Intuitive Understanding Of Euler's Formula

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161

Hi Paul! Thanks for the comment, great question.

That’s something that bothered me as well, give the section “Example: 3^i” one more gander. I wanted to work through why a real number to an imaginary exponent always stayed on the unit circle.

Here’s my intuition. It might sound mystical, but there’s one circle, the unit circle. Every other circle is just a scaled variation of it (larger or smaller). Fine. Similarly, there’s only one number, 1.0, and every other number is a scaled / rotated version of it. Also fine.

How about growth? Well… there’s only one type of growth: e^x. Every other type of growth is just a slowed-down or sped-up version of this “unit” growth (e^x represents 100% continuous growth).

When we have a setup like 2^x, we’re saying “Ok, let’s grow a little slower than e”. We can substitute 2 = e^ln(2), and realize 2^x is the same as e^x, but it grows more slowly (at ln(2) = 69.3% interest, instead of e’s 100% interest).

An imaginary exponent modifies your growth rate, turning it perpendicular (which forces you to spin in a circle – you’ve strapped the rocket on sideways). Making 2^x grow perpendicular just means “Well, now you’re growing in a circle, but at a rate of 69.3%, compared to e^x’s 100%”. And as you might guess, 2^i will go around the circle, but a shorter distance than e^i would have. In fact, e^i will go 1 unit around the circle, while 2^i will go .693 units around the circle.

Hope this helps!

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Thanks heaps Kalid, for taking the time to discuss this issue.

After posting my second comment, my feeling is that my intuitive understanding leads me to look for something unitary, simple, or ‘pure’, to define complex multiplication and exponentiation. That seems intuitive.

So, how about this:

We could think of multiplication as being an operator with two ‘sub-operations’ - the ‘Real’ ‘operation’ that scales the magnitude, PLUS the ‘imaginary’ operation that adds to the angle - or rotates.

To apply this multiplication in two parts it is convenient if the two ‘operations’ that need to be performed are ‘orthogonal’ operations - where orthogonal simply means that they do not affect each other, and thus can be computed independently and the two results combined to get the final answer.

In this view of multiplication, the imaginary rotation operation is orthogonal to the familiar scaling of real multiplication - by DEFINITION - it’s the way we must change the angle while leaving the magnitude unchanged.

Similarly, the scaling of the magnitude is DEFINED as the operation that does not change the angle, but only scales the magnitude.

Together the two are combined to perform the familiar operation of complex multiplication.

Therefore, our ‘pure’ imaginary ‘operation’ must trace a perfect circle - only because we have defined the operator as only capable of adding to the angle, but never changing the magnitude - a fixed radius is a perfect circle by definition!

But, I do not feel that the ‘unit circle’ is the one ‘mystical’ circle (in your words), rather it is just the consequence of repeatedly using the purely imaginary operation - a perfect circle with fixed radius that goes around for infinity with increasing angle, analogous to the way that the real axis goes on to infinity with increasing scale.

However, I found that thinking about ‘growth’ when looking at e^i caused me to go off-track - and that word ‘growth’ caused me to incorrectly intuit that the graph of increasing powers of i would be a spiral, not the perfect circle.

Now, the way I am thinking is this: the ‘circle of any radius’ is intrinsic to the multiplication by the ‘pure’ i, because we define that to be ‘adding the angle but leaving the magnitude alone’.

Further, the very special ‘unit circle’ actually has nothing to do with the special number ‘e’, rather it is derived independently by the property that anything to the power of zero must be 1. So the circle is locked to a length of 1 unit as a natural property of exponentiation. This works regardless of the real number we choose as the base, not just by choosing e.

In fact, thinking about e as the base wrecked my intuition of what was happening when we raised real numbers to multiples of i. Perhaps the unit circle is ‘deeper’ concept than ‘e’. However the base ‘e’ is still special because it relates the other special number ‘pi’ by way of rotation FREQUENCY: [i times 2 times pi] for each full rotation.

I hope that makes sense.

Please do not take offense if I appear to contradict you. All I am really discussing is my own evolving intuition of complex powers, and ‘pi’, and ‘e’. Of course this may be different to your, or anyone else’s intuition.

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Aha, now I get it…!

I just revisited the famous Fourier transform using my new intuition about e^ix being the ‘unit of all oscillating things’.

The Fourier transform makes intuitive sense now because I can see the oscillator, e^ix, in the definition.

What the Fourier transform achieves is the decomposition of a waveform, any waveform, maybe piece of a Mozart symphony or a Lady Gaga song - it doesn’t matter, into it’s constituent ‘pieces’, where each ‘piece’ is a pure tone.

