We have already written about the amazing property of the number e, which can help when blindly from a finite number of options you need to choose a single option. Today, let us remember about Euler’s identity – by right the most beautiful equation, in which the number occupies an important place ebut not only that. Imagine for a second that you know almost nothing about mathematics, just start to discover its endless beauty – and enjoy.
We all know about number π – the magical ratio of the circumference of a circle to its diameter. The number π can be roughly represented as a fraction 22/7. Number feature π is that in its decimal notation, the decimal places never end. Its approximate value is 3.141592653589793238 … That’s why π is called an irrational number – it cannot be written as a finite number of digits after the decimal point. And here’s another interesting irrational number – e… The number e is “Euler’s number” (from Euler). Here are the first few digits of the number e: 2.7182818284590 …
Not only is this number irrational, it is used in literally all areas of mathematics. It is used in logarithmic functions as the base of the logarithm. We call such a natural logarithm and write it like this: ln x…
What does this entry mean? In natural logarithm f(x) =ln(x) Is the degree to which you want to raise the number e, To obtain x… How to calculate the value e, you ask? There are several ways.
Here is one of them: e Is the limit of the sequence, the common term of which is (1+ 1 /n)ⁿ… Here’s another one: the area of the area under the graph. y= 1 /x from x= 1 to x=e is equal to one unit square.
Another way to define e: look at row 1 + 1/1! + 1/2! + 1/3! + 1/4! +…. The sum of this series is e…
To make you feel how important a number can be e, consider an example. Suppose you have a bookcase with only one specific book on each shelf. Suppose also that someone drops all the books on the floor and you start picking them up from the floor one by one and putting them back on the shelves. The odds of each book ending up on its own shelf is approximately 1 /e… The more books, the closer this probability will be to 1 /e…
Let’s talk now about another interesting mathematical object. It’s called simply: i… Let’s figure out what it is.
If you multiply 2 by 2, you get 4. That is, 2 squared is 4. The square of a positive number is a positive number. But if you square –2, you also get 4, which is a positive number. In other words, no square of a real number can be negative. This is where the concept of an imaginary number arises.
The number √-1 is written with a letter i… i means an imaginary unit. That is, the entry √-5 can be replaced by the entry √5 i… Hence it follows that i² = -1. Number i forms a set of complex numbers, that is, combinations of real and imaginary numbers. For example, writing 8 + i√5 is a complex number. The plane of imaginary numbers is used to render complex numbers.
Mathematics began to study the properties of complex numbers from about the middle of the 18th century. Euler was once having fun with Taylor’s wife … oh sorry … with Taylor’s side. Taylor series:
This madcap just wondered how the Taylor series would behave if we substituted the number i (and that, quite a normal thought for any madman).
And here’s what he started to get:
But i² = -1 (Wow! I’m already starting to get overwhelmed by scientific excitement!) Let’s group all the members of the series containing i…
Guess what will happen next? Some members of the series containing i, are reduced into one group, while others that do not contain an imaginary part, that is, without a number i, – to another. Two Taylor series are obtained: one for the cosine, the other for the sine.
We got the famous Euler formula. Different meanings x and e^ (ix) can be flipped onto the complex plane. For example,
It is a complex number that can be represented on the complex plane. Continuing to plot points e^ (ix) for different values x, you get a circle.
If you need to know the radius r at any point (for example, at point 5 + 7i), the value is calculated x and the real part is taken re^ (ix). Finally, if we substitute the value x = π, we get:
(insofar as cos π = −1 and sin π = 0).
Combining the three most extraordinary mathematical symbols, we get the magic equation:
Here it is in front of you – according to mathematicians, the most beautiful equation in all of mathematics. It’s called Euler’s identity…
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