Pinkie Pie guide to Irrational and Transcendental Numbers · 9:25pm Jun 2nd, 2015
Hi there! Pinkie Pie 
here. I’m writing this blog post as it’s all about math and Pineta thought that maybe I could explain it in a way that you would find easier to understand. I'm not sure that makes sense—everyone seems to have trouble understanding me. I mean even my friends don’t always understand me. Even I don’t understand me sometimes. Anyway hopefully I can make you smile and maybe teach you something new. This post is all about this logo which has been around the forums for some time and on a lot of T-shirts:
You get it? Pink-E-Pi! That’s me! That’s me! I thought that was really cute, and it’s super cool ‘cause e and pi are the two coolest irrational numbers in mathematics. Let’s talk about them!
Pi (π) is the one which everyone learns about in mathematical kindergarten: 3.14159265358979323846264338327950288419716939937510582... The ratio of the circumference of a circle to the diameter. It's an irrational number so it has an infinite (∞) number of digits, and you can't write it as the ratio of two whole numbers. It's also a transcendental number, so it's not a square root or cube root or other sort of root. The square root of two is irrational, but not transcendental. Do you know what a square root (√) is, dear reader? It's a number which when you square it—multiply it by itself—you get the number which it is a square root of, like three is the square root of nine. I mention that 'cause for a long time I thought a square root was part of a square tree. There weren't many trees on the rock farm where I grew up so how was I to know? But then my sister Maud told me that it's not part of a tree, it's a type of number which can be irrational but not transcendental.
But I digress, let's get back to circles! Of course the tauists like Walabio prefer to use tau (τ) instead of pi, which Maud explains in The Tao of Two Pie is the ratio of the circle circumference to the radius. Using tau makes a lot more sense in many ways, although pi is far more popular. But why stick with one circle constant when you can play with both and enjoy twice the infinities and three times as much pie?
So what about e? This one's a bit different, and is not quite so quick and easy to explain. But I, Pinkie Pie, shall take on the challenge. So go and get some popcorn, take a seat and prepare for the big fight: PINKIE PIE VS MATHEMATICAL IGNORANCE: AN EXPONENTIAL ADVENTURE! Are you ready?
ROUND 1: DING DING DING! (Hit a metal saucepan with a spoon as hard as you can for the sound effects, unless you've got a proper bell, in which case use that). This is the one which economists like mylittleeconomy like. It's about bank interest rates, which, I know, is usually a bit dull, but bear with me.
Suppose you put 1 bit into an account which pays 100 percent interest per year. Then after one year you have your 1 bit, plus 1 bit interest. Total: 2 bits. Yay! But what if the interest is paid every half-year? Then after six months, you accrue 0.5 bit interest, and in the next six months, you get interest on 1.5 bits, so at the end you have 1.5 plus 0.75 equals 2.25 bits. So this half-year compound interest beats the simple annual interest. But why stop there? Make friends with your banker (bankers need more friends, give her a big hug), and see to it that your interest is paid four times a year—then you end up with 2.4414 bits. Go and see her every day, and the total comes to 2.7145 bits. Now let's go really crazy and calculate the interest every hour for a year. Net total: 2.7181 bits. At this point you notice that this isn't actually giving you much more pocket money, but that was never the point. This isn't about making bits, it's about having fun with math. So how many bits do we get if we calculate the interest, not just every hour, every minute, or every micro second, but every infinitesimally small moment for an entire year?
Answer: 2.71828182845904523536028747135266249775724709369995... You guessed it—it has an infinite number of digits! The wonderful beautiful irrational and transcendental number we call e.
DING DING! END OF ROUND ONE. Time for a break. Go and get some snacks. Back soon.
ROUND TWO! DING DING! Me again! Do you know why the number e is called after the letter e? It's ‘cause it's the number whose exponential function is its own derivative. Do you want to know what that means? Yeah, I bet you do!
Twilight once said that she thought one shortcoming of ponykind is our inability to understand the exponential function. I said it wasn't that difficult a function to understand—it just goes whoosh! She said that ponies don't understand the way it whooshes and just how much it does it and maybe she's sort of right. So how do we explain an exponential function? For this we need PARASPRITES! At least, we need the IDEA of parasprites in our head. We really don't actually want any REAL parasprites. No way! You remember what happened when the parasprites came to Ponyville? First there was one, then three, then fifteen... and they just keep multiplying until there were millions and they took over the entire town and I had to drop what I was doing and put together a one-pony band at short notice to get rid of them, which was really tough especially when my friends didn't help, but I guess that's not really their fault as they didn't understand what I was doing.
Anyway the important thing is that the number of parasprites grows at a rate which is proportional to the current number of parasprites. So as it gets bigger, the rate at which it gets bigger, gets bigger. And the rate at which the rate at which it gets bigger, gets bigger. So this is why it can go from really small to really really really really really big very quickly.
