I had the same question last year and though about it a lot. I think we don’t need it because a negative logarithm does have a solution when you consider complex numbers. In fact, every functions (sin, cos, log, …) can be expressed as a complex exponential. So there is no need to define as ln(-e) as it would be redundant.
Note that the square root is a power function. Power functions are pretty common compare to logarithm so it makes a lot more sense to define the imaginary part from this function. I think we could define the imaginary part from the logarithm function but it would end up more complex. After all, we use power functions much more often then logarithm .
This is a great article. It has explained a lot of what goes through my mind about complex numbers. How would you describe a 3-dimensional system of numbers? What new numbers would you use and what would their role play?
@KT: Great question. The problem with a fixed translation is that the amount varies: going from 1 to -1 in two operations requires steps of 1, but going from 9 to -9 requires steps of 9.
With a rotation, the size of the input doesn’t matter: we turn 90 degrees with each step, and always face backwards. It’s a good point and I’ll see if I can clarify that. (Move from forward to backward in two steps, no matter how large the original number is.)
@Matt, @Alex: Interesting question. There might be a way to define i as the solution to a negative logarithm.
i = e^(i * pi/2), so
which is a bit of a recursive definition but perhaps can work? I haven’t taken formal Algebraic structures classes to know exactly how these are all constructed, but there may be a way. But technically, yes, the logs of negative numbers can be solved with complex numbers.
@Joseph: Great question. You can extend to 3d and more (4d are quaternions). In some dimensions problems arise (in 3d there’s the problem of “Gimbal Lock”) but it’s a fun thing to think about!
Your posts are fantastic. This is the way math should have been taught in my high school!!. It’s a bit late now, but I am trying to catch up! A basic question for you.
In your example, how does multiplying by (1+i) rotate something by 45 degrees? Since multiplying by i rotates by 90 degrees, I am thinking we should multiply by i/2 to rotate something by 45 degrees. What is wrong in my thinking? thanks
which is two operations: scale by 1/2, then rotate by i. But what we really want is for the rotation in i to be divided by 2, not to reduce the size of the number.
The square root splits an operation into 2 equal steps. For example, 9 represents a jump from 1 to 9. The square root of 9 is 3, which represents two equal steps: 1 to 3, and 3 to 9.
So, if we want to jump to i (90 degrees) in two steps… we’d take the square root of i! Whoa, what a weird concept:
$$\sqrt{i} = 45-degree turn$$
We can visualize this with the number (1 + i), which is a 45-degree angle (it’s equally strong on both sides, a perfect diagonal, just like a 45-degree angle).
Technically, the number (1 + i) is a scaled up version of the square root of i, since its magnitude is sqrt(1^2 + 1^2) = sqrt(2). But the direction is the same.
This is definitely a tricky thing to keep straight, you might enjoy this article:
Something which vexed me though was the following paragraph:
“If we multiply by -i twice, we turn 1 into -i, and -i into -1. So there’s really two square roots of -1: i and -i.”
Looking at the figure, this doesn’t fit in with my mental image of the situation at all - in my mind multiplying by -i “turns the dial” 90 degrees - so this would (per the figure) turn 1 into -1 rather than -i.
When you say “What transformation x, when applied twice, turns 1 into -1?”, I completely agree that this would be two transformations of either i or -i, but this seems to be in conflict with the first statement I quoted.
I’d really appreciate a clarifying response (keep up the good work!)
Hi Nicolay, thanks for the comment. Positive i is a 90-degree turn counter-clockwise, and negative i is a 90-degree turn counter-clockwise. So, in both cases, it takes 2 steps to go from 1 to -1. Hope that helps!
“Positive i is a 90-degree turn counter-clockwise, and negative i is a 90-degree turn counter-clockwise.”
Yes, I completely agree with that. What I don’t understand is how you can do TWO turns counter-clockwise and go from 1 into -i [1]. In my head, that would go from 1 to -1, not from 1 to -i.
