What is Euler's Number?

They keep talking about compound interest, does anyone know what compound interest is?

Yes, I know, of course.

We deposit our money in a time deposit account at the bank, and in all the accounts they call interest on maturity, it is stated annually. 

For example, if they say that you will get 10% interest, it means that one year after the date you deposit the money, you will get your 100 liras back and they will pay you 10 liras interest on it. The 10% interest here is the annual interest rate.

In this case, if you make your maturity period 6 months, after six months you will be paid half of the annual interest rate as interest in addition to your 100 liras, that is, in our example, since we say 10% interest, they will give you half, that is, 5 liras, instead of 10 liras.

Now, if we have 105 liras at the end of 6 months, if we deposit 105 liras in the time deposit account again with 10% interest for a maturity period of 6 months, we will be entitled to receive 5 liras 25 kurus, which is half of 10 liras 50 kurus at the end of six months.

In this case, at the end of one year, we will receive 10 liras 25 kurus for 100 liras, excluding the principal.

This means that a 6-month term account with a 10% annual interest rate has an annual compound interest rate of 10.25%.

This is called compound interest.

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That's good!

With this account, I would invest my money in monthly maturities. I would probably be much more profitable in this case. In fact, let's put it in daily maturity, the daily maturity account will earn much more interest income!

Yes, banks know this, so they adjust their interest rates according to the maturity periods.

In such a volatile inflation environment, daily maturities are naturally the best, but unfortunately daily interest rates are much lower than annual maturities.

Everyone is smart, not only you!

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So, if we shorten the maturity period in this way, how much more can the compound interest be than the normal interest rate under normal conditions with shorter maturities?

Actually, this is a math question. The answer is what we call the "e" number. The Euler number is 2.71828.

When our annual interest rate is 100%, the maximum compound interest rate we can earn on our principal as the maturity periods get shorter is 171.8%. 

If we invested our money for one year at 100% interest for a term of one year, our 100 liras would be 200 liras, but if we invested our money for one year at 100% interest for less maturities by shortening the maturity periods, our 100 liras would be 271 liras at the end of the year.

Good rate!

But what can we do? 

Even if we can reduce the maturity period to seconds, this is the maximum gain that can be achieved, not more.

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Euler's number is one of the special numbers of mathematics, like pi. At the same time, like "pi", it is a non-rational number, meaning that it has infinitely many fractions, no matter how hard you try. It is also a transcendental number. The definition of transcendental numbers is the name given to complex numbers whose exact result cannot be deduced by a ratio and whose result cannot be deduced by a formula.

This Euler number is named after the Swiss mathematician Leonhard Euler, but it was actually Jakob Bernoulli who first discovered this number.

Bernoulli realized the existence of such a fixed number in 1683 when he was working on the maximum possible compound interest rate for the aforementioned 100% annual interest rate.

However, he did not raise this issue. 

Forty-eight years passed and in 1731 Leonhard Euler, in a letter to Christian Goldbach, referred to it as the "e" number. 

In fact, the number "e" was not mentioned by Euler or Bernoulli, but by the Scottish mathematician John Naiper, who worked on logarithms in 1618, even before Bernoulli, in some calculations in his book.

Today, however, the letter "e" has become a mathematical symbol and is known as Euler's number.

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Yes, this number is a mysterious number!

It is also interestingly involved in logarithmic calculations, e-based logarithms, that is, logarithmic calculations symbolized by "ln", have a special place in mathematics. 

The number "e" is also widely used in derivative and integral calculations.

Even though it is not the square root of a negative number, it has an interesting place in the world of complex numbers, which are derived because they are very useful in calculations. 

For example, Leonhard Euler also found an equation called e^(i x "pi")=-1. 

This equality is also shown as e^(i x "pi")+1=0.

This equality is called the Euler equation.

It is described as the great equation of mathematics because it contains the five mysterious numbers of mathematics.

There is 0, which is still conceptually quite a mysterious number.

There is 1, again conceptually a very special number, one, unified, whole!

There's "pi", a number that I still can't understand how we can't find the end of it, after all, even though what we call a circle and the diameter of a circle are things that can be physically demonstrated, the number "pi", which is the ratio between them, has no end! How this can be, it's really impossible to believe.

Then there is the number "i", the relative square root of the number -1, which in reality cannot be squared! The number "i" is the value on which complex numbers are based.

And finally the number "e", the mysterious number that was discovered from the compound interest calculus, but which has found a place in almost every formula of mathematics.

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That's why Euler's equation is called the great equation of mathematics.

There are other similar equations, but I think that's enough for today.

There are many more such mysterious numbers in mathematics, for example, there is the golden ratio, the number of nature, then there is the silver ratio, the super golden ratio, the Laplace limit, the Gauss constant, the cosmological constant...

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But for me, the Euler number is a really special number, because when you know that number, you know that no method can make a lot of money unless what you are doing is gambling. 

And gambling is a risk, and it's also a bad habit.

I think we should teach Euler's number and its source in our schools if we don't want an army of poor losers pretending to be awake.

Stay with math! You cannot be wrong!

Love and regards to everyone from Moscow.