A Visual, Intuitive Guide to Imaginary Numbers

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Hi Alonzo, happy to comment. I think you’re on the right track: we started with very basic interactions (addition), came up with a concept for repeats of this interaction (multiplication as repeated addition), and realized this more complex interaction could have other properties of its own (scaling). I think the true mechanisms of multiplication probably dawned on people over time (like genetics; previously it was the “bloodline”, where tall people had tall children, and we later realized it wasn’t the blood so much as the DNA inside which determined hereditary traits… we got more specific and nuanced in our understanding).

Looking back, we can now see that multiplication is a more advanced interaction which seems to transfer properties (like dimensions! inches x inches = square inches) whereas addition does not (inches + inches = inches). So, I think your story seems to make sense – humanity has had a gradual unfolding / clarification in the meaning of math.

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Thanks Reza, glad you enjoyed it.

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@Justaguy: Thanks.

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@Alonzo: No problem, happy to help

  1. (1 + i) is a 45 degree angle because it’s equal parts real and imaginary, similar to how a NE (northeast) trajectory is at a 45-degree angle because it’s equal parts North and East.

Think of “i” as an indication of direction – we’re used to most numbers being on a single dimension, but “i” is an indication that a number is in this new direction. Confusingly, i is often written by itself (instead of 1i) so it gets mixed in. The magnitudes being the same is similar to “1 mile East is the same distance from me as 1 mile North”, i.e., both are a single mile away. Similarly, 1 and 1i are both “1 unit” away from 0, but in separate directions. This is what the magnitude is meant to measure: how far are you from zero?

Yep, it’s best to think of “i” as a type of symbol indicating direction [similar to a negative sign], and in many cases you’ll see numbers written as “a + bi” (i.e., the number is a in the East direction, and b in the North direction). Because we’re sloppy sometimes, we might just write 3i instead of (0 + 3i) to indicate 3 units only in the i direction (i.e. 3 units in the North direction).

  1. I should have clarified: when I wrote “Multiplications can do two things: scale (change the size) or rotate (change the orientation).” I meant that multiplication can do one, the other, or both :).
  • “times 3” means scale up by 3
  • “times negative 1” means rotate 180 degrees (if you were going forward, go backwards)
  • “times negative 3” means rotate 180 degrees AND scale up (if you were going forwards 20mph, go backwards at 60mph).

Imaginary numbers give us a new type of rotation: we can rotate “partway” (i.e. 90 degrees, not the full 180), and we can still scale, so there is some complex number which is “double your speed and rotate 90 degrees counter-clockwise”. (In this case, rotating 90 degrees counter-clockwise is “times i” and doubling your speed is “times 2”, so “times 2i” would have both effects).

You raise a very, very good question about multiplication. I think we’re still discovering what multiplication in it’s purest form is, just like we uncovered what “numbers” really were. We started out thinking numbers were things you could count (fingers, pebbles) and then realized “Hey, there are halfway numbers, like fractions!”. So we had 1/2, 1/3, etc. Then we realized there are certain numbers that are partway but are not fractions (like sqrt(2)) and we got the real numbers (some sequence of decimals). Then we realized that numbers could be negative (why can’t they be less than zero) or complex (why are they stuck in one dimension?).

Each time we discovered new numbers, the meaning of multiplication expanded a bit. With pebbles, multiplication might mean repeated addition. With real numbers (like 13.45 hours * 54.12 miles per hour) we might think of “scaling” or similar (vs. repeated counting). With negatives and complex numbers, because they move in different directions, multiplication implies motion in that new direction to.

To me, the essence of multiplication is “applying” one number to another. When you apply a number, you transfer its properties to the result. So multiplying by a negative number gives the “negativeness” to the other number. Multiplying by a complex / imaginary number gives the “rotation-ness” to the result. Multiplying by a large number gives the “largeness” to the result. Without getting too cute, the essence of multiplication is how to apply the “essence” of one number to another. Hope this helps!

