История чисел (на английском)

Grankina Natalia (8122). 
Summary 1. 
 
Many interesting things about the numbers can be found in articles from the magazine “Scientific American”. 
 
In the introduction the author tells us about the relationship between mathematicians and numbers. It is generally accepted that a mathematician is a man who is good at numbers, but mathematicians don’t think so. They say that they have as much difficulty as anybody else in reconciling their bank statements, but we are talking about something else. Whatever they say, they know all about Liouville numbers, Fibonacci numbers, transfinite numbers, hyper-complex numbers, and so on. 
 
We can now proceed directly to the story about the numbers:

 
History of number systems originated many centuries ago, when the Babylonians came up with numbers for the accounts. Since then much has changed. For example, to date, physicists use quaternions, matrices and transfinite numbers. Author perfectly described it this way: “the complexity of a civilization is mirrored in complexity of its numbers”.  
 
Hereinafter described in more detail: the numbers system can be divided into five stages:  
1) natural numbers,  
2) integers,  
3) rational numbers, 
4) real numbers,  
5) complex numbers. 
 
In the following paragraphs, the author tells about each of these five stages separately and in detail. 
 
In the passages about natural numbers describes: the options for their writing (Roman, Arabic and Greek), problem of expressing large numbers, jump from the finite to the infinite. On the latter point, the concept of an infinite series of numbers was a supreme act of the imagination, because it ran counter to all physical experience and to a philosophical belief that the universe must be finite. 
 
Further, the author tells us about the transition from natural numbers to integers (recall that the integers include negative numbers, positive numbers and zero). It turns out, that this step was very difficult to make. Only with the publication of Girolamo Cardano's “Ars Magna” in 1545, that is relatively recent, negative numbers were fully incorporated into mathematics. 
 
It was all a bit easier with rational numbers (simply put fractions). They were discussed as early as 1550 B.C. in the Rhind papyrus of Egypt. This is understandable: the fractional numbers are used in everyday life almost as often as positive. Easier to imagine half of the apple, than a negative number of them. 
 
The next stage of abstraction is the system of real numbers, which includes rational and irrational numbers. We have already talked about the rational numbers; it's time to understand the irrational. The simplest example of an irrational number is a square root of two. This number was found by Pythagoreans. On the one hand, it is the length of the hypotenuse of an isosceles right triangle. On the other hand, it was proved that can not be represented as a fraction. It took centuries to understand that irrational numbers are nothing more than a non-periodic infinite sequence of digits after the point. Irrational numbers are infinitely many, but the most famous of these are  and e. 
 
From the standpoint of schoolchild equation x² = -1 has no solution, since any number squared must be non-negative. Nevertheless, being in the field of complex numbers, we meet with an amazing imaginary unit i = , and then we get two solutions of our equation ±i. In general, the complex numbers are as follows: a + bi and they are very useful in describing various physical phenomena. In the last paragraphs describes different formulas related to the i. The Euler's formula is particularly beautiful, in my opinion, which links together , e and i. 
 
Finally, I would like to quote Goethe: "It is often said that figures rule the world, at least there is no doubt that the figures show how it is managed."