euler’s formula

Preface

Whenever I do a post like this, I hope I don’t make a silly mistake; but then again, posts of this nature contain content copied and pasted from other sources.

I first encountered the following equation in high school. The class was further mathematics, and only had 3 people. I believe it was for the A-levels module FM1 (sadly, this was ages ago, FM stands for further mathematics). Our teacher was teaching the Taylor series and he proved the equation that way. Later, in this post, I will just prove the equation via differentiation.

I don’t remember encountering this equation ever again, not even at university (though I didn’t study mathematics, sorry for being so cryptic, maybe later I will tell you what I studied at uni). Actually, maybe we touched it at university, but it didn’t stick in my mind that much.

Recently, I was researching mathematical proofs (because this is what I do in my life now), and came across this neat equation again.

 

Main point: Euler’s formula

e^{i\theta } = cos\theta + isin\theta discovered by Swiss mathematician Euler, born in 1707.

 

Beauty

It is beautiful for many reasons. When you set \theta = \pi, you get

e^{i\pi}+1=0.

This equation contains:

  • the basic arithmetic operations (each only occuring once):
    • addition,
    • multiplication, and
    • exponentiation
  • the five fundamental mathematical constants:
    • 0 (additive identity),
    • 1 (multiplicative identity),
    • π (3.141592653),
    • e (2.718281828),
    • i (square root of negative 1, basic unit of imaginary part of complex numbers)

 

Proof

By differentiation.

  • define function f(\theta)=e^{-i\theta}(cos\theta+isin\theta)
  • with product rule, it is easy to get the function’s derivative f'(\theta)=0
  • this means the function is constant in \theta
  • evaluate f(0)=1
  • this function is 1 everywhere
  • rearrange this to get Euler’s identity

QED.

 

Picture

euler's identity

Euler’s equation uses polar coordinates (distance, angle). e^{i\theta } = cos\theta + isin\theta separates the complex growth into its components: real growth (distance from origin) and imaginary growth (anti-clockwise rotation).

The equation in that picture is explained by starting at (1, 0). Operating (1, 0) with e^{i\pi}. This explains why e^{i\pi} is -1.

But there is more! Let’s not stop there… since this is periodic (goes around in a circle), this also applies to anything that is +2\pi.

It also shows why e^{\pi/2} = i.

Then we can show that i^i = e^{-\pi/2} = 0.20787... a real number!

Other usefulness of this equation is it replaces a lot of dreadful trig formulas you had to memorise in high school like double angle formulas – yuk! Just substitute in 2x in the theta and do some expansions, comparing imaginary parts with imaginary parts etc… and you will get what you want!

 

Salt

You might want to check other sources to gain a better intuitive understanding of the equation – it covers a lot of mathematical ground so it is worthwhile doing that. I warn you now that I do not have a complete understanding myself so take everything above with a grain of salt.

lies

I feel pained whenever I catch myself blatantly telling a lie to someone else. And I did yesterday when my neighbour asked me how the winter was for me.

I blurted some detail as a reply that was an outright lie. My neighbour might have sensed that it was a lie. I definitely knew it was a lie. It was not some innocent reply like “it was okay”. Anyways I added so much detail to that answer as if to compensate for something.

Doesn’t matter what I said. I lied and I added way too much detail that makes no sense. I could have just said the winter was ok. But I blabbled to fill the silence and those blabbles were lies.

What is wrong with me…

Human nature.

Maybe that is why I don’t talk much with other people. It’s because all we do is tell lies unless we talk about some abstract thing like mathematics. I have a feeling that we all tell lies because we inherently do not accept imperfection yet we, undoubtly, are.

I way around this is to either accept your imperfections (though my hypothesis is that we inherently cannot) or just accept that we tell lies. Is there another option?

minimalism challenge day 8

  1. a lot of running bibs – lots of sentimental value, maybe I will regret it later, I kept this forever, but whatever
  2. running competition certificates – whatever
  3. university fee receipts – university is super expensive

I cleared out most of my “important” documents. What is left are my high school exam certs, a school report, and various university letters, some IDs.

5 books:

  1. the philosopher’s handbook (rosen)
  2. digital dice (nahin) – computational solutions to practical probability problems – zzz…
  3. how to calculate quickly (sticker) – someone’s last name is called sticker?
  4. dictionary of statistics (oxford) – why buy dictionaries anyway – have you not heard of the internet?
  5. the art of thinking clearly (dobelli) – nobody needs to tell me how to think

minimalism challenge day 7

I realise the last post was a bit dark.

I decided to open up my “important” papers cabinet file thingy and see what things I can throw away. I bet everything in it isn’t important – well, nothing is important anyways – they are mostly receipts and things of sentimental value.

