Euler constant...e!

e!

We have all used this mathematical constant in our math classes. Do you know how it came into existence and why it’s so famous?

The big, famous constant, e!, it's one of the famous mathematical constants, one of the most important, goes along with pi, golden ratio, and square root of two, etc.

So e is an irrational number, and it's equal to 2.718281828……. and goes on.

The problem with e is that 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 its 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?

Let's look into its history, how it came to existence and where it 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 and earning interest on his money.

So imagine you've got one rupee in the bank and you have a very generous bank and they're going to offer you 100 percent interest every year. Wow, thanks a lot, bank! so 100 percent interest, so it means after one year, You’ll have two rupees. So you've earned one rupee interest and you've got your original rupee. So, you now have two rupees.

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

Now is that better or worse?

Well, let's think about it. You're starting with one rupee and then I'm going to offer you fifty percent interest every six months. So after six months, you now have one rupee and 50 paise and then you wait another six months and you're earning fifty percent interest on your total, which is another seventy-five paise. And you add that on to what you had so it's two rupee twenty-five paise

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 one-twelfth, that's your interest and then you're adding that onto the original rupee that you've got.

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 rupees sixty-one paise.

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 rupees sixty nine paise.

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

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 for every day.

So it would get better if you did it every second, or every nanosecond. But Jacob Bernoulli wanted more than that, let’s see what he did.

What if every instant I'm earning interest. Continuous interest. What does that look like? That means if I take make n in the formula tend to infinity I would have 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. You know Euler, he works everything out.

And the value was 2.718281828459... and so on.

You may have noticed we got very close when we did the compounding for each day. So making the interval smaller we will get closer and closer to e. Now Euler called this irrational number e. He didn't name it after himself, although it is now known as the Euler constant.

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.

This is a fraction that goes on forever, continuous fraction. Because there's a pattern, and that pattern does hold 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 23 decimal places. To do that, he had a different formula

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 so far so good but why is e a big deal? It's because e is the natural language of growth. Okay, let's draw a graph y equals e to the power of x.

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 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 also 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. 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

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.