Numbering systems

A numbering system is a systematic method for representing numbers using a particular set of symbols. The most commonly used numbering system is the decimal system, based on the number 10, which is called the basis or radix of the system. The basis tells how many different individual symbols there are in the system to represent numbers. In the decimal system these symbols are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The range of these numbers varies from 0 to (10 − 1). This is a particular case of a more general rule: Given any positive basis or radix N, there are N different individual symbols that can be used to write numbers in that system. The range of these numbers varies from 0 to N − 1.

In the computer and telecommunication fields, three of the most frequently used numbering systems are the binary (base 2), the octal (base 8), and the hexadecimal (base 16). The binary system has only two symbols: 0 and 1. Either of these symbols can be called a binary digit or a bit. The octal system has eight symbols: 0, 1, 2, 3, 4, 5, 6, 7. The hexadecimal system has 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A stands for 10, B for 11, C for 12, D for 13, E for 14, and F for 15. The reason for choosing single letters to represent numbers higher than 9is to keep all individual symbols single characters. See also Bit.

All the numbering systems mentioned so far are positional systems. That is, the value of anysymbol depends on its position in the number. For example, the value of 2 in the decimal number 132 is that of two units, whereas its value in decimal 245 is that of two hundreds. In the decimal system, the rightmost position of a number is called the ones (10⁰) place, the next position from the right is called the tens (10¹) place, the next position the hundreds (10²) place, and so on. Observe that the powers increase from the right. The power of the rightmost digit is zero, the power of the next digit is one, the power of the next digit is two, and so on. These powers are sometimes called the weight of the digit.

Conversion to decimal numbers

In any positional system, the decimal equivalent of a digit in the representation of the number is the digit's own value, in decimal, multiplied by a power of the basis in which the number is represented. The sum of all these powers is the decimal equivalent of the number. The corresponding powers of each of the digits can be better visualized by writing superscripts beginning with 0 at the rightmost digit, and increasing the powers by 1 in moving toward the left digits of the number.

Decimal numbers to other bases

The conversion of a given decimal number to another basis r (r > 0) is carried out by initially dividing the given decimal number by r, and then successively dividing the resulting quotients by r until a zero quotient is obtained. The decimal equivalent is obtained by writing the remainders of the successive divisions in the opposite order in which they were obtained.

Binary to hexadecimal or octal and vice versa

The table shows the decimal numbers 1 through 15 written in binary, octal, and hexadecimal. Since each four-bit binary number corresponds to one and only one hexadecimal digit and vice versa, the hexadecimal system can be viewed as a shorthand notation of the binary system. Similar reasoning can be applied to the octal system. This one-to-one correspondence between the symbols of the binary system and the symbols of the octal and hexadecimal system provides a method for converting numbers between these bases.

To convert binary numbers to hexadecimal, the following procedure may be used: (1) Form four-bit groups beginning from the rightmost bit of the number. If the last group (at the leftmost position) has fewer than four bits, add extra zeros to the left of the bits in this group to make it a four-bit group. (2) Replace each four-bit group by its hexadecimal equivalent. A process that is almost the reverse of the previous procedure can be used to convert from hexadecimal to binary. However, there is no need to add extra zeros to any group since each hexadecimal number will always convert to a group with four binary bits.

A similar process can be followed to convert a binary number to octal, except that in this case three-bit groups must be formed. Individual octal numbers will always convert to groups with three binary bits.