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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.

• [Brady]: Sometimes here on Numberphile we can make more videos than we'd otherwise be able to

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• Okay, I'm gonna go for e, e.

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

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# e(オイラーの数) - 数マニア (e (Euler's Number) - Numberphile)

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林宜悉 に公開 2021 年 01 月 14 日