phrase that all new - it is nothing like a well-forgotten old, fully applies to the binary system.It turns out that in ancient China have used something resembling our "unity-toe", though not for arithmetic and writing texts for the Book of Changes.The closest to an understanding of the different number systems were Incas: they used and the decimal and binary systems, however, last only for text and encoded messages.We can assume that even then, 4 ths. Years ago, the Incas knew how to make a translation from binary to decimal system.
modern version of the binary system was proposed by Leibniz only some 300 years ago, and after a half-century, George Boole left his name in the memory of future work on the algebra of logic.Binary arithmetic together with the algebra of logic was the foundation of the current digital technology.It all began in 1937 when he proposed a method of symbolic analysis of relay and switching circuits.The work of Claude Shannon has become "mother" for the relay computer performs binary addition already in 1937.And, of course, one of the objectives of this "great-grandfather" of modern computers has been translated from binary to decimal system.
's only been three years, and another type of relay "computer" to send commands to the calculator of complex numbers using the phone line and teletype - well, just old internet in action.
What are binary, decimal, hexadecimal, and, generally speaking, any N-ary system?Nothing complicated.Consider a three-digit number in the decimal system our favorite, it is represented by means of 10 characters - from 0 to 9, with regard to their location.Determine the number of digits that are at positions 0, 1, 2 (the order goes from the first to the last digit).At each of the positions can be any of the numbers of the system, but the magnitude of this number depends not only on his mark, but also the place position.For example, for the number 365 (respectively, positions 0 - figure 5, reference numeral 1 - figure 6 and position 2 - figure 3) the value of a zero position - a 5 in the first position - 6 * 10, and the second - 3 *10 * 10.It is curious that, starting from the first position, contains a number of significant digit (0 to 9) and the base system to the extent equal to the number position, iewe can write 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 location includes meaningful numbers from the set of the system, and the multiplier of the base system to the extent equal to the position of the number (bit number it is the number of positions, but one more).
terms of representation of its binary form of puzzles for its simplicity - only 2 numbers in the system - 0 and 1. But the beauty of mathematics is that even in a truncated form as it may seem, binary numbers are the same full and equal likeand a "tall companions."But how to compare them, for example, with a decimal number?As an option, you do not hurry, translated from binary to decimal.The task can not be called difficult, but the hard work requires attention.So, let's begin.
Based on the above, on the order of representation of numbers in any system, and bearing in mind the simplest of them - binary, take any sequence "of ones-tac-toe."We call this number VO (in Russian IN), and try to find out what it is - translated from binary to decimal system.Let it be VO = 11,001,010,010.At first glance, the number of the number.Let's see!
The first row contains the number itself in an extended form, and the second write out as the sum of each item in the form factors - significant digit (here the choice is small - 0 or 1) and the number 2 to the power of the positional number in the decimal system, we're doingtranslated from binary to decimal.Now, in the second line you just need to perform a calculation.For clarity, we can add yet a third line with the 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 = + 1024 * 1 1 * 0 + 512 * 0 + 256 * 128 + 1 * 64 + 0 * 32 + 1 * 16 + 0 * 8 + 0 * 4 + 1 * 2 + 0 * 1.
calculate the "arithmetic" in the third line, and we have what we were looking for: VO = 1618. So what else is great?And what is the number - the most famous of all of which are known to people, it is linked proportion of the Egyptian pyramids, the famous Mona Lisa, musical notes and the human body, but ... But with a little refinement - knowing that the good should be a lot of his Majesty the caseHe gave us the number to 1,000 times the present value of - 1.618.I think that all went.And incidentally translated from binary to decimal help of an infinite sea of numbers "catch" the most remarkable - it is called "the golden proportion".
