Intuitive Understanding Of Euler's Formula

pi will never go to where it’s been. This is the same as a spiral that never touches itself. I suck at math, i wish somebody could use this equation to solve for pi.

I appreciate the insight you are sharing and it has helped a great deal. One thing i am still in the dark about is what is the point of the e operator in the first place when having an imaginary exponent.

Basically, why doesn’t 1 ^ i rotate me counter clockwise around the unit circile.

Hi Dan, great question. e is a good choice of base because it represents 100% continuous growth: e^i will rotate you 1 radian around the unit circle.

Any other base will also rotate you, but by different amounts. 2^i will rotate you slightly less than 1 radian around (ln(2) = .693, so you’ll move .693 radians around the unit circle).

1 as a base is interesting, because it doesn’t represent any growth (1 to any power = 1). So, 1^i = 1. (Another way to put it: when saying you have 1 as your base, you are saying your growth rate is ln(1) = 0, which means you aren’t changing at all).

I see what you’re saying. Thanks Kalid.

This was a great audio and follow up. I haven’t listened to or read the entire stream yet, but I will. I wanted to contribute this. I teach AP Calc at the HS level here in the US. I have some pretty bright students. While two of them were in PreCalc with me, after we had done some work with DeMoivre’s theorem, they posed this question of what the function f(x)=i^x would be like (what is base i ?). This question led to one of the most robust and interesting explorations that I, and they, made. One young man programmed the solutions to show that they were like the spiral helix of a DNA strand. Really cool.

Thanks John! I love hearing how other people (especially teachers) are exploring it. I’ve come to see i as the epitome of rotation, so when we use it as a base we end up rotating around. Great stuff.

Incredible article!

I found that trying these ideas out with other imaginary numbers really helped to make clear what is so special about the definition i^2=-1. By other imaginary numbers I mean what if you take i^2=0 or i^2=1 and then everything between and beyond. When it equals 0 instead of falling back into a circle you just keep going up from 1; and when i^2=1$ you start growing upwards and then accelerate out more and more and you get a hyperbola (hence the hyperbolic sine and cosine). There’s a really neat wolfram demonstration here: http://demonstrations.wolfram.com/TransformationsOfComplexDualAndHyperbolicNumbers/
Basically as i^2 gets really negative the e^it flattens out like a pancake, when it gets close to 0 the circle bows up into two verticle lines at 1 and -1, and from there it stretches out into a hyperbola. I hope that made some sense.

More than 30 years before Euler published his famous equation, Roger Cotes wrote essentially the same formula, except he expressed it using the natural logarithm (ln) instead of the exponential function.

From Roger Cotes’ notes:

ln (cos x + i.sin x) = ix

Now, Cotes’ equation is truly extraordinary, but I cannot see how Cotes came to this conclusion, more that 30 years before Euler (in fact while Euler was still a small boy in a far off land).

This earlier version is not as simple as Euler’s, and Cotes did not offer any proof of his result (Euler provided that proof later). Also, Cote’s version does not evoke any intuition for me at all - it just seems to appear out of nowhere, without any motivation.

This anomaly also highlights the fact that, historically speaking, logarithms were discovered before exponentials, even though exponentials are the more obvious concept, being merely repeated multiplications.

I wonder how this happened. Why did Cotes’ strange version of Euler’s equation appear more than 30 years before Euler’s?

It is a pity that Roger Cotes died so young - what would the history of mathematics be if he had instead lived to be an old man?

Hi Paul, really neat question. In my head, I see logarithms as giving the “input” that leads to some effect. So, ln(10) is asking "What amount of time is needed to grow from 1 to 10, assuming 100% continuous growth? (ln(10) = 2.302 time periods)

In my head, when I see

ln(cos(x) + i.sin(x)) = ix

I think “What input would lead to the pattern cos(x) + i.sin(x)?”

My intuition is that only an imaginary growth rate (i*x) would lead to circular motion (cos(x) + i.sin(x)). It is interesting that in general, logarithms were discovered before exponents: I see this as thinking about the cause (time, or an interest rate) before working out the effect (what the growth actually looks like).

I made an animated, geometric proof of Euler’s formula, using the intuitions that I learned from this website, and from the amazing slideshow at http://slesinsky.org/brian/misc/eulers_identity.html (already linked above I believe), and from some basic rules about vectors (basic physics) and limits (basic calculus).

Check it out if you enjoy being enlightened as much as I do, and believe that most ‘complex’ concepts, once the abstractions are removed, are really not all that complicated. I ‘understand’ Euler’s formula now as well as I believe anyone can ‘understand’ anything, I can explain it in terms of other things that I ‘understand’, and ultimately on things that are so familiar to my experience that I can take for granted.

https://www.desmos.com/calculator/ohwnsgnef2

The contents of this site should be taught in schools parallel to “standard” course.
I knew for a long time that the trigonometric terms of Euler’s formula represent helical motion is ‘x’ is taken as an axis, which simply reduces to circle if ‘x’ is considered and angle.
But I never took the time to analyze the exponential part in this way, Great insight.

