• Member Since 28th Oct, 2012
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Pineta


Particle Physics and Pony Fiction Experimentalist

More Blog Posts440

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    Pineta · 12k words  ·  49  0 · 827 views

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Jan
17th
2016

The Pinkie Pie Guide to Imaginary Numbers and Euler’s Equation · 2:00pm Jan 17th, 2016

Hi there! It’s me—Pinkie Pie—again. I’m back because my blog post The Pinkie Pie Guide to Irrational and Transcendental Numbers was the most popular thing on Pineta’s blog in 2015, and that seems reason enough to follow it up with The Pinkie Pie Guide to Imaginary Numbers and Euler’s Equation Are you ready?

(Actually it’s not entirely true that I was Number One. Technically I was beaten by Sociology of Marine Bronies, but that’s just a short boring post pointing out a boring research article. I guess it just got lots of hits ‘cause it contains certain keywords which must have triggered an enhanced surveillance by the NSA or something like that.)

Anyway in this blog post I’m going to continue talking about math. In my last post I explained all about the irrational numbers e and pi(π). If you haven’t read that yet then go and do so, then come back here and get ready for the next super-exciting mathematical instalment in which I will tell you all about imaginary numbers and Euler’s equation. To understand this you are going to need to know a teensy bit of algebra and trigonometry and have a lot of imagination.

Okay, the first thing to get straight is that Euler’s equation is the most super awesome amazing beautiful elegant mathematical equation EVER. It’s like the chocolate caramel vanilla ice cream cup of equations. With extra sprinkles and a cherry on the top. I mean just look at it:

Why is it so cool? Well to start with it has all the coolest numbers in mathematics in it: e and π and i. (Those of you who are saying ‘what about tau?’—yes I know what you’re thinking—I’ll get onto that later, but I’m going to use pi now). So I explained about e=2.718281828459045235360287471352662497757247093… and π=3.141592653589793238462643383279502884197169399… before. Now let’s talk about i.

PART 1 - THE PINKIE PIE GUIDE TO IMAGINARY NUMBERS

i is a square root of minus one. This means when you multiply i by itself you get −1. This sounds pretty weird as you probably think when you multiply a positive number by itself you always get a positive number (say 2×2 = 4). And when you multiply a negative number by itself you also get a positive number (say −2 × −2 = 4). But you can’t multiply any real number by itself to get a negative number. So how can we do it with i?

Because i is not a real number. It’s an imaginary number. We have to imagine that it exists. You know some boring old mathematicians said imaginary numbers didn’t exist and they were just something made up by other mathematicians. Seems rather silly to me. Imaginary numbers are kinda like imaginary friends. Just because they’re imaginary, it doesn’t mean you can’t have a lot of fun with them. You just need to invite some real friends along to the party to finish the cupcakes.

Got that? Good.

Now we have imagined the square root of minus one, we can also imagine the square root of any negative number. The square root of −4 is 2i as 2i × 2i = −4). Now let’s go a step further into our imaginary mathematical world and find the square root of i itself. It turns out this is part-real and part-imaginary.

(1/√2 is another irrational number: 0.707106781186548…)

Here’s the proof:

This is what we call a complex number as it has both a real part and an imaginary part. We can now replace the boring old one-dimensional number line which we use to show real numbers with a two-dimensional complex plane. Every point on the plane is a complex number. The distance along the horizontal axis gives the real part, and the distance up or down the vertical axis gives the imaginary part.

Okay! Now you know everything you need to know about imaginary and complex numbers. Well maybe not everything—actually you might need to know a lot more about them, but this is all you need to know to understand the rest of this blog. Anything which you need to know about complex numbers beyond this is your own problem. But before we can get on to Euler’s Equation, we need to do a quick bit of revision on trigonometry.

PART 2 - THE PINKIE PIE QUICK REMINDER ON TRIGONOMETRY - SIN AND COS AND ALL THAT STUFF

Now since we’ve drawn a triangle, we can do some trigonometry. Triangles are fun. Do you remember about sine and cosine functions, dear reader? If you’re forgotten then here is the definite summary of Everything You Need to Know About Triangles:

Now look at the triangle on the complex plane with the point √i and you can see that the sides of the triangle are a=1/√2 and b=1/√2, so we can show the long side of the triangle is just c=1 and sin x = 1/√2 and cos x = 1/√2. This gives the angle x = 45 degrees or π/4 radians (where 360 degrees is 2π radians).

Now note that for this complex number, side a - the real part - is equal to cos x, and side b - the imaginary part is equal to sin x. Remember this - you will need it in a bit. Now we are ready for the really exciting bit.

PART 3 - THE PINKIE PIE GUIDE TO EULER’S EQUATION AND WHY E TO THE I PI EQUALS MINUS ONE

So why is so super cool? Because it shows that there is this special mathematical connection between three of our favourite numbers that until now seemed totally unrelated. We first found out about π when drawing circles. Then learned all about e when playing games with bank interest rates and parasprite populations. Then we imagined i so we could have a square root of a negative number. Then we put them all together like this and WOW! Isn’t it just the cutest bit of math EVER? You let your imagination run away exponentially and it comes back with pi—sort of.

