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  • [James]: 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 plus] 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.

  • [Brady]: Him or Gauss?

  • [James]: 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.

  • [Brady]: We were pretty close when we were doing it daily, weren't we? It was already two seventy one at daily.

  • [James]: 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.

  • [Brady]: Why'd he call it e then?

  • [James]: 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.

  • [Brady]: Bit of a coincidence!

  • [James]: 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.

  • [Brady]: Would you have called it g if you discovered it?

  • [James]: I would not have called it g. No, I would've hoped somebody else would've called it g

  • and then I would have accepted that.

  • Euler proved that this was irrational.

  • He found a formula for e which was a new formula. Not this one here, a different formula.

  • And it showed that it was irrational. I'll quickly show you that.

  • He found that e was equal to two plus one over

  • one plus one over two plus one over one plus one over one plus one over four plus one over one plus one over

  • one plus one over six... and this goes on forever.

  • This is a fraction that goes on forever, continuous fraction. But you can see it goes on forever

  • Because there's a pattern, and that pattern does hold.

  • You got two, one one four, one one six, one one eight.

  • So you can see that pattern goes on forever, and if the fraction goes on forever

  • it means it's an irrational number.

  • If it didn't go on forever, it would terminate, and if you terminate you can write it as a fraction.

  • And he also worked out the value for e. He did it up to eighteen decimal places.

  • To do that, he had a different formula to do that, I'll show you that one.

  • And this time, he worked out e was equal to one plus

  • one over one factorial plus one over two factorial plus one over three factorial

  • plus one over four factorial... and this is something that's going on forever.

  • It's a nice formula, if you're happy with factorials. Factorials means you're multiplying all the numbers

  • up to that value. So if it was four factorial, it'd be four times three times two times one.

  • Okay, why is e a big deal? It's because e is the natural language of growth.

  • And I'll show why. Okay, let's draw a graph y equals e to the x.

  • So we're taking powers of e. So over here at zero, this would cross at one.

  • So if you took a point on this graph, the value at that point is e to the power x.

  • And this is why it's important. The gradient at that point, the gradient of the curve

  • at that point is e to the x. And the area under the curve which means the area under the curve

  • all the way down to minus infinity is e to the x.

  • And it's the only function that has that property.

  • So it has the same value, gradient, and area at every point along the line.

  • So at one, the value is e because it's e to the power one. The value is 2.718, the gradient is 2.718

  • and the area under the curve is 2.718. The reason this is important then, because it's unique

  • in having this property as well, it becomes the natural language of calculus.

  • And calculus is the maths of rate of change and growth and areas, maths like that.

  • And if you're interested in those things, if you write it in terms of e, then the maths becomes much simpler.

  • Because if you don't write it in terms of e, you get lots of nasty constants

  • and the maths is really messy. If you're trying to deliberately avoid using e,

  • you're making it hard for yourself. It's the natural language of growth.

  • And of course e is famous for bringing together all the famous mathematical constants with this formula,

  • Euler's formula, which is e to the i pi plus one equals zero.

  • So there we have all the big mathematical constants in one formula brought together.

  • We've got e, we've got i, square root of minus one, we've got pi of course, we've got one and zero

  • and they bring them all together in one formula

  • which is often voted as the most beautiful formula in mathematics.

  • I've seen it so often, I'm kinda jaded to it, don't put that in the video.

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  • Okay, I'm gonna go for e, e.

[James]: We're gonna talk about e!

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A2 euler interest formula pound brady factorial

e (Euler's Number) - Numberphile

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    林宜悉 posted on 2020/04/04
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