So let's write down a mathematical function and do a teeny bit of calculus. If we start with one parasprite, and their number doubles every hour, then after x hours we have two to the power x, that is
parasprites. Which one day later is over sixteen million! Now a more general way to write it is using the exponential number e:
where N is the number we start with at time zero, and r determines the precise rate at which they grow. Why do we use e? Because it has the useful property that it doesn't change when you differentiate it. If
then
.
This means the rate at which the exponential function changes is always equal to its value. If you draw a graph of it, then at any point x, the gradient dy/dx is equal to y. This makes it one of the easiest functions to differentiate, and we can do all sorts of interesting calculus with it.
DING DING! END OF ROUND TWO. Time for tea and cakes.
ROUND THREE! DING DING DING DING! In this final round we will look at a classic math problem which shows one of the random areas in which one of our favorite irrational constants appears: The Hat Check Problem. This is a fun game you can play if you have lots of friends with hats. At least with some friends. It doesn't work with Applejack 
as she always wears the same hat and she's very attached to that hat.But it's perfect for Rarity 
and her Canterlot friends as they all have lots of hats, and they are forever getting new ones. At least I thought so but when I tried to get Rarity to play it at the Grand Galloping Gala, she wouldn't do it. I'm sure Discord would have joined in but we didn’t have the chance.
So, how to play? It's dead simple. You take a hat from each of your friends, then you mix them up, and give everyone a hat back chosen at random. So everypony gets a new hat! Fun! Except if you choose them at random, then chances are somepony will get their own hat back—boring! But let’s calculate the probably that nopony gets their own hat back? With two friends, it's 1/2. With three friends, it's 1/3, with four, 0.375. And if you play it with a whole big gianormous infinite universe of friends—it's turns out it's 1/e
DING DING DING! END OF ROUND THREE! That's enough from me for now. I hope you enjoyed that and learned something, and next time someone asks you to pick a number between one and ten, you can give them a totally irrational answer. Maybe another time I'll talk about imaginary numbers and Euler's equation, which is the coolest equation ever.
Take care
Vectors by Shho13, Synthrid, Juicy-Cactus, pikn2, scrimpeh.
Now read:The Pinkie Pie Guide to Imaginary Numbers and Euler's Equation
I so want to hear Pinkie actually say all this. Well explained, and well done.
Using tau makes much less sense! τ already has a lot of uses. Most importantly (for this criticism), torque. Using τ for torque and τ for angles is just begging for confusion. What is the work done by some (possibly variable) torque in one full rotation? \int_{0}^{τ} τ(θ)dθ ? Ew.
As far as 3.14... vs 6.28... goes, I prefer 3.14. They both show up in a lot of places, but I'd rather have a factor of 2 than a factor of 1/2.
I can't believe nobody's posted this video yet.
I like math in the first place, but this was way more fun to read than I'd have guessed! I'm out of practice on calculus so I'll just have to take your word about the derivatives... It looks familiar at least, the stuff you did with e. e always kinda confused me. This was a good way to explain it! I liked the fact about the slope always equaling the y value at the point the slope is taken ^u^ Fun!
Now we just need to go into what happens when you calculate e^(pi*i)
That equation made me realize how fascinating mathematics really are.
Not to add to or subtract
from the mathematical insights of the revered PDP, but, um, what about defining e as the argument to the natural logarithm (integral from 1 to x' of dx/x) that produces 1?
Please don't butcher me Pinkamena
A most enjoyable dissertation on irrationals by my most favorite imaginary irrational of all. Thanks go to both Pineta and eπ for this one.
(Plus, the listing of digits extends past the boundary of the blog's text box, which is perfect on so many levels.)
3118623 The fact that you're criticizing this based on a single factor, and ignoring all the other factors to arrive at your judgement that it is, on balance, far less sensible, makes me sad.
It also makes Maud sad, but I'm pretty sure you can't tell.
See, now I can't use this in class.
3119345
The fact that you assume that because I didn't make an exhaustive list I didn't consider other factors makes me sad.
3119472 If you had considered other factors and thought they were significant, then you should have included them; to do otherwise is to either assume they are patently obvious to all others (which is rather patronizing and quite prone to error if it turns out you made a mistake yourself, since there's no way to catch that) or to simply throw the discussion by leaving out major reasons for considering things the way you do. And if you're not even really trying very hard to convince those you disagree with why they are mistaken, why would anyone listen to you?
Thus, if someone says, "there are lots of reasons why X is good" and you reply with "but it isn't, because Y", the only assumption anyone can safely make is that you consider that Y is, on its own, a sufficient reason for X not to be good. But if this is not really the case at all, and not even you would attempt to argue that, then again, why did you frame it that way?
3119482
I didn't figure people would take it super seriously, so I included my primary reason for disliking tau as a symbol (tau is overloaded enough...pi is also overloaded, but somewhat less so and at least it is less prone to direct conflict). I didn't bother listing arguments for 3.14 vs 6.28 because my feelings about the notation are stronger than my feelings about the choice of circle constent. I'm not bothering with the arguments now because I'm writing these replies between games of Heroes of the Storm.
