refer to negative numbers. I’m sure the OP has no problem with that. His problem is the usefulness of _I_, the square root of negative one. ---
**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 9, 2013, 9:42pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/11 "2013-12-09T21:42:54Z")
> [@Measure\_for\_Measure](#): > > I haven’t found a satisfactory answer for this question: all presented applications are outside of my purview. > > That said, here’s one approach. You can’t think of negative fish either. They don’t exist. But negative numbers do. And their applications can be used when plotting positions in space. You can also have 10 fish, _take away 2_, and come up with eight. The negative fish here would be the ones you take away. Once someone has accepted the use of negative numbers to indicate position to the left of an origin point (and positive numbers to indicate position to the right of the same origin point), you can introduce use of complex numbers to specific positions in a plane, with positive imaginary numbers indicating “up” and negative ones indicating “down”. With this interpretation, multiplication by “i” means go from the current position to one that is the same distance from the origin but 90 degrees counterclockwise from the current position. ---
**Author:** ![Snarky\_Kong](https://avatars.discourse-cdn.com/v4/letter/s/a183cd/32.png) [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)\ **Post date:** [December 9, 2013, 9:45pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/12 "2013-12-09T21:45:31Z")
> [@barbitu8](#): > > Some of the posts in this thread refer to negative numbers. I’m sure the OP has no problem with that. His problem is the usefulness of _I_, the square root of negative one. Everyone here is aware of that. Did you read those posts? ---
**Author:** ![Fridgemagnet](https://avatars.discourse-cdn.com/v4/letter/f/22d042/32.png) [@Fridgemagnet](https://boards.straightdope.com/u/Fridgemagnet)\ **Post date:** [December 9, 2013, 10:04pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/13 "2013-12-09T22:04:41Z")
It’s a very useful mathematical tool that, while having no real-world meaning, is key in solving a lot of practical problems. I’ve mostly used it for designing analogue electronic filters, but also handy for describing resonant systems (such a spring/damper car suspension), control loops (such as temperature regulation in air conditioning) or general transfer functions (such as used engine oil percolating through a compost heap, say). Also, mathematicians, scientists and engineers would be in trouble without [Euler’s formula](http://en.wikipedia.org/wiki/Euler%27s_formula). Square i and you get -1. This helps keep the equations smaller (e.g. i^3 is -i; i^4 is 1, I^5 is i, etc). A scientific equation using i for describing some real-world thing might not seem useful, but with a bit of mathematical manipulation it can be transformed into elements that are tangible like frequency, magnitude and phase. ---
**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 9, 2013, 10:20pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/14 "2013-12-09T22:20:26Z")
> [@Fridgemagnet](#): > > while having no real-world meaning, 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. ---
**Author:** ![Blaster\_Master](https://avatars.discourse-cdn.com/v4/letter/b/cab0a1/32.png) [@Blaster\_Master](https://boards.straightdope.com/u/Blaster_Master)\ **Post date:** [December 9, 2013, 10:26pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/15 "2013-12-09T22:26:22Z")
As others have pointed out, there are definitely real world applications for i, so I won’t bother to reiterate them, but I also don’t think that that’s the issue here. Math, in order to make intuitive sense, has to be able to have some kind of line connecting a new concept back to ones that already make intuitive sense. The unfortunate part about imaginary numbers is the choice of the name, which implies that they don’t have real world applications. Worse, while it is helpful to have real world applications for mathematical concepts, most of the i is useful for aren’t something that is really appropriate for when the mathematical concept. So, instead, we get stuck saying “trust us, it’s useful.” The natural explanation, comparing complex numbers to negative numbers, I think ultimately just confuses the issue. I have 5 ducks, that’s a positive number, then I have 3, so that difference is conceptually equal to -2 ducks, that’s a negative number. But even knowing that i is the square root of -1, that doesn’t give us a useful conceptual analogy, because subtraction isn’t a place where that concept would be used. Instead, I think a better analogy would be considering space. We can all imagine one dimension as a number line, two as a cartesian plane, and three dimensions as x, y, z. We intuitively can carry on that concept to four or more dimensions and come up with useful applications. For