Hi Kalid,

I understand your intuition for ln(cos(x) + i.sin(x)) = ix, which is essentially knowing Euler’s formula, and understanding that ln(x) is the reverse function of e^x, and therefore cos(x) + i.sin(x) therefore would be what what ‘causes’ ix to appear as the result.

My personal amazement is that apparently Roger Cotes could see that too, long before Euler’s elegant equation was published.

It seems that in Cotes’ era (early 1700’s), logarithms were heavily employed by mathematicians, but there was much less understanding of the exponential function, until Euler shed his intellectual light on it decades later.

So, how did Cotes understand his natural log version of the equation? Today it is seen as the reverse of Euler’s equation, but what intuition could Cotes possibly have?

We are lucky to have any of Cotes’ notes, published by his relative after his early death, but I cannot help but feel that Cotes understood much, much, more than we have on the public record.

I feel this particularly because his natural log version of Euler’s equation only makes sense when we see it is the reverse of Euler’s. Can we reasonably conjecture that Roger Cotes already knew and understood that e^ix = cos(x) + i.sin(x), long before Euler?

No less than Sir Isaac Newton, a friend of Roger Cotes, wrote of him:

"If he had lived we would have known something."

The early death of Roger Cotes was more than just a personal tragedy - it could well have set-back mankind’s advances in science and mathematics by many decades.

This one makes me cry.

Thank you that’s the closest anyone’s ever come to explaining it to me I think I’ll reread at my leisure! Good to know the identity has practical uses and isn’t just magic!

When we consider only addition the entire maths can fit into a single dimension. We pour in multiplication and it gets 2D. Couldnt there be a 3D model? What growth would be more superficial than that of multiplication? Moreover can we think of addition as dealing with the value of observation, multiplication as dealing with the frequency of observation and all the other operators as just functions. negative number corresponds to imaginary number.Negative numbers taught us to extend the number line behind the origin. Imaginary numbers taught us to extend the number line above the origin in another dimension. but somehow I feel like that maths isnt absolute instead it too is relative. How could negatives and imaginary numbers exist if there was only one observation??? We get the concept by comparing several numbers which cant fit into any known pattern so we introduce(-x) and sqrt of (-x).
Please answer my question, could we have 3d maths???

@Arkadeep I look at dimensions this way, if you multiply two things, it’s best to imagine it as square. If you multiply three things, it’s best to imagine as cube.

for example, 2x2 = 4 , corresponding to vertices of a square in cartesian plane.
extending it, 2x2x2 = 8, corresponding to vertices of a cube in 3-D.
even more, 2x2x2x2 = 16, corresponding to vertices of a hypercube (look up 4D cube on wikipedia, that object is very amusing.)

In short, i think of multiplication as not just 2D, but operating in as many dimensions as there are numbers being multiplied, multiplying is like stretching an object into higher-dimensional object.
point(0) -> line(1) -> square (2) -> cube/cuboid(3) -> hypercube like objects (4) … and so on.

If we represent numbers using such objects (orthogonal in ‘n’ dimensions, cuboid like) prime numbers will always be lines, composite ones could be represented by ‘n’ dimensional objects depending on count of factors (except 1).

This is a way of looking at multiplication, as extending or stretching an object whose count of vertices represents a number to a whole new dimensions, not just 1D to 2D, but from N-D to (N+1)-D.

And of course there is higher dimensional mathematics, and actually it’s a subject of current mathematical research due to it’s role in String theory and it’s derivatives. read up kaluza-klein theory, it actually is based on 11-D objects.

Hi Kalid, i was delighted ! – Have visited hundreeds of pages abouth math, and in every one i found an attitude very usual among mathematicians: this tendency to surround things with a halo of mistery… and intentionally hiding the intuitive understanding. There is something selfish in this: the message is that this is too much for your small brain… I saw this king of frightening sentences in renowned textbooks. And some of these people showed up in this blog!! Please ignore them and continue. Your clevernes, combined with your generosity is simply invaluable.
Thanks again !!!

It is just incredibly awesome. I spent time getting on that and I suspected that i get lots out of it!

@jcj “a halo of mistery… and intentionally hiding the intuitive understanding.”

That’s one way of looking at it, but I think it’s quite ironic that you need a mysterious motivation (why would anyone intentionally do what you’re describing?) to create a mysterious mystery of evil doers…

I prefer to think that people use whatever language suits their purpose of “understanding more, in a shorter time, by using shared symbols to replace concepts”.

Don’t be afraid of the mathematicians… they’re people just like you and I. Be nice to them, they’re just doing their thing, and more power to them. :slight_smile: