The phrase that everything new is nothing but the well-forgotten old fully applies to the binary system. It turns out that even in ancient China they already used something resembling our “unit-zero”, though not for arithmetic, but for writing the texts of the book of Changes. The closest to understanding the different number systems were the Incas: they used both decimal and binary systems, though the latter was only for text and encoded messages. It can be assumed that even then, 4 thousand years ago, the Incas knew how to translate from binary to decimal.
The modern version of the binary system was proposed by Leibniz about 300 years ago, and after another one and a half centuries George Bull left his name in the memory of his descendants with work on the algebra of logic. Binary arithmetic, together with the algebra of logic, has become the foundation of modern digital technology. It all began in 1937, when a method of symbolic analysis of relay and switching circuits was proposed. This work of Claude Shannon became a “mother” for a relay computer that performed binary addition in 1937. And, of course, one of the tasks of this "great-grandfather" of modern computers was to convert from binary to decimal.
Only three years passed and the next model of the relay “computer” sent commands to the complex number calculator using a telephone line and teletype - well, right, the ancient Internet in action.
What are binary, decimal, hexadecimal and, generally speaking, any N-ary system? Nothing complicated. Take a three-digit number in our favorite decimal system, it is displayed using 10 characters - from 0 to 9, taking into account their location. We will determine that the digits of this number are at positions 0, 1, 2 (the order goes from the last digit to the first). At each of the positions can be any of the numbers in the system, however, the value of this number is determined not only by its outline, but also by the position. For example, for the number 365 (respectively, position 0 is the number 5, position 1 is the number 6, and position 2 is the number 3), the value at the zero position is just 5, at the first position - 6 * 10, and at the second - 3 * 10 * 10. It is curious that starting from the first position, the number contains a significant digit (from 0 to 9) and the base of the system is to the extent equal to the position number, i.e. we can write that 345 = 3 * 10 * 10 + 6 * 10 +3 = 3 * 102 + 6 * 101 + 5 * 100.
Another example:
260974 = 2 * 105 + 6 * 104 + 0 * 103 + 9 * 102 + 7 * 101 + 4 * 100.
As you can see, each positional place contains a significant number from the set of this system, and the multiplier from the base of the system is equal to the position of this number (the number of bits is the number of positions, but +1 more).
From the point of view of representing a number, its binary form is puzzling in its simplicity - only 2 numbers in the system are 0 and 1. But the beauty of mathematics is that even in truncated form, as it might seem, binary numbers are as full and equal as their more "tall comrades." But how to compare them, for example, with a decimal number? As an option, you need to make, and not rushing, a translation from the binary system to decimal. The task cannot be called difficult, but this painstaking work requires attention. So, let's begin.
Based on what was said above about the order in which numbers are represented in any system, and bearing in mind the simplest of them, binary, we take any sequence of “one-toe”. We call this number VO (in Russian, VO), and try to find out what it is - a translation from binary to decimal. Let it be VO = 11001010010. At first glance, a number is like a number. We will see!
In the first line, we will arrange the number itself in a stretched form, and the second we will write as the sum of each position in the form of factors - a significant digit (here the choice is small - 0 or 1) and the number 2 in the degree equal to the position number in the decimal system, we do the translation binary to decimal. Now in the second line you just need to perform the calculations. For clarity, you can also add the third line with intermediate calculations.
VO = 1 1 0 0 1 0 1 0 0 1 0;
VO = 1 * 210 + 1 * 29 + 0 * 28 + 0 * 27 + 1 * 26 + 0 * 25 + 1 * 24 + 0 * 23 + 0 * 22 + 1 * 21 + 0 * 20;
VO = 1 * 1024 + 1 * 512 + 0 * 256 + 0 * 128 + 1 * 64 + 0 * 32 + 1 * 16 + 0 * 8 + 0 * 4 + 1 * 2 + 0 * 1.
We calculate the "arithmetic" in the third line and have what we were looking for: VO = 1618. Well, what is so wonderful about that? And the fact that this number is the most famous of all that people know: proportions of the Egyptian pyramids, the famous Gioconda, musical notes and the human body are associated with it, but ... But with a little clarification - knowing that there should be a lot of good, His Majesty the case gave us this number 1000 times greater than the present value - 1.618. Probably everyone got it. And along the way, the translation from the binary to the decimal system helped to “catch” the most remarkable from the endless sea of numbers - it is also called the “golden ratio”.