1: “If we multiply by -i twice, we turn 1 into -i"
Ah! Now I see. That sentence wasn’t clear, 1 to -i is the result of the first rotation by -i. The second rotation by -i would turn -i to -1. [The full sentence is: “If we multiply by -i twice, we would first turn 1 into -i, and then -i into -1.”]
I’ve updated the text to:
“If we multiply by -i twice, the first multiplication would turn 1 into -i, and the second turns -i into -1.”
The problem for me is the justification for the complex plane and the rotation concept in the first place. The number line was a logical outgrowth of the discovery of zero, negatives, rational, and irrational numbers and just filled in our ‘blind spots’. It is coherent and intuitive and corresponds to our experiences in everyday life.
The complex plane does not have that intuitive feel. Rather it feels like a gimmick that happens to work for describing cyclic phenomena. I have a feeling that this may be a ‘blind spot’ for me and I am waiting for the lights to go on.
Hi there, Kalid. First off, I just wanna thank you for the great article, it demystified a lot of the hocus-pocus surrounding complex numbers. It did raise a couple of questions in my mind, however:
Multiplication by i. I understand the concept of it rotating a number by 90 degrees (or pi/2 radians!), but what does its coefficient do? For example, say we had a complex number z = 3 + 4i. What does the 4 do? My raw intuition tells me that it would represent a full rotation. Is this correct? If not, what does the coefficient actually mean?
Is it possible to combine 3 complex planes at right angles in order to be able to express rotation in any direction? Again, my intuition tells me it’s possible, but I can’t reconcile it with the fact that a garden-variety 3-D system uses three axes, since a complex plane contains a real and an imaginary axis; it would seem that there wouldn’t be enough “room” (I hope that conveys my meaning properly. Probably not.) to include the necessary 3 imaginary axes and the 3 real ones.
@Dan: Good question. At the lowest level, the complex numbers are an algebraic relationship that are consistent. I.e., the rules of multiplication, division, square roots, etc. work out, and don’t lead to contradictions.
However, as humans, that isn’t always enough to accept the concept :). The notion of a complex plane, a 2d geometric metaphor, maps to the properties of the complex numbers. However, it isn’t the only way to see them – there are other metaphors (like matrices) that might work better. My brain stops itching when I see the complex plane interpretation but others may work for you. If you find them let me know!
For multiplication, the complex parts rotates you, and the real part scales you. So when you multiply by “4i” you are saying “scale up 4x, and rotate”. The real coefficient is just a scaling factor. If you want to rotate twice, you’d multiply by i^2. (Which is -1, a 180-degree rotation.). Check out this article for more:
Great question. Yep, it is possible to represent 3d rotations, but you you actually need 4 dimensional numbers (3 imaginary axes and one real one). These numbers are called quaternions, and I’d like to do a follow-up on them.
Kalid- I’m elated I stumbled upon your blog. I am a person who questions everything and once I started learning complex numbers, my natural reaction was: Egad, Aren’t there enough concepts and theorems and whatnots to CHOKE our minds- and as if it weren’t enough already, we have to learn imaginary concepts that don’t even EXIST? What the hell man. (cough cou-) (…)Wow you just silenced the choking of my mind! xD
My teacher had actually mentioned about the rotation thingy and I was a little mind blown. Now all that’s left has exploded into a thousand jittery pieces, and- wait am I making this too ‘complex’?
I just noticed something.
So does this mean that complex numbers and vectors are not only related, but very much interconnected- so much so that they almost mean the same thing? Just that they seem so far apart in our minds that we can’t see the big picture?
And one more thing.
I’m 'tan’kful 'co’s I 'sec’retly ‘cot’ on a lot of meaning as’sin’ed to this topic. (Seems like you’re not the only with a thing for puns)
Can’t find anything with cosec. racks my brainsboggles my mind cough - Uh- My mind started choking again! cough ‘co- * 1 sec’ I think I found it!
AHA!! x’D
(keep up the good work!)