# e (Euler’s Number) – Numberphile

James: We'll talk about (e)! The famous constant (e), well, it's one of the famous mathematical constants. One of the most important constants, such as pi, the golden ratio, and square root of number 2. Mathematical constants are the most important types of constants, and (e) is one of them. (e) is an irrational number, and it is equal to: 2.178281828 … etc. The problem with the constant (e) is that it is not a geometric identifier (using geometry). Now, the constant pi (pi) is a geometrically defined constant. It is the ratio between the circumference of a circle and its diameter. It is something known to the ancient Greeks. Many of the geometric constants are ascribed to them. But the constant (e) is different, it is not geometric, it is not geometrical.

It is a mathematical constant related to inflation and the rate of change, but what is related to them? Let's look at the problem in which the constant (e) is used for the first time. So, we're going back to the 17th century, this is Jacob Bernoulli, and he was interested in compound interest, which is to earn interest on your money. Imagine you have a pound in the bank.

It is a very generous bank, as they will offer you 100% annual interest. Thank you very much, Bank! 100% interest, so after a year … You'll have two pounds. You earned pound interest, and you have the original pound in the bank … So, you got two pounds. What if I offered you 50% interest every six months? Is it better or worse? Well let's think about it … Well, you start out with 1 pound, then I'll offer you 50% interest every six months. So, after six months, you'll have a pound and a half. Then you will wait another six months, and you will earn 50% interest on the total amount of your cash … Which is another 75 piasters … And you would add that to what you already had, and you would have a pound and a quarter. This is better! it's better.

But what happens if you do this more regularly? What if I add interest every month? I'll offer you 1/12 interest every month. Is that better? Let's think about it. After the first month has passed, it will be multiplied by the following number: 1 + 1/12 1/12, this is your interest, and you will add the value of this interest to the pound that you already own. You will do this for the first month … Then in the second month … You will take this value and multiply it again. At the same value. And in the third month you will hit it again …

and again. You will do this twelve times a year. So, to calculate the total value in a year, You will raise this value to the power of 12. And the value will be 2.61 pounds. So this is actually better. The fact is that the more interest is added regularly … The higher the result, the better. Let's try this every week. What if you added profit every week? How better would this be? What I mean is that you are earning interest at 1/52 of the original amount per week.

And at the end of the year … You have 52 weeks, and the total amount is 2.69 pounds. It gets better and better. In general, you can notice a pattern forming … in general the final shape will look like this: You multiply by (1 + 1 / n) ^ n. I hope you notice this pattern happening. So, here, n = 12 if you were adding interest every month, and n = 52 if you were adding interest every week. If you add interest daily, the value will be equal to 1 * (1 + 1/365) ^ 365 This equals 2.71 pounds. Well, and it gets even better if you add the interest every second or every nanosecond. What if I constantly add interest? In every moment I earn continuous interest. What will this look like? This means I have this mathematical formula here … It will become (1 + 1 / n) ^ n, as n tends or tends to infinity.

This will be the continuous, never-ending benefit. Now, what does this mean? What is this value? This is what Bernoulli wanted to know. He couldn't solve it, but he knew that the value was between two and three, so Euler was able to solve it fifty years after Bernoulli had thought. Euler solves everything! Brady: Is it Gauss's mother? James: Euler or Gauss, if you say it's Euler or Gauss, you'd be right. And the value that Euler found was 2.718281828459 … etc. Brady: We were very close to this value when we added interest daily, right? The value was 2.71 when the interest was added each day. James: You're right, we've been getting closer to value, right? We've been getting closer and closer to that value. So when adding daily, we were really close to it. But if you did it forever, you'd have an irrational number. Euler called this value (e). He did not release his name to it, however it is known as the Euler number. Brady: And why did he name it (e) then? James: It was just the letter. He might have already used A, B, C, and D for something else. is not it? So he used the next letter for them. Brady: A strange coincidence! James: I think it was an amazing coincidence! I don't think he was cocky and he called her by the first letter of his name. But it's an amazing coincidence, the value is called (e), which is Euler's number. Brady: Would you have called it G if you were the one who discovered it? James: No, I wouldn't call it G, but I would have hoped someone else would call it C …

I would have agreed to this. Euler proved that it was an irrational number. He found a new formula for calculating (e), not this previous formula, but a different formula. And it turned out to be an irrational number, so I'll show you this very quickly. It's found that (e) equals 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 it goes on indefinitely. This is a fracture that lasts forever, an infinite fracture. And you can see that it lasts forever Because there is a pattern, and this pattern remains true. You have 2, then 1, 1, 4, 1, 1, 6, 1, 1, 8, etc. Thus you can see that the pattern continues forever, and breaking it is an infinite fraction. This means that it is an irrational number. If it doesn't last forever, it becomes finished, and if it becomes finished you can write it in fractional form. Euler was also able to calculate the value of (e), and he reached eighteen decimal places for this number.

To do this, he used a different formula. I'll show it to you. This time, he found that (e) equals 1 plus … 1 over the 1 factorial plus one over the 2 factorial plus 1 over the 3 factorial Plus 1 over the fourth factorial. And that goes on forever. It is a good formula. If you don't know the meaning of a factorial, then a factorial is the product of all the numbers … The precedent for a certain number, so if it is a factorial of 4, it equals 4 * 3 * 2 * 1. Well, why is this so important? Because (e) is the natural language of inflation.

And I'll show you why. Well let's draw a graph, where y = e ^ x (Y = e ^ X) Here, we take exponents of (e). So at zero, x = 0 (X = 0), the line will intersect at 1. So if you take a point on this graph, the point value will be equal to (e ^ X) Therefore, this is important. The slope at that point, the slope of the curve … At this point it equals (e ^ X), and the area under the curve, which means the area under this curve … Even negative infinity is equal to (e ^ X). And this is the only function that has this feature. The value, slope, and area at each point are equal. So, at number 1, the value is equal to (e), because the value at this point is equal to e ^ 1.

The value is 2.718 and the slope is 2.718 … And the area under the curve is 2.718. The reason this is so important is that it is a unique property. Because it has this property, (e) it becomes the natural language of differentiation. Differentiation is the mathematics of rate of change, inflation, and areas. And if you're interested in these things, if you write with (e), the math gets much simpler. Because if you don't use (e), you get a lot of complex constants And then the math gets messy. If you are trying not to use (e), You make things harder. It is the natural language of inflation. Of course, the constant (e) is famous for putting all the famous mathematical constants together using this formula … Euler's formula, which is as follows: (e) s pi (Pi) plus one equals zero (e ^ + 1 = 0) Here we get all the important mathematical constants in one formula. We have (e), and we have the imaginary number (i), which is the square root of negative one, we have the (pi), and the numbers 1 and 0.

And they were collected in one formula … Which most people see as the most beautiful formula in mathematics. I've seen her a lot, and I'm kind of bored with her. Don't put this in the video! .