Intuitive Understanding Of Euler's Formula

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141

I was hoping you’d share your opinion of what ‘i’ means to you.
Mathematicians say that imaginary numbers system is orthogonal to real numbers systems but i see no suggestions about an inherent meaning of arriving at a square root -1 when deriving an equation related to a ‘real world’ phenomenon (e.g., electricity).

Further, cos(x) + sin(x) is all that one requires to ‘rotate’ around a circle, point by point.
The inclusion of ‘i’ as a coefficient of sin(x) does not induce a rotation.
however, multiplying 2 COMPLEX numbers does result in rotation: “multiplication by a complex number of modulus 1 acts as a rotation.” - wikipedia

So, I ask, are you convoluting the vector form of the equation of a circle [cos(x) + sin(x), where x stands in for an angle of rotation] which gives the path of rotation around a point (a circle) versus the general form of the Euler equation which includes i as a coefficient…which isn’t the same as ‘1’ as a coefficient…i.e., it changes the meaning of the equation.

I ask, not accuse, because it has been many years since i finished my engineering degree, and so haven’t played with these in a while.

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misanthropope, I appreciate the content, though not the tone, of your suggestions (I hope you come to realize that snarkiness does not lend wisdom or gravitas to a message). If the title didn’t make it clear, the post isn’t for those exploring the deepest nuances of the equation (see: Wikipedia, MathWorld, your typical professor, etc.). Sure, we can substitute i = i^5 = i^9… or explore the equation with calculus, baffle newcomers, and feel macho and clever. My goal was to be helpful.

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Hi Kalid, thanks again for your great articles. I have one question on this topic.

I understand that imaginary growth rotates around the unit circle, and I understand that using sin/cos we can do the same. What I don’t understand (maybe I’m missing a point here) is that the rotation we get by plugging a number into e^i.x gives us the exact same rotation as plugging in the number into cos(x)+i.sin(x). Why do these two ways of rotating are exactly ‘in sync’, rotate at the same ‘speed’?

I hope you understand my question, basically: why are we sure that for example e^i.2 = cos(2) + i.sin(2) ? I see this must be true for the special case of euler’s jewel, where we plug in pi, but that does not guarantee that this is the case for the general formula (‘the intermediate steps’ so to speak). Might be a dumb question but the reason why this is true doesn’t click for me currently. Thanks!

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e represents the process of starting at 1 and growing continuously at 100% interest for 1 unit of time.

Shouldn’t it be - over an infinite amount of time. e raised to anything approximates to the expected result only when the time approaches infinity. Not nitpicking - I am just trying to reconcile this with your other article on e. Great stuff, keep posting - now I can explain these things to my daughter without having to hide behind mysticism.

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@Pete: Thanks for the feedback, I’ll clarify that part!

@George: No worries, great question. e is just a number, about 2.7. So we can rephrase it like this:

If you have 100% yearly interest, but compound as fast as you can (i.e., get your interest every microsecond and add it to the principal), you’ll turn 1.0 into 2.7 at the end of the year.

Now, we can repeat this process for any number of years (not just 1), and that becomes 2.7^x (where x is the number of years we compound). So if we let this system run for 2 years we’d get 2.7 * 2.7 = 7.29, and so.

e is the exact amount of interest we get (a tad bit more than 2.7), and we wrap it up as a letter (just like pi, so we don’t have to worry about our decimal accuracy, we can say “Ok, that exact amount of interest you get, let’s just call it e and work from there”).

Hope that helps clarify! We can of course wait any number of years, but e (2.7-ish) is what we have after waiting a single year. (I’ve said year here, but it’s really whatever time period we’re looking at. If you get 100% continuous interest per month, then e is your growth after 1 month).

146

that’s great

147

Hi Kalid,

… and another thing … :wink:

I’ll admit that it seems like a cheat to to use a rotation definition to avoid explaining why circles and especially unit circles appear on Argand diagrams for purely complex exponents, but I have a background thought process that led me this way.

Allow me to explain:

We are accustomed to thinking of the multiplication of real numbers as a pure scaling operation, and it makes perfect sense to do so, BUT, if we think of real multiplication in a slightly different way, then maybe we can intuitively understand complex multiplication without fearing the apparently enormous gulf between real and complex numbers.

Specifically, look at multiplying by a negative number, say -2. Our old-fashioned brains think of this is just ‘going the other way’ on the real number line. Fair enough.

BUT, if we think of a real number as a vector, or an arrow from the origin to a point on the real number line, it is intuitive that the arrow of the negative number is made to ‘point the other way’ using the already familiar concept of rotation by 180 degrees (or pi radians).

So if we think of multiplying by 2 by -2, it could be broken into two distinct (orthogonal) steps. Step 1 - changing the magnitude: scale 2 by 2 to get an arrow pointing from the origin to 4, then Step 2 - rotation around the origin: rotating the new arrow by 180 degrees, leaving the arrow pointing to -4.

The new way is still the correct answer, and it seems a bit weird, but it subtly introduces the concept of complex number multiplication. The only remaining step is to consider rotations that leave the vector pointing to ‘a position not on the real number line’, and draw that position on the diagram I know as the Argand diagram.