But I know—so far it’s just me saying how cool it is—now I need to actually show you how we know that e to the i pi equals minus one. Na-na na-na-na na—Let’s go!

Except—what does that actually mean? e raised to the power of i times π? That’s a bit weird. I mean we know and . But how do we raise e to an imaginary power?

To do this you need to use a Taylor series. Taylor serieses are really cool as they let you write any math function (of x) as the sum of an infinite number of x-es, x-squareds, x-cubeds and so on. Like this:

And it just keeps going on. There are an infinite number of terms but I’ve only written down six as it would take like forever to write them all down and that’s far too boring a way to spend eternity. I also don’t have time to explain why this works, but you can trust me, it does. Pinkie promise. By the way, in case you’re wondering what the ! sign means—you multiply the number before it by every integer between itself and one, for example 4! = 4×3×2×1 = 24.

Now we can calculate e raised to any power. So we can write an expression for as

Where we remembered that , so , and . Now let’s shuffle them around a bit and put all the real numbers together on the left, and their imaginary friends on the right.

Now this may not look like it’s going anywhere. But now comes the trick. You get out your table of Taylor Series and see that these two terms in brackets are the Taylor Series for sin(x) and cos(x). So…

So is a complex number with real part cos(x) and imaginary part sin(x). This is where angle x is in radians (Remember 2π radian equals 360 degrees). Now in part 2 we showed that a complex number makes a triangle on the complex plane, where the real part forms one side, which is equal to cos(x) and the imaginary part forms another side, which is equal to sin(x) (if the third side is of length 1). This is exactly what we have here, so is a point on the complex plane which forms a triangle with angle x.

So as x goes from 0 to π, traces out a circle of radius 1, starting out at the real point 1, then going complex, and after half a circle it comes to the point −1.

So there you have it. The Pinkie Pie introduction to imaginary numbers and why . But before we finish let’s look at it from the tau perspective. If you have read The Tau of Two Pie, then you will know that the Tauists like to write things in terms of τ=2π instead. This is because Tauists don’t like half-measures. Pi only gets you halfway around the circle, but tau goes the full way, so Euler’s equation for Tauists is . So some say that this version is even more beautiful than . I’m not so sure about that. I quite like having the minus one on the right hoof side as it makes it clear that there’s something imaginative going on, whereas makes me think τ=0. But maybe I’m just too smart to understand aesthetic stuff. Reminds me of that time when Rarity was making my dress for the gala and I thought it would be better with lots of lollipops, balloons, candy, and streamers, and Rarity wasn’t so sure and it turned out she was right as she always is with dresses.

Anyway now we’ve come full circle and it’s time for a We-learned-all-about-imaginary-numbers-and-Euler’s-equation party. So let’s get out lots of lollipops, balloons, candy, and streamers and PARTY!

Comments ( 12 )

Poor Maud, feeling all left out because she wanted to demonstrate the brute force method of breaking this problem.

SOHCAHTOA

[Maud] Its Calculus. I threw it. [/Maud]

Now you can go back to all those high school calculus functions, and see Why you had to learn the functions using sines, cosines, and now you can see the easier way of defining them.

Strange physics occur when you play with even more complex numbers. As for teh mathematician who hates i, he is forgetting the basics of maths. All numers are imaginary. They are currently the standardised symbology of western civilisation that represents the unity entity and collective representatives in a compressed format. thats why algebra and substitution works.

x^2+(1/x)^2 has a real solution but uses complex values to cover all working out. All four intermediate complex roots.

Can we please have a like button for blog posts? Please? :raritydespair:

Aside from that, I think we'd all benefit, Pinkie, if you would explain quaternions as expertly as you have Euler's equation.

3692877
The hell with quaternions, I want Pinkie to explain category theory.

You write the best blogposts ever!

If my life ever takes a 180 and I end up becoming a math teacher, I think I may have to distribute this as a handout :V

Fantastic exposition.

Tau brings one all around the circle. The radius defines the circle. ¡Tau is the circle-constant!

I once had a piist try to claim that pi is just as good because if constant diamater defines a circle. I then pointed out curves of constant diameter. Maybe, you can right a post about curves of constant diameter.

Quaternions would be an interesting subject.

That's a remarkably good explanation of the whole subject, I think. You (and Pinkie!) have a knack for this kind of a thing. :pinkiehappy:

Even though I had trouble wrapping my head around this I think this is one of my most favorite Pinkie pieces I've ever read. I just wish could understand math. My head's not so good at it. Everything sorta wriggles around in my noggin and slips off of it like an ice cube off a tray.

3736547
Thanks. I wouldn't dismiss your math skills just because of this piece. Whatever Pinkie may say, it is actually quite difficult to get your head around Euler's equation without doing a full course on complex numbers and other topics. I watched quite a few youtube videos by math popularizers to see how they go about it. I don't think any of them really succeed in fully explaining it to a non-mathematician audience

3737183 Oh, I don't feel that bad about it. More than anything I love the way you capture her personality. You make her believable. I like this PInkie very much.

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