字幕表 動画を再生する 英語字幕をプリント [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 thanks to excellent sponsors. And today we'd like to thank "The Great Courses Plus," which is a fabulous service. 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Just to go and have a look. It's a great way of showing your support for Numberphile and showing the people at The Great Courses Plus that you came from here. Our thanks again to them for supporting this video. Okay, I'm gonna go for e, e.