Clearly, any rotation that isn’t 180 degrees will end up in this new ‘space’, but now it becomes a simple exercise to demonstrate that squaring a new ‘special point’ we can now locate at a magnitude of 1 and an angle 90 degrees off the real axis, results in the real number -1. Logically this ‘special point’ must be the square root of -1, or the famous ‘i’!!!

Using this thinking maybe ‘i’ makes a sort of intuitive sense, that is hinted at by thinking of multiplying by our usual negative numbers as being done by a special ‘180 degree rotation’ operation on vectors along the real axis.

I will admit that since I am an Engineer, it is very easy for me to introduce things that others might find incomprehensible, such as vectors, so I was wondering if you felt that this explanation meets your own standard of ‘intuitive’ for explaining the existence of i, the famous square-root of -1, and the multiplication of complex numbers?

Thanks again for paying attention to my ramblings, I look forward to any comments you may have.

Kind regards.

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@Appreciative: Thank you! Very glad the material helped :).

149

“3 is the end result of growing instantly (using e) at a rate of ln(3). 3 = e^ln(3)”

This line reads confusingly. It’s two sentences but could be read as (ln(3).3 - e^ln(3)
I’m sre some people could interpret the dot as an operator such as multiplication.

"3 is the end result of growing instantly (using e) at a rate of ln(3):-

3 = e^ln(3) "

is a lot clearer.

150

Sorry- there’s no way to edit these comments - that should read as follows:-

“3 is the end result of growing instantly (using e) at a rate of ln(3). 3 = e^ln(3)”

This line reads confusingly. It’s two sentences but could be read as (ln(3).3 = e^ln(3)
I’m sure some people could interpret the dot as an operator such as multiplication.

“3 is the end result of growing instantly (using e) at a rate of ln(3):-

3 = e^ln(3) “

is a lot clearer.

151

pardon me, that’s horrendous. why not start by reading what euler wrote about the topic?

euler’s identity has a much more fundamental origin than the trig function formula which generalizes it. one could, for instance, demonstrate that the derivative of [exp(ix)/{cos(x) + i sin(x)}] is zero, with a pretty minimal set of assumptions and zero hand waving.

i^i has an infinite number of distinct values. that fact (and it is a fact) ought to make you rethink some things. i strongly recommend beginning with powers (and especially fractional powers) in the complex plane. it’s a precalculus topic and your misapprehensions there are, for want of a better phrase, compounding.

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To the author:
You have a great way of presenting information… You clearly have a deep understanding of the subject, so much so that you can explain it in such ‘simplified’ and physical forms. You would make a great teacher (if not already).

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Hey Kalid, Thanks for the pointing to this article. I’m going to need to go back a couple of times! But this was exactly what I was looking for. Thank you.

I’m currently working for a Grant tasked with enhancing calculus education for engineers (by making things less abstract, something I know you care about…). If you are interested, I’d love to hear your thoughts. (You should have my email from this comment). But either way, you’re very good at what you do, keep it up!

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Thanks JD, very glad it helped! I just reached out to your email :).

155

Very nice job, helped me quite a bit to understand better what is going on!

156

Hi again Kalid,

There is an excellent video on youtube with clear graphics, showing visually what the the Laplace transform does, how it relates to the Fourier transform and control systems theory.

I am still digesting it’s contents, plus other videos, as I try to get a better intuitive grasp on how the Laplace transform works…

157

Hello Kalid, Happy 2014 I have often thought that many things can be considered as a metaphor for human existence, or at least an aspect of human existence. So as I was reading your post, I "imagined " the inside of my skull to be a “3D” complex plane if you will ( a spherical 4 pi r^2 version ) where the Complex plane has a Real axis , and an Imaginary axis, but also possesses a third axis ( similar to X Y Z ) , and along this third axis can be plotted a point that is part real, part imaginary and part part awareness. As a metaphor…many times

158

Sorry Kalid, I accidentally hit the "post comment’ button, so I will start again

Hello Kalid, Happy 2014 I have often thought that many things can be considered as a metaphor for human existence, or at least an aspect of human existence. So as I was reading your post, I “imagined ” the inside of my skull to be a “3D” complex plane if you will ( a spherical 4 pi r^2 version ) where the Complex plane has a Real axis , and an Imaginary axis, but also possesses a third axis ( similar to X Y Z ) , and employing this third axis can be plotted a point that is part real, part imaginary and part awareness. As a metaphor………many times I find myself taking the real and mixing it together with the imagined , but when I apply real concentration / awareness to what is going on, rather than focusing on those two only , I find myself focusing on the “actual” within me, and this “rotates” me within even further, and I experience “growth” that is neither linear nor circular. Neither sensation nor thought. I once read that true learning has occurred, if the student comes away feeling lighter than before he/she felt before the lesson. I know I have extrapolated “away” from the mathematical topic being discussed, but all learning should hopefully bring us closer to our heart"s desire. May I just say a big thankyou To you Kalid, for taking the time to share your mathematical insights in not only a clear and good humored fashion, but also the quality of your presentations are first class. I will leave you with a quote…" The essence of mathematics lies in its freedom."
Sincere Regards Bluetone

159

G’day Kalid,

I like your explanations - very much actually, and I enjoyed the read - thanks for that!