The pure tone at middle-C is said to be about 256 Hertz (cycles per second), and this is expressed by the formula e^ix, by plugging 256 times pi times 2 into x;

Pure middle-C (256 Hertz) = e^i.2.pi.256

Hitting middle-C on a piano keyboard produces a tone (partly) expressed by the complex exponential formula with 256 plugged in. But it also produces a bunch of softer tones nearby, certainly at the harmonic tones 512 hertz, 1024 hertz and other frequencies too.

If we recorded the striking of the middle-C key on a piano, and express that recorded waveform using the Fourier transform, we would get the breakdown of the piano’s middle-C sound in terms of all its characteristic tones. Of course this would include the loudest tone located at 256 hertz, but also lots and lots of softer tones at other frequencies.

The waveform of a harp playing middle-C, or even a human voice singing, can be broken down into a similar set of tones that are characteristic of that harp or that singer.

Relating this back to our complex exponential function, the Fourier transform expresses the tones that make up any waveform, and their relative ‘loudness’, as a sum of all of those ‘units of oscillation’, or A.e^ix, scaled with different frequencies (x = 2.pi.f in Hertz) and amplitudes (A).

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Hi Paul,

Awesome, glad things are clicking! Yes, that’s it: the Fourier Transform gives the amplitude at each possible frequency. You might like this video:

(He plays a middle C on the piano, you can see the spike in the amplitude at the desired frequency. As you move up an octave [double the frequency] the spike moves. That piano isn’t particularly tuned, so the spike is noisy, but the idea is there)

As you mention, a harp vs. human voice has different characteristic subtones – a pure 256Hz sine wave sounds very bland, like an emergency broadcast tune (http://www.youtube.com/watch?v=21ZELdFob38).

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Hi Kalid,

Until now, the Fourier transform definition just looked like a butt-ugly integration formula. I could ‘do the math’ like a robot, in your words, but before now I didn’t understand why it worked.

Long, long ago my university professors told me that the Fourier transform would break a waveform down into it’s constituent frequencies, and I just took their word for it without really understanding why it worked.

Now that very same integration formula actually looks like something straightforward and intuitive - a sum of oscillators - because I see e^ix for what it actually is - the unit oscillator!

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Exactly! I had the same struggles when first learning. Another way to put it: The Fourier Transform projects a signal onto the unit oscillator (e^ix) and runs through every possible frequency (coefficient on x), and seeing what the overlap is there. This is similar to taking a dot product of a vector, projecting one vector onto another, and seeing how they overlap. If the projection is 0, there is no overlap, i.e. no component at that frequency.

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That’s exactly what I see the Fourier transform as now - a kind of scanner that tells us how much of a given frequency is found in a sample - the total of what we find in scan of at a specific frequency tells is the ‘amount’ of that frequency in that sample.

Repeating for different frequencies gives us the full picture - the relative amount of each frequency in the sample waveform we are analysing.

Naturally, here in the real world, we use a software library or a semiconductor chip to actually do the scanning for us - FFT libraries etc. I am not so crazy as to actually do those integrations by hand!

Now… onto my other old friend - the Laplace transform…

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I like the “scanner” analogy. Ah, I’m actually looking at the LaPlace transform now too. My key intuition is that it’s just a generalization of the Fourier Transform to spirals, not circles.

A spiral has both a rotational frequency and a decay (or growth) factor. This can be represented by a complex number (a + bi, which means growth factor ‘a’, rotational speed ‘b’) compared to the Fourier Transform, which assumes a growth factor of 0 (i.e., the circle components never change size). One advantage of this general approach is you can model signals which decay (or grow to infinity), especially physical systems (like a spring which oscillates while losing energy to friction).

