Binary and Decimal System Numbers Converters. Base 2 and Base 10 Conversions. Counting in Binary

Binary and decimal system numbers available converters on this website, base 2 and base 10 conversions:

  • DEC to BIN: Numerical conversions of unsigned base 10 to base 2, explained.
  • BIN to DEC: Numerical conversions of unsigned base 2 to base 10, explained.
  • DEC to BIN: Numerical conversions of signed base 10 (positive and negative integer numbers) to base 2, explained.
  • BIN to DEC: Numerical conversions of signed base 2 to base 10 (positive and negative integer numbers), explained.
  • DEC to BIN: Numerical conversions of decimal system integer numbers to binary in one's complement representation or in two's complement representation
  • BIN to DEC: Numerical conversions of binary in one's complement representation or in two's complement representation to decimal system numbers
  • DEC to BIN: Float and double decimal system numbers to 32 / 64 Bit Single / Double Precision IEEE 754 Binary Floating Point Standard Representation
  • BIN to DEC: 32 / 64 Bit Single / Double Precision IEEE 754 Binary Floating Point Standard Representation to Float and double decimal system numbers
  • Go down this page to find the menu with all the converters.

Counting in binary system

Introduction: the binary system code. Binary numbers. Base two numeral system

  • In mathematics and in digital electronics, a binary number is a number written by using the binary numeral system, which is a base 2 numeral system.
  • In this system, numeric values are represented by using only two different symbols: typically 0 (zero) and 1 (one).
  • The base 2 system is a positional numeric system with a radix of 2.
  • Binary is the way computers hold information, by using only the two symbols: the 1's and the 0's.
  • Examples of binary numbers: 01, 10, 001, 010, 011, 100, 101, 110, 111, etc.
  • The decimal numeral system (some call it denary system) we're all familiar with is a base-ten system, meaning that it uses ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
  • Examples of numbers in the decimal system: 1, 2, 3, 10, 101, 304, 579, 2746, 54206, etc.

Counting in binary system

  • After 0 and 1, next it comes 10.
  • In fact, when counting, whenever we reach a number that is made up entirely only of 1's, an extra 1 digit is added to the left of that number (the leftmost bit) and the rest of the bits are cleared (all 0's).
  • This way, after the number 111 it comes the number 1000.
  • This is very similar to what is happenning in the decimal system: when we reach a number that is made up entirely of only 9's, an extra digit set on 1 is added to the left of the number, while the rest of the bits are cleared (0's). After 999 next it comes 1000.

Here are the first 32 (33) decimal system positive integer numbers (base 10) and their binary number representation equivalents (base 2):

  • 0(10) = 0(2); 1(10) = 1(2); 2(10) = 10(2); 3(10) = 11(2); 4(10) = 100(2);
  • 5(10) = 101(2); 6(10) = 110(2); 7(10) = 111(2); 8(10) = 1000(2);
  • 9(10) = 1001(2); 10(10) = 1010(2); 11(10) = 1011(2); 12(10) = 1100(2);
  • 13(10) = 1101(2); 14(10) = 1110(2); 15(10) = 1111(2); 16(10) = 1 0000(2);
  • 17(10) = 1 0001(2); 18(10) = 1 0010(2); 19(10) = 1 0011(2); 20(10) = 1 0100(2);
  • 21(10) = 1 0101(2); 22(10) = 1 0110(2); 23(10) = 1 0111(2); 24(10) = 1 1000(2);
  • 25(10) = 1 1001(2); 26(10) = 1 1010(2); 27(10) = 1 1011(2); 28(10) = 1 1100(2);
  • 29(10) = 1 1101(2); 30(10) = 1 1110(2); 31(10) = 1 1111(2); 32(10) = 10 0000(2)
  • As you can see, there are 32 distinct numbers that can be represented on 5 digits or less (from 0 up to 31).
  • This can also be calculated, since 32 = 25.
  • The volume of distinct numbers that can be represented on 8 digits is: 28 = 256. From 0 up to 255. 255 in binary is 1111 1111.
  • Because the binary system uses base two, as opposed to the decimal base ten, the numbers written in the binary code get more digits, but they both obey the same principles.

» Alphabet letters binary coding

» 0 and 1. How computers work. Bits and bytes.



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