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Hello Kalid. Over the weekend, I gave some thought to what you wrote in your last post and I would like to see whether my understanding is on the right track. The following is a story I am trying to put together in my head by collection various pieces of information (I am sure there are gaps):

It sounds like the Addition (along with Subtraction) of objects represents one of the most natural concepts to humans. Mathematics provided a formal definition for Addition (perhaps there were multiple definitions) to capture this natural observation and - from there- realized inherent properties and conclusions based on these definitions (and it was useful). But at some point, someone decided to define a more complex operation on objects, yet - it still had addition at its core. In the early days, the complex extension was in the form of repeated addition, thus we have the early definition of the term multiplication - and it was useful. However, overtime folks realized different forms/definitions of complex operations that also had addition at it’s core - and they were also useful. Because these operations seemed to share an enhanced complexity involving addition - they were also called Multiplication. Of course, based on the various definition, we would get various inherent properties and interpretations (though some were shared among the different definitions)

So perhaps a pattern was established with these various forms of “complex addition” such that the operations result in a complex interactions between the objects involved. This complex interaction forces the very nature of their interaction to have a strong influence on what the result looks like. This leads to descriptions of Multiplication like you stated above, " the essence of multiplication is how to apply the “essence” of one number to another".

So to generalize, Addition for your system asks, “how” do you want to define a simple union, collection, joining for the object in your system. Multiplication asks, “how” do you want to extend the use of the addition operation to define a more complex interaction among the objects in your system. Now your definitions will probably result in observed properties, and some of them may actually be useful. But, whether or not the properties or the system as a whole turns out to be useful - you get to define it. The questions of “how” associated with Addition and Multiplication asks you some common questions associated with defining a Mathematical system. Thus you gets statements like those from above: “And if it’s your own type, you need to describe the behavior of the “x” symbol (which you may call Multiplication)” - you get to define it."

I suppose you could generalize it even further by saying Addition is “how” you - the creator of a system - would like to define a simple interaction between your system’s objects and Multiplication is “how” you - the creator- would like to define a more complex interaction between your systems objects. But, at this point, it seems like the abstraction has gotten to a point where meaning and the ability to distinguish starts to be come lost.

So Kalid, would it be possible for you to comment on this story I am trying to create for myself. Perhaps some things are factually incorrect. Other things are consistent with how things really work. In other words, as someone with much more knowledge on this subject that myself, would you critique the story ?

Thanks,
Alonzo

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Hi Richard, thanks for the note! Glad you’re enjoying the style, I try to write in the way I would want to be taught (in a casual, informal manner).

Great question on the size. When multiplying, we give all “properties” to the result. When you multiply something by -2, not only does it become negative, it doubles as well.

Imaginary / complex numbers will “give” their angle, but also their size. For simplicity, I made a 45-degree angle using a triangle of sides 1 and 1. But, the length of the diagonal is sqrt(2), or about 1.4 [by the Pythagorean theorem]. So, when multiplying, the final result was sqrt(2) or 1.4 times bigger than the original. In this case, we were only interested in the angle anyway, but for consistency we could divide by sqrt(2) to bring it back to the original size. I might need to make this more clear though, thanks.

Ah, good old negatives, tricking us into the new millenium :). Although to be fair, it’s easier to think “lower” means smaller. Maybe a better wording would be “find a colder temperature” (not lower) and hopefully it’d be more clear.

Glad you’re enjoying the site!

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@prakash: Awesome, glad it’s clicking. Great question by the way.

When you multiply by a complex number, you also scale by its size. i has size 1, and i/2 has size 1/2. So that’s why multiplication by i/2 halved the magnitude (if you like think of i/2 as i * 1/2, and do the multiplications separately).

To keep the original distance the same, you want to multiply by (1 + i)/sqrt(2). (1 + i) has size sqrt(2) [by the Pythagorean theorem], so we divide by that amount to bring its size back down. There’s more in the follow-up article on complex numbers. Hope this helps.

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@Douglas: Thanks for the comment! (Great English by the way). Really happy you’re enjoying the site, I like connecting with other people who want to find the intuition behind concepts. Math is only cold and static if we don’t really understand it :).

Good question – if we have an angle like 58 degrees, we’d need to use sine/cosine/pythagorean theorem to figure out the x and y. In a world where we somehow know that the angle is exactly 58 degrees, we would also have calculators to compute sine and cosine.

In a world where we just measured the angle (measuring the trajectory on a map, for example) we can just use the measurements directly. That’s the really neat thing about imaginary numbers!

