Intuitive Understanding Of Euler's Formula

Euler's identity seems baffling:

$$e^{i\pi} = -1$$

It emerges from a more general formula:

This is a companion discussion topic for the original entry at

[…] This post was mentioned on Twitter by GreyMatters and Pi Guy, Kalid Azad. Kalid Azad said: New Post: Intuitive Understanding Of Euler’s Formula […]

[…] pretty good and actually intuitive explanation of the Euler’s formula: Intuitive Understanding Of Euler’s Formula. With beautiful pictures as you can see […]

This is a phenomenal article! Took me quite a bit of time and multiple readings to get my head around it, but now I get it. I think. :slight_smile:

Hey cool that makes a lot of sense. I already thought of imaginary numbers that way, but the growth pulling in a circle is very straightforward.

Are there complex equations that move in this way that intersect the real set of numbers in cool ways? Like I could imagine some complex equation that loops around, intersecting the real set of numbers (the real numberline) to create the set of primes or something like that.

@Aditya: Thanks! Yes, it took me a while to really see the equation, there may be a nicer way to go back and streamline how it was presented – I’d like to avoid the need for people to have multiple readings :).

@Brandon: That’s an interesting question – actually, the Mandelbrot set is like that to some degree, where there is a complex (2d) function which gives rise to some pretty amazing patterns. I don’t know of any others off the top of my head though.

I <3 BetterExplained - keep 'em coming!

This gets easier if you've already got the hang of the physics concept that to move in a circle you must keep accelerating. If you accelerate in one direction, you will get faster and faster, but if you keep accelerating in a new relative direction, your speed will be the same, but you'll move in a circle (your velocity changes.)

Acceleration is a kind of growth, and so it logically follows that if you grow in a relative direction, you'll rotate but not speed up.

@wereogue: Thanks for the support! Yes, the physics interpretation definitely helps see this relationship, and I like the way you put it – our growth/change is really an acceleration. Our velocity is always perpendicular to our position (and acceleration perpendicular to velocity) which gives us a circle. It’s funny how much overlap there is between math and physics :).

Thank you for showing us that Maths can be easy and simple. You are an example to today’s mathematicians, what you are doing is really inspiring. I always try to think how to make things easier to teach, learn and do. But you really make it so well!

Keep it doing it!
you can go very far!

(Excuse my English)

Great article, Keep going on…

Great article and great video!

I was wondering about what you used to generate the graphics, they look great :).

Cheers from Romania,

I didnt bother to follow your argument because the topic doesnt interest me but I like your attitude that math is just logic and “common sense” and theres way too much hocus pocus and 'mysticism" that often creeps in imho.

Thanks for making me excited about math again.


I’m still digesting it all, but i just have to say: “continuous perpendicular growth will rotate you” is just plain sexy. Wow, it makes sooo much sense to me :slight_smile: Keep it up sir!


Kalid, this is extremely impressive. I’ve been trying to understand this for a long long time. I found it difficult to see past the numbers and symbols and wanted to understand it ‘visually’ so I knew what was trying to be achieved. You have described it all beautifully and for the first time I am really understanding how it all fits together. I so wish you were teaching me in school 25 years ago. Thanks again.

@Kalid’s Friend: It really bothered me for a long time also – Euler’s formula was used everywhere but I didn’t have a gut feel for it! I’m really happy it was able to help :slight_smile:

@Mariano, @Mithun: Thanks for the kind words!

@stuart: Yes, I think everything should be understood / explained intuitively, and not accepted as mystic.

@mark: You’re more than welcome.

@Sebastian: Haha, I like that phrase too – whatever it takes to make it click :).


Thank you for this!

Would it be accurate to say that if you traced out the complex growth curve just as you did the real and imaginary growths, you would get a spiral?

By “tracing out”, I simply mean that if you are given e^(ax+bi), you simply put points at specified intervals to the answer. For example the following would give you 3 points on the way to your answer:


Oops, on my previous post, please assume x=1.

@lewikee: Yep, you got it – imaginary exponential growth rotates you in a circle, and regular exponential growth grows you (e^a/4, e^a/2, etc.). You end up spinning around the circle but getting further and further away, making a complex spiral.

If you put

parametric plot (e^(t/4) * cos(t), e^(t/4) * sin(t))

plot on wolfram alpha

into wolfram alpha you can see an example (I separated e^ix into cos(x) + i*sin(x) to get the x and y components, and put them separately into the plot – I also scaled down the regular exponential growth to make a tighter spiral).