28 March 2015
We Already Have Pi Day, But Why Not 60 Seconds for Euler?
Posted by Dan Satterfield
Fareed Zakaria of CNN has a very good piece in the Washington Post about STEM and I must say that it makes a lot of sense, although you might not think I would from the title. Check out “Why America’s obsession with STEM education is dangerous” and then come back here for more.
Ok, so your back and I hope you agree with what he said, because I do, and while you likely have an intrinsic understanding of the beauty of a painting like Van Gogh’s Starry Night, or the music of Rachmaninoff. I wonder if you also have an appreciation for the beauty of science and maths? So, if you’ll let me, I’ll try to show you just a glimmer of its beauty as well, but fair warning: If you are a geek like me, you’ll quickly see where I’m heading here,but If not then I’m writing this especially for you.
Let’s say that the economy has gotten so good that the Bank of England announces it is going to pay 100% interest on savings accounts starting January 1 next year. Being a skinflint like me you decide to invest 1£ of money (or 1$ at Bank of America, which makes the same offer to compete) and at the end of the year, they call you (like that would happen) and tell you that they have credited your account with an extra dollar of interest, so now you have 2£! Now, your brother-in-law (who works for the prestigious law firm of Dewey, Cheatem, and Howe) says, you are dumber than dirt because you could have gotten more!
How so you ask, and rather sheepishly, since he’s always making you look foolish and he usually ends up being right. He tells you, with that superior tone you hate, that you should have made them pay your interest once per month! That way, at the end of one month you would have been given 1£ divided by 12. and each month after your interest would be calculated on not just your original pound, but the interest that they had paid you previously. It turns out that it’s not much, but if you have 1 million pounds invested, your suddenly talking real money. Big money! All because you had the bank pay your interest at the end of each month they had your money.
Now, you suddenly have an idea (that hopefully will make you look smart in the eyes of your brother-in-law) and suggest they pay the interest at the end of each day, no better yet- EACH HOUR of the day, for the entire year! How much more money will he have than the 2£ at the end of the year? The answer is you will have 2.71 pounds, and IF you insist they compound it every second, and then every tenth of a second all the way to continuously, you will end up with 2.71828 pounds in your account at the end of the year. Now you realize you should have made a bigger deposit, because your brother-in-law put 100,000 pounds in, and is vacationing in the south of France with his 271,182.8 pounds!
Rather interesting, is it not, that if you compound interest continuously you end up with this strange number?? It turns out this number has a name, and it’s called Euler’s number. Euler’s Number is an irrational number like PI, and goes on forever, but 2.7182818284590452353602874713527 is good enough for just about every purpose including working at the Large Hadron Collider. But why is this number, this number? Why is pi 31415926535… ?
It turns out that this number shows up so commonly in maths and physics equations that it is just given the letter e, like we use π for 3.14…
Furthermore this number turns up in the equations that describe our universe, like Einsteins equations of relativity and the those of quantum mechanics. It’s an important number, One that is so important that we are left asking why is little e SO important?
No one knows.
If there are other universes, do they have the same number or do they have a different set of vital numbers? You’re not asking me are you?? Those are the kinds of things that keep Stephen Hawking up at night!
One last thing, that is just downright beautiful though. It’s called Euler’s Identity and it’s below:
If you take the number e to the power of “pi times “i”” (square root of -1) and add one- you get zero. i is called an imaginary number (it’s a complex number), and it too shows up in all kinds of weird places, like for instance Schrödinger’s equation in quantum dynamics. This equation is true by the way, and you can find a proof here if you do not believe me, but think about this for a second. e, i , π, and 1, and zero; the five most important numbers in math. A complex number, real numbers, and an imaginary number as well, and all related by that single, simple and elegant equation!
Don’t ask Stephen Hawking, he doesn’t know that either!
If that doesn’t make your knees wobble just as much as walking up to Van Gogh’s Sunflowers at the National Gallery in Trafalgar Square, or hearing da,da,da,dahhh, then something is very wrong with you. In fact, I think this deserves as much recognition as π, so I suggest that on 2 July (2/7*) at 1828 GMT, we all say “e to the i pi plus one equal zero, and have a toast of wine to Leonhard Euler!
* If you are wondering why I said July second instead of Feb. 7th, it’s because only in America do we write the date backwards. The rest of the world writes it from shortest period to longest, i.e. day, month, year. Oh, and for those on the East Coast of the U.S. that will be 2:28 PM EDT.
prof prem raj pushpakaran writes — let us celebrate eDay (Euler’s number or Napier’s constant) on February 7 !!!!