Name: e(オイラーの数) - 数マニア (e (Euler's Number) - Numberphile)
Uploaded: 2021-01-14T10:38:01.000Z
Duration: 10 min 42 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

現在進行形

The big, famous constant, e! Okay, it's one of the famous mathematical constants,

最上級形容詞

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.

So, 100 percent interest, so it means after one year,

So you've earned one pound interest and you've got your original pound.

What if I offered you instead fifty percent interest, every six months?

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,

and you add that on to what you had so it's two pounds twenty-five

It's better. So what happens if I do this more regularly?

I offer you one-twelfth interest every month

So after the first month, it's gonna be multiplied by this.

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,

and your third month you would multiply it again,

you actually do that twelve times in a year.

So it's actually better. In fact, the more frequent your interest is

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.

Every instant I'm earning interest. Continuous interest. What does that look like?

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.

[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.

[James]: It was just a letter. He might've used a, b, c, and d already for something else.

[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

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

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

字幕リスト動画再生

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

access

nasty

constant

pattern