There is just one thing that I do not get, and it seems to me to be central to the whole issue.

Your explanation flubbed one very important point - Why is the complex magnitude fixed when raising real numbers to purely imaginary powers?

Your explanation of purely imaginary powers reads, in part: “Surprisingly, this does not change our length …This is something I want to tackle another day”.

It is also very surprising to me, and it seems to me to be the key to understanding complex numbers intuitively.

Let me restate what I am after in my own words:

“What is an intuitive way of saying that ANY real number (say 2) raised to ANY purely imaginary power, say 2i, MUST lie on a perfect circle, centred at the origin, on the complex plane?”

Your article has focused on the real number, e, raised to different, purely imaginary exponents, like (i x pi/2), and (i), and (i x 2 x pi) etc. Fair enough. But if I use ANY other real number instead of e, like 2, or 2 million, and raise them to the same imaginary exponents, then they ALSO lie (at different places) on the SAME UNIT CIRCLE on the complex plane! Wow! Intuitively, why must this be the case?

Of course our starting point must be that any number raised to the zero exponent, including (i x zero), must be the real number 1, plus I also accept, intuitively that a purely imaginary exponent will rotate just like any complex multiplication.

But here is the crux: what stops the resulting magnitude from increasing from away from 1, (or decreasing below 1) to create a spiral on the complex plane?

It ‘feels’ like the action of purely imaginary exponents is on a fixed-length ‘rod’ of length 1 tied to the origin, but I lack any intuition for why the rod’s length cannot vary as it rotates. I understand intuitively why it starts at 1, and why it rotates around the origin, but the surprising part is that it does not ever move away from or towards the origin in a kind of spiral, instead of the perfect circle.

The standard explanations I have found seem to prefer Euler’s proof using the power series expansion for e: relating it to the power series expansions for sine and cosine. For my taste, a power series is not ‘intuitive’ - it’s hard for me to get a good ‘intuitive meaning’ derived from a sum of the infinitely differentiated functions. Possibly you agree?

Can you please help me here? Why is the action of raising to purely imaginary powers a ‘rod-like’ rotation around the origin? Why is it, intuitively, that the magnitude cannot be anything but 1, regardless of the real number we choose, or the real number we multiply by i to raise the exponent?

My current thinking is that it must relate to the subtle difference between multiplication and exponentiation. These closely related concepts are not the same, and the ‘special nature’ of exponentiation somehow forces a fixed magnitude when using purely imaginary powers. Not much to go on so far!

I sincerely appreciate your input on this problem.

For complete disclosure - I am an old Electrical Engineer, and all this maths stuff I have been doing for years. I have been trained to apply the rules, but in some cases I just behave like a robot instead of using my intuition, and your article has inspired me to fill in the gaps in my intuition.

Many thanks in anticipation.

160

G’day again Kalid,

Further to my previous question - perhaps the answer to my own question lies in the meaning of any complex multiplication by purely imaginary units (i times something).

It seems intuitive to me that just multiplying by i causes a ‘pure’ rotation around the origin, and continuing this pure rotation defines a perfect circle.

Further, exponentiation (raising to a power) is a ‘kind-of’ multiplication, so it seems reasonable to conclude that exponentiation by purely imaginary numbers will retain the purely circular characteristic - purely circular rotation around the origin must be what happens when we multiply or raise to the power of a purely imaginary number.

The subtle difference between multiplying by i and raising to the power of i is simply this: Multiplying rotates in a perfect circle, retaining the original magnitude, but while exponentiation retains the purely circular characteristic, it also forces the magnitude (radius) of the circle to be exactly 1, because anything raised to the zero power is 1, and this must always be one point on the resulting perfect circle.

Euler’s equation simply means that the ‘special’ real number ‘e’ is the rotation that repeats every 2 times pi times i. Other real number bases higher that e rotate ‘faster’, and numbers lower than e rotate ‘slower’ than 2 times pi times i, but still on the same circle. The real base determines the rate of rotation.

Euler’s identity is simply the special case when we use the base e, raised to purely imaginary numbers, it rotates through real the axis at -1 (going half-way around the circle), at exactly [pi times i]: e^i.pi = -1.

Or, said another way: all real numbers raised to purely imaginary powers rotate on a circle that repeatedly cuts through 1 and -1 on the real axis - it’s just that the base e first returns to the real axis when the exponent is exactly i times pi, and begins to repeat then circle when the exponent is exactly i times 2 times pi.

This ‘intuition’ is a sort-of cheat - I have defined my way out of this problem by saying that purely imaginary multiplication means rotation in a pure circle. Complex multiplication in general is the combination of this pure rotation for the complex angle, plus a translation for the magnitude.

So, do you feel that I have answered my own question? Does my explanation count as ‘intuitive’ in your view?

At least I have avoided referring to a power-series definition of e (not very intuitive), plus I have been more general, in that I am not restricting myself to imaginary powers of e, but discussing the ‘shape’ of all imaginary powers of any real number.