Euler's Equation:
e ^ τ i = 1
Ponies measured across and around circles. This lead to the wrong Circle-Constant. It stuck because, in 2D, a = ( τ / 2 ) r ^ 2 . This seems to support the wrong circle constant, but one needs to understand the generalized case:
n = ( ( τ d - 1 ) / d ) r ^ d
Half Tau appearing in the equation of the area of a circle is just a case of 2s canceling. Indeed, musing Half Tau leads to mathematical errors:
Let us suppose that one needs to figure out the volume of a sphere:
v = π r ^ 3
¡Wrong!
It has a 3 and 4, so it is:
v = ¾ π r ^ 3
¡Wrong!
v = 4 / 3 π r ^ 3
¿Why?
If wr would have used the 1 True CircleConstant, τ (Tau), we could have avoided the error:
m = τ ( d - 1 ) / d r ^ d
We get:
v = ⅔ τ r ^ 3
The mathematics are easier if one uses the 1 True CircleConstant τ (Tau).
I really like your explanation of e.
3119731 Nice! Thanks. That 4/3 always bugged me.
3130313
Yes, τ (Tau) makes the mathematics much simpler. A circle has τr, thus ¼ of a circle is τ/4 instead of 2/π (the multiple multiplying and dividing by 2 leads to errors). The interior of an n-sphere is τ((d-1)/d)r^d. Euler's identity is e^τi=1. one makes 1 turn all around the circle:
* e ^ 0 / 4 τ i = + 1
* e ^ ¼ τ i = + i
* e ^ 2 / 4 τ i = - 1
* e ^ ¾ t i = - 1
* e ^ 4 / 4 τ i = + 1
With π, one only makes it half-way around the circle (e^πi-1. πists find this so ugly that they pretty it up with numerology:
e ^ π i + 1 = 0
¡Pure numerology!
¡τ (Tau) is the 1 true CircleConstant!
3119706 I don't think that's as useful, though. You lose the information that e^(πi) = -1.
3133621
The point of e ^ ( m i ) = n is that it allows one to mathematically describe rotations. Like a clock, 1 and 1 turn have the same value (1) Indeed all integer turns have the value 1. e ^ ( π i ) = - 1 is problematic because one only goes half-way around the circle. πists hide this with numerology:
e ^ ( π i ) + 1 = 0
The fact is that π fails. Although τ works just fine:
e ^ ( 0 / 4 τ i ) = 1
0th of a turn and we start at 1.
e ^ ( ¼ τ i ) = + i
1 quarter of a turn and we are at 1.
e ^ ( 2 / 4 τ i ) = - 1
At half-way around the circle, we are at negative 1.
e ^ ( ¾ τ i ) = - i
At 3 4ths of the way around the circle, we are at - i
e ^ ( 4 / 4 τ i ) = + 1
At 1 turn we are back at 1.
If one wants to find the sign of the current value, just plug the value into the equation like thus:
e ^ ( m τ i ) = n
A πist has to multiply by 2:
e ^ ( 2 m π i ) = n
The reason πists have to multiply by 2 is because they use only half of the 1 True CircleConstant τ.
3133848
Do we need to use e^(i*pi/4) to know which way it went around the circle?
I don't understand this fully; I'm just noticing that e^(i*tau) = 1 allows that either e ^ ( π i ) = - 1 or e ^ ( π i ) = 1, so it's destroyed information.
3135492
Well, Euler's Formula allows one to mathematically represent rotation. This is useful in engineering and physics. The magnitude represents how far away the baseline one is and the sign represents whether one is above or below the baseline We start with a UnitCircle:
A UnitCircle has a radius of 1. One starts at a height of 1. One goes through i at a quarter turn, -1 at an half turn, -1 at 3 quarters of a turn, and back to 1 at a full turn. ( e ^ ( i τ ) ) = 1 one makes a complete turn. τ is equal to 1 circle. τ is the 1 true circleconstant. ( e ^ ( i π ) ) = -1. π only goes halfway around the circle. π is only half the circleconstant.
More About τ And Euler's Formula
3135881 The most-informative version, then, is e^iτ/4 = i, since that (and not the others) tells you which way you're going around the circle.
3136053
Positive values have one going counterclockwise, while negative values have one going clockwise. One can measure turns per unit-time such as Hertz (cycles per second). This video explains this well:
3136053
3136317
Going round in circles...
orig08.deviantart.net/364c/f/2015/161/7/0/circle_by_pinetapony-d8wss49.png
3137879
¡That is cool! ¿Did you make it yourself?
3136317 Yes; my point is that you can't derive that from either e^(i*tau) = 1, or from the standard identity.
3138407
I see what you mean now:
The standard complex plane has the reals run horizontally with negative vales on the left and positive vales on the right with the imaginary numbers running vertically, with positive values above and negative values bellow. It is all just convention.
3137931
With a little help from friends...
orig13.deviantart.net/a4d6/f/2015/162/5/7/work_in_progress_by_pinetapony-d8wuiaz.jpg
3138998
¡Miss Pinkamena Diane Pie does good work!