We're gonna talk about e! The big, famous constant, e!
Okay, it's one of the famous mathematical constants, one of the most important, goes along with pi, and I don't know, golden ratio, and square root of two, constants in maths that are the most important constants, and e is one of those constants.
So e is an irrational number, and it's equal to... 2.718281828, something, something, ...
The problem with e, is it's not defined by geometry. Now pi, is a something that is defined by geometry, right, it's the ratio of a circle's circumference and it's diameter.
And it's something the ancient Greeks knew about. And a lot of mathematical constants go back to the ancient Greeks, but e is different. e is not based on a shape, it's not based on geometry.
It's a mathematical constant that is related to growth, and rate of change, but why is it related to growth and rate of change?
So let's look at the original problem where e was first used.
So we're going to go back to the seventeenth century, and this is Jacob Bernoulli, and he was interested in compound interest, so, earning interest on your money.
So imagine you've got one pound in the bank. And you have a very generous bank and they're gonna offer you 100 percent interest every year. Wow, thanks alot, bank!
So, 100 percent interest, so it means after one year, you'll have two pounds. So you've earned one pound interest and you've got your original pound. So, you now have two pounds.
What if I offered you instead fifty percent interest, every six months? Now is that better or worse? Well, let's think about it.
Ok, you're starting with one pound and then I'm going to offer you fifty percent interest every six months. So after six months, you now have one pound, fifty and then you wait another six months and you're earning fifty percent interest on your total, which is another seventy-five p, and you add that on to what you had so it's two pounds twenty-five.
Better! It's better. So what happens if I do this more regularly? What if I do it every month? I offer you one-twelfth interest every month. Is that better? So, let's think about that.
So after the first month, it's gonna be multiplied by this. One plus one-twelfth. So one-twelfth, that's your interest and then you're adding that onto the original pound that you've got.
So, you do that, that's your first month, then for your second month, you take that and multiply it again by the same value, and your third month you would multiply it again, and again. You actually do that twelve times in a year. So in a year, you'd raise that to a power twelve, and you would get two pounds sixty-one. So it's actually better. In fact, the more frequent your interest is, the better the results.
Let's start with every week. So if we do it for every week, how much better is that? What I'm saying is you're earning one over fifty two interest every week. And then after the end of the year, you got fifty two weeks and you would have two pounds sixty nine. So it's getting better and better and better.
In general, you might be able to see a pattern happening here. In general it would look like this: You'd be multiplying by one plus one over n, to the power n. Hopefully you can see that pattern happening. So here n is equal to twelve if you do it every month, fifty two if you do it every week.
If you did it every day, it'd be one pound multiplied by one over three hundred and sixty five to the power three hundred and sixty five. And that's equal to two pounds, seventy one. Right, and so it would get better if you did it every second, or every nanosecond.
What if I could do it continuously? Every instant I'm earning interest. Continuous interest. What does that look like? That means if I take this formula here one plus one over n to the n, I'm gonna n tend to infinity.
That would be continuous interest. Now what is that? What is that value? And that's what Bernoulli wanted to know. He didn't work it out. He knew it was between two and three. So fifty years later, Euler worked it out. Euler, he works everything out.
Him or Gauss?
It's either Euler or Gauss. Say Euler or Gauss, you're probably going to be right. And the value was 2.718281828459... and so on.
We were pretty close when we were doing it daily, weren't we? It was already two seventy one at daily.
You're right, You're right. We were getting closer, weren't we?
We were getting close and closer to this value. So already we're quite close to it. If you did it forever though, of course you would have this irrational number. Now Euler called this e. He didn't name it after himself, although it is now known as the Euler constant.
Why'd he call it e then?
It was just a letter. He might've used a, b, c, and d already for something else.
Right? So you use the next one.
Bit of a coincidence!
It's a lovely coincidence! I fully believe that he's not being a jerk here, naming it after himself. But it's a lovely coincidence that it's e for Euler's number.