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Hello Kalid,
I am trying to be intuitive …so my question is this
Is it mathematically valid to distinguish between the " imaginary zero " that is the origin of the imaginary axis as opposed to the “real zero” which is the origin of the real axis ? They magically occupy the same place at the same time on the complex plane. As it stands , Euler’s identity is written in such a way that it regards the origins of the Imaginary axis and real axis to be identical. But how can they, when one origin is real and the other is imaginary ? If one was to write Euler’s identity with respect to " both" origins …We would have a “shadow " of the identity …
(е^πі/ 0!)^2 ⁼ 1 ( which reads( Eulers identity over Zero factorial Squared is equal to 1. As you know 0! is 1. and usually denotes 1 x " the potential” of something to occupy a particular space, however in this case, we are actually factoring “two” zeros, the real zero and the imaginary zero, and taking advantage of the fact that 0! equals 1. As a result, this “reflective shadow " of the identity, it is equal to 1 not -1. This is perhaps an unforeseen consequence of mixing " dimensions”. The imaginary is a different dimension to the real dimension is it not ? And even if this “equation” is perhaps “token”, and somewhat redundant, it is at least acknowledging that we now have an exception to the rule that says " two things cannot occupy the same place at the same time. " In the complex plane, we can cleverly combine real with imaginary (a + bi) , however if the imaginary number in a complex number is zero and the real number in that complex number is also zero, then the complex number should be equal to 0! not just plain old zero, because when both are zero, there is no longer a “combination” of real and imaginary parts, and for a point on the complex plane to “exist”, it must be a combination of both real + imaginary numbers. Kalid, in your view of things, does this line of reasoning have any mathematical merit and is my equation mathematically sound ?
Regards Bluetone :slight_smile:

Regards Bluetone

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@Bluetone: Great question! I think it comes down to a matter of definition; I found a discussion here: http://www.physicsforums.com/showthread.php?t=206108.

My theoretical math knowledge isn’t very well refined, so I’m not sure of the formal / analytic definition of the reals, imaginaries, etc. You may be aware, in some systems there is the concept of a “signed” zero (is zero positive or negative?), which has some of the similar issues: http://en.wikipedia.org/wiki/Signed_zero. I also found the idea of http://en.wikipedia.org/wiki/Semi-continuity which might be related (if you consider the real and imaginary axes overlapping, which one gets the filled-in dot? :))

In general, the definition that leads to the most practical conclusions is the one we usually go with (such as defining 0! = 1).

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Thank you for your reply Kalid.

How do we teach our kids to be “intuitive” I wonder ?

Kind Regards
Bluetone

172

Hello again Kalid,

I believe that the key to understanding the transform methods (Fourier, Laplace and Z-transforms) lies in their motivation - we need to ask - why were they invented, and what do they do for us?

They are not like the exponential function taken alone, which is something more akin to a discovery, like discovering an natural element, say carbon. The transforms are more like tools or processes - they are there to be used as the need might arise. Fashioning useful things using or containing carbon, in my analogy.

The Fourier transform has probably the simplest motivation to understand: it is used whenever we want to decompose a signal into it’s constituent parts. This could be for many reasons, like we want to design a mobile phone or a GPS satellite system for example.

The Laplace is another great workhorse transform used just a much as the Fourier transform, but for more complex, and maybe less intuitive reasons.

Now that I think about it, it is worthwhile discussing some the reasons I think that the Laplace transform is used:

  1. The Laplace transform converts calculus into algebra. Transforming a differentiation becomes a multiplication operation in the Laplace transform, and an integration becomes just a division. So even massive systems of differential equations are reduced to mere algebra with the magic ‘s-variable’ and these can be quickly simplified and factorised.

  2. Directly related to this, the Laplace transform gets rid of those ugly, ugly transcendental functions like sines, cosines, hyperbolics, logs etc. Once again we are dealing with mere algebra with the s-variable, instead of apparently intractable differential equations.

  3. Probably of most importance to me is related to System Theory: When the fully factored Laplace functions are graphed, we can easily see how close a system is to instability (the vertical axis) - an oscillation that keeps growing out of control. The fairly methodical process of deriving the Laplace transform equations for a System can allow an engineer to see what changes could be made to ‘pull-back’ from the axis: to make a system relatively less prone to instability.

That last one seems like a lot to handle, and it probably counts literally as ‘rocket science’ - because that’s what rocket scientists (and engineers) have to do - keep the rocket (system) under control even though the environment around changes - and sometimes it can change radically.

You see, the things that are used to build springs, walls, rockets etc all change their characteristics as they get hotter or colder for example. Having a manageable set of equations, or a least a way of deriving the equations, is crucial to answering questions like: will something shake apart when the rocket nozzle reaches it’s designed max temperature? What will happen as the metal walls stiffen in the dead cold of space? etc.

So, here goes my first guess for an intuitive understanding of the Laplace transform in terms of its relationship to Euler’s famous equation:

  1. Maybe Laplace reduces calculus to algebra because ‘e’ to the power of anything equals itself after differentiation and integration - it ‘survives’ the process. e^x is always there, regardless of how many times calculus operations are performed.

  2. Because Euler’s equation lets us write any sine, cosine, hyperbolic function etc in terms of the base e, the ugly transcendentals disappear and are replaced with the more manageable ‘e^x’ - so even calculus with transcendentals can be reduced to mere algebra in the s-variable.