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I just want to try to give you a brief introduction into Geometric Algebra.
In physics we learn that numbers for them self are meaningless. You always have to accompany them by the unit. 5 seconds are quite different from 5 Volts and 5 inches but also from 5 years. What Geometric Algebra does is mainly taking this second part into account. So i.e. in 2D space we can find 2 different directions e1, e2 (think here of unitary vectors that have their meaning (inches, m, ym, …) already coded in their length. e1 is i.e. 1m in one direction) These two directions we want to use as a base to describe our problem. We then can describe any point in our plane as an instruction how to go there from the origin. (go 5 times in e1 and 2 times in e2: x = 5e1 + 2e2). If we are in a nice 2D space we of course can go the other way like x = 2e2 + 5e1, so addition is commutative. You will then think about addition, subtraction and … multiplication. And then you will come to the point to ask what is the meaning of multiplication? Usually multiplication of 2 distances is something like an area and so it’s here. There appears an element that’s something like ‘weight times e1 multiplied with e2’ that I want to write as: Ae1@e2. A is just a scalar, the ‘weight’ and corresponds to the area. But what about these entities? e1@e2 is called Grassmann product of e1 and e2 and it’s anticommutative so e2@e1 = - e1@e2 what means it does matter which way around you go that area to describe it. So e1@e2 has some kind of ‘orientation’. (In physics we often describe planes with normal vectors that are oriented perpendicular to the plane - forget about the vectors but think of their properties - they can be oriented in two opposite directions to describe the same plane.) You then will find that this Grassmann product also does a rotation of 90 degrees if you multiply it with a vector in the plane and then you also understand why it’s important to know the direction. (Here you find already all the ingredients of complex numbers.) And finally you will find that squaring this aera-element corresponds to a 180 degrees rotation. Square(e1@e2) turns any vector into it’s opposite.

Conclusion:

  1. Geometric Algebra treats numbers AND their entities (units, directions, … their meaning)
  2. when it comes to multiple dimensions (multiple units, directions, meanings) i can be associated with the result of the Grassmann product i = e1@e2 which by itself is the unit element (an oriented plane) of the multiplication of the entities of the two different dimensions.
  3. The Grassmann product e1@e2 has a clearly understandable meaning and i has been revealed as a powerful workaround that we already could get rid of.
  4. We could and should introduce a new aera of ‘meaning’ and ‘understanding’ into physics by application of the Geometric Algebra that’s already well defined and widely used in computer graphics. But we are already on the run. So that’s what’s really mindblowing!

PS: I used a unusual notation here to avoid confusion. e1@e2 is usually written as e1^e2. As this might be confused with ‘to the power of’ I have chosen my unusual notation.

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thanks so much…I’m a beginning high school math teacher… looking to explain complex numbers so students can better understand it.

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I was pretty lost when thinking about the concept, but it makes much more sense now. Thank you!

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well thanks a bunch! After realising i did not know what complex numbers were I thought it would be useful to learn. Excellent layout and presentation of knowledge. Now, on to prove the Riemann conjecture!

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Just found this by way of the Fourier article linked by a Facebook friend. Good stuff!

Check out Isaac Asimov’s Adding a Dimension, if you haven’t already. You seem to have the same approach as he did. There’s even a chapter in it about complex numbers (including a conversation much like you had about negative numbers…).

I mastered complex numbers years ago, but I’ll read this again - never know when you’ll pick up a fun factoid. (Prob’ly spend more time on the Fourier article…)

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我晕，看不懂

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Got it ! i.e what is the root of i . I may come back to you as something arose when I was trying to figure it out, that I cannot see a solution to but I will try before I bug you. Thank you again,Kalid. I will be back.
:slight_smile: from Ireland.

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Hi Ian, glad it helped! Give yourself some credit, imaginary numbers aren’t easy! It took the math field a few decades after their invention to really accept them :).

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Hi Carrie, really glad it helped!

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Hi there, I just wanted to say Thank you. I love maths but have never studied much beyond secondary school level. You have a great way of explaining things it makes it all very tangible. I am sitting back with the square root of i problem and slowly but surely getting there. I know I should see it already, but I will eventually :slight_smile:

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I graduated with a BSc in electrical engineering and I haven’t understood complex numbers like I do now
Open a university
Great thanks

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Hi Paul, thanks for the comment! Haven’t seen that book yet, I’ll put it on my reading list (and man, I wish I’d seen that complex number analogy earlier in school).

Definitely agree about the factoids. I’ve realized I’m never “done” with learning about even basic concepts. They’re like atoms, where you break them into electrons, then quarks, then strings (?) and so on. There always seems to be another level to explore.