instance, adding t gives you space-time, or we could add a w and get a fourth spatial dimension, or whatever. I think it’s fairly easy to see how we could conceptually expand that to any arbitrary number of dimensions for various applications, but those are a couple simple ones. However, despite knowing that there’s intuitive real world applications for hyperspace, we can’t meaningfully model even very simple concepts. A hypercube is virtually impossible to wrap one’s head around. So this is how I think about complex numbers. Take a coordinate like (3,4); I intuitively know what that is and how to find it on graph. But I could also sort of think of the axes such that “x” is a unit of left-right-ness and “y” is a unit of up-down-ness. Thus, with just a bit of a difference in notation, I could think of it as 3x + 4y. So, if we can have an arbitrary number of extradimensions, why can’t one of them be i instead of z or t or whatever? In just the same way that a particular dimension is useful in certain applications, like t is useful when time is relevant, and ignored when it’s not, or z is useful when dealing with three spatial dimensions and ignored when it’s not, i is useful in applications where the square root of negative numbers comes into play. Thus, taking the same concept above, you get something like 5 + 6i. Beyond that, the difficult part is that complex numbers don’t follow the same sorts of rules that your rationals do. But really, it’s the whole point that the rules they follow work differently that makes them a bit different. It’s been a few years since I worked on it, but I used them extensively in my work on my PhD thesis because I was working on modeling human movement, so they showed up a lot when working with angular velocity and such. I imagine that, considering both are related to the right-hand rule, that’s probably how it comes up in electromagnetism too, but it’s been far longer since I did any of that. ---
**Author:** ![Martini\_Enfield](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/martini_enfield/32/11123_2.png) [@Martini\_Enfield](https://boards.straightdope.com/u/Martini_Enfield)\ **Post date:** [December 9, 2013, 10:31pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/16 "2013-12-09T22:31:50Z")
This thread is clearly maths related and therefore might as well be in Martian (which is why I’m resisting the urge to say “i is not a number, it’s a letter!”, since apparently letters can be numbers in bizarro world), but i is often used in book prefaces and the like, we’re you’ll see the pages numbered i, ii, iii, iv and so on - a holdover from Roman numerals. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [December 9, 2013, 10:48pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/17 "2013-12-09T22:48:45Z")
> [@Martini\_Enfield](#): > > … I’m resisting the urge to say “i is not a number, it’s a letter!”, since apparently letters can be numbers in bizarro world… Can’t "i’ be both? The letter “x” can represent a number; the Greek letter “π” (pi) is a number… ---
**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 9, 2013, 10:51pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/18 "2013-12-09T22:51:19Z")
> [@Blaster\_Master](#): > > The unfortunate part about imaginary numbers is the choice of the name, Indeed. Gauss pretty much says exactly this: [QUOTE=Karl Friedrich Gauss] That this subject has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question. [/quote] ---
**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:** [December 9, 2013, 11:02pm UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/19 "2013-12-09T23:02:40Z")
> [@Giles](#): > > Can’t "i’ be both? The letter “x” can represent a number; the Greek letter “π” (pi) is a number… And let’s not forget _e_. ---
**Author:** ![Arjuna34](https://avatars.discourse-cdn.com/v4/letter/a/eb8c5e/32.png) [@Arjuna34](https://boards.straightdope.com/u/Arjuna34)\ **Post date:** [December 10, 2013, 1:23am UTC](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920/20 "2013-12-10T01:23:45Z")
> [@ZenBeam](#): > > From that page: > I’m an electrical engineer, I work in electromagnetics, and none of that looks familiar. > > The part lower down, about sinusoidal variation of voltage and current, where taking the derivative d/dt becomes multiplying by j omega is a very important advantage, and used all the time in circuit analysis and electromagnetics. > > Maybe someone else has seen the part I quoted in actual use, but to me, it looks like a garbled description of sinusoidal variation. I would avoid using that part unless someone else can tell you who actually uses that. I’m also an electrical engineer, and the section you quoted doesn’t make any sense to me either. Looks like a very mangled (to the point of being wrong) explanation of how complex numbers are used to express impedance. [Next page](https://boards.straightdope.com/t/practical-uses-for-the-number-i/675920.md?page=2)