Let’s have a look, shall we.

There are various labels in this cabinet files: *name of my university*, *running*, *personal*, *work*, *receipts*, *health*, *certificate*.

I will just take EVERYTHING out. Here goes nothing…First thing I can throw away is this filer thingy – I don’t know the proper name for this thing that keeps papers and has different… and looks like an accordion. Doesn’t matter – it has collected a lot of dust – I am sure it has a lot of utility left, but I don’t care it’s GONE!

  1. accordion file thingy

6 more to go. Lots of business cards – they can definitely go away.

  1. various business cards
  2. various medical records, immunisation record
  3. various receipts from shopping in Romania – no need to keep those
  4. various receipts from shopping in Hungary – no need to keep those
  5. gamescom ticket 2011 – a ticket to gamescom, I went there to meet a few people online, which was super awkward
  6. a lot of oyster cards – four in total, I am not going back to the UK ever again, so what’s the point of keeping these.

minimalism challenge day 6

  1. Pair of blue shorts that are obviously cheaply made and its dye is leaking whenever I wash them – it’s leaking to the white rope that tightens the waist.

I cannot think of anything to throw away at the moment. Maybe a pair of shoes. Or two. Or three. Some books I never read, but bought because – why? I don’t know but I will keep for sure one. Or two. Or three.

This seeming freedom to choose is giving me a lot of angst. This anxiety is killing me slowly, chipping me little by little, but not fast enough.

I will go out and get something sorted at the bank, get some food, and hopefully I can sort everything out (I doubt it) when I get back.

***

I am back. I was not able to get rice and soy sauce. I was carrying too much.

1 + 1 = 2

I am just trying out latex on wordpress to see if it works. I note that it is not aesthetically pleasing here since the background is offwhite… Okay it is simple to type out the code for 1 + 1 = 2. But for something more complex like integrals, I wonder if there is an easier way to write it and then via a software convert into the latex code for wordpress. Why? I just like simple mathematical proofs, and there is a list of them that I want to include in this blog. Maybe. I am already stuck at understanding the proof that \pi is irrational.

Apparently to safely dispose of an old credit card, you need a magnet to demagnetise the magnetic strip… seriously?! I will just cut it up in pieces and then throw them away in several goes.

  1. Old credit card
  2. Old credit card
  3. Asia miles card
  4. British Airways card

Cards are just a scam for you to spend more, and I doubt I ever saved any money from those air miles cards.

  1. Interational driving permit that expired

I don’t really like driving anyways. I just ate a bag of candy and a box of biscuits in succession. I don’t feel good. I don’t feel that bad either.

Ok. Why do I do things that I know is bad for me but I do them anyways? It is hard to see the long-term consequences. And what are they? We are all going die anyways. That is just a hypothesis, because when you want your life to end, and it doesn’t, sometimes you feel that you will never die and are immortal.

minimalism challenge day 5

I just realised I bought laundry detergent sans parfum ni colorant. I intitally just bought it because I like its white design (rather than the hideous orange/blue Tide ones). I don’t see why I should not start buying cleaning products without unnecessary colors and perfumes from now on since I am adopting minimalism – this is not going too far, plus it might do some good for the environment and myself.

Also, I have three bottles of perfume that I am reluctant to throw away because they were expensive, but I rarely use them anymore – I could spray into the detergent – which would kind of defeat the purpose of sans parfum, but at least my perfume actually smells nice. And it helps me make use of them. 2 birds one stone kind of thing. My life is such a bore.

  1. Eye dropper
  2. Eye dropper
  3. Eye dropper
  4. Lacoste loyalty card
  5. Indigo loyalty card

Eye drops – ever since I had lasik, my eyes became really dry – and I have a feeling that is why my eyesight is still so terrible. But of course they never work – it only soothes and hydrates for the moment the drops touch your eyes and then after a while – dry again!

The loyalty cards – I never shop anymore unless out of necessity. And those cards do not have my identity information – I think, so it is easy to throw away. Expired credit cards take more effort – you’d be paranoid that people would somehow make use of it so you would have to cut it up in tiny pieces and throw them in stages.

minimalism challenge day 4

Three more exercise books full of my notes on actuarial mathematics.

  1. Exercise book
  2. Exercise book
  3. Exercise book

Screw actuarial mathematics that are meanlingless in my current life. More set theory, which may or may not be even more meanlingless but I do have to fill a void in my existence somehow.

  1. Starbucks gold card. Because Starbucks is a scam – only good for their clean toilets and wifi. Everything else is overpriced/bad.

WordPress doesn’t know how to continue numbered lists like Microsoft Word. At least I don’t.