How to convert to decimal number system?

Most people on our planet use decimal notation when counting, but computers use binary. Some tribes at the dawn of human development used the duodenal and sixtide. It was from them that we left 12 hours on the dial and 60 minutes per hour.

Sometimes you need to transfer a number from one system to another. This article will take a closer look at how to convert to the decimal system from some other popular systems.

The principle of building numbers from numbers

First of all, you need to understand what the number system is and its basis. The number system is a way of representing numbers as a combination of certain digits. The base of the system is the number of digits used in it. For example, in the decimal system with a base of 10, only 10 digits - from 0 to 9. In hexadecimal, respectively, 16 digits, which are denoted by Arabic numerals 0 - 9 and Latin letters A - F instead of digits 10 - 15. For example, 2F7BE16- the number of hexadecimal system.With such a record, the lower index denotes the base of the number system. The key difference between systems with different bases is the "value" of the number 10. In hexadecimal system 1016will be equal to 1610, and in binary 102equal to just two. 10016will be calculated as

10016= 1016* 1016= 1610* 1610= 25610.

It is also necessary to distinguish between the concept of "figure" and "number." The number is denoted by one character, and the number - and can be several. For example, the number 910in binary will look like 10012, and the digit 9 in the binary system does not exist as such.

Translation algorithm

To translate the number into the decimal system, you need to learn how to apply a simple algorithm.

  1. Determine the base of the number system. It is denoted by the subscript after the number, for example, in the number 2F7BE16base is 16.
  2. Multiply each digit of a number by a base to the extent equal to the number of the digit from right to left, starting with zero. Including 2F7BE16E (equal to 14) is multiplied by 16 to the zero degree, B (number 11) - by 16 to the first degree, and so on: 2F7BE16= 2*164+15*163+ 7*162+ 11*161+ 14*160. 
  3. Add the results.

2*164 +15*163 + 7*162 + 11*161 + 14*160 = 19449410.

Consider the examples of how the most popular - hexadecimal, octal and binary systems are converted to decimal.

  • 57368 = 5*83+ 7*82+ 3*81+ 6*80 = 303810
  • 10010112= 1*26+ 0*25+ 0*24+ 1*23+ 0*22+ 1*21+ 1*20= 7510
  • 2F7BE16 = 2*164 +15*163 + 7*162 + 11*161 + 14*160 = 19449410

Of course, it is inconvenient, irrational, and reluctant to count each time manually. There are many calculators that can transfer numbers from system to system.For example, the standard Windows calculator in the Programmer mode (Alt + 3 keys or the View menu) can work with base systems 2, 8, 10 and 16.