  3. Both 1 and 2 allow us to have systems of equations, manageable with algebra and expressed with s-variable, which engineers and scientists are trained to use for analysis and design of whole systems.

Understanding 1. and 2. above require an intuition for how e^x works, and 3. provides us the motivation for even bothering to go through the steps to derive each Laplace transform.

But I think I will need to chew on this problem further for a while…

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@Bluetone: Excellent question – I think people are naturally intuitive, we just need to encourage it and show that the rigor is a supplemental way to understand something, but not a replacement for really understanding it yourself. (It doesn’t mean “go with your gut instinct”, it means “be sure you truly understand it, and put a concept into your own terms”).

@Paul: Thank you, this is great! I like your description of the breakdown, it’s exactly the high-level insights I need when diving in. Looking forward to that video too!

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Is there a practical (by that I mean real life application) use for Euler’s Identity? I’m looking to use it as the basis of my mathematical modeling project but I need something to model and also something to test and gather data. Any ideas would be appreciated!

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@Dylaan: Euler’s identity is often used whenever you need to model a circular path or repeating pattern, since it’s a convenient way to build a circle in a single function (instead of having to separate out sine and cosine).

As an example, the Fourier Transform uses Euler’s formula to decompose signals into repeating cycles, which is used to analyze data (http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/).

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Hi Dylan, Kalid,

I am hunting for the nearly same thing - the ‘real life application’ that demonstrates the truth of Euler’s equation (not Euler’s identity, which is closely related).

The nearest I have come up with is they physical example of a ruler held to a desk, with about half of the ruler poking over the edge of the desk - when the ruler is ‘struck’ it vibrates for a second and settles back to motionless.

Technically this physical motion of the ruler is called a ‘damped sinusiod’, and this can be mathematically shown as a simple exponential [e^-kt] multiplied by a sinusiod [sin(wt+p)].

The oscillation - the sine part, can be expressed using Euler’s equation, and this type of solution is commonly found in mass-spring-damper systems studied by engineers.

BUT - saying that solutions to the equation of motion for a vibrating ruler is based on Euler’s equation is a long, long, way from being what I would call intuitive. By itself this is not what I wanted to demonstrate a ‘real-life’ example of Euler’s equation.

(Actually I am cheating a bit - the ruler’s motion also contains harmonics of the principle frequency (w) - but that is not really the problem I have with using it as an example.)

I feel that the same is true for the Fourier transform example - it is not intuitive, even though I know it is true. As an Engineer I have lived and worked with the mathematics for a very long time, so what is self-evident to me is definitely not what a layperson would call intuitive.

The real ‘surprise’ of Euler’s equation is that any oscillation we find in nature, which we can express as the familiar sine wave, can also be expressed using the imaginary number ‘i’. Writing the word sine just ‘hides’ the imaginary part, and rewriting the same sine using the complex exponential formula [e^it] reveals the true nature of every sinusoid.

Euler’s equation tells us that all sinusioids are essentially complex exponentials [e to the power of i] but while it is mathematically true, the hard part is to demonstrate this idea intuitively.

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Thanks Paul. For me, the truth of Euler’s Equation emerges once I realized that it was just another way to build a circle.

Method 1 is to laboriously compute the grid coordinates of every point on the circle (using sine and cosine). Method 2, Euler’s Formula, is to simply rotate a line around the center.

The tools needed to create this rotation are the number 1 (our starting point), continuous growth (e^x) and rotation (i)… and putting them together in the proper order :). [Remembering that radians measure distance moved, so using them instead of degrees, etc.]

That made things click for me, and I didn’t need a deeper meaning (beyond that it can also make a circle), since circles are applicable to most problems. Many trig facts, such as sin(a+b), become easy to compute with Euler’s Formula vs. grinding through the trig identities. I hope to write on this soon.

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[…] a follow-up, we’ll learn about graphing, complements, and using Euler’s Formula to find even more […]

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You are my hero. i envisioned the universe to be a spiraling spiral, and you have shown me that this is what this equation is. i feel i have found the end of the rainbow. thanks.

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Actually a shrinking spiraling spiral. This is how i think of eternity in the universe.

Existence is a byproduct so to speak of the great nothing. If you think about it, the only thing there really is is the great nothing that can never be measured or understood. because it can’t be understood it has to be ever changing randomly. Maybe it could be represented as pi. I see an ever shrinking spiral that keeps spiraling so that it doesn’t touch itself. if it touches itself, then there would be two points the same, which means it could be understood if measured at those two points, but because that’s impossible, it just keeps shrinking.