1.5 Numbering Systems

Hexadecimal Value | Binary Value | Decimal Value

0 | 0000 | 0 1 | 0001 | 1 2 | 0010 | 2 3 | 0011 | 3 4 | 0100 | 4 5 | 0101 | 5 6 | 0110 | 6 7 | 0111 | 7 8 | 1000 | 8 9 | 1001 | 9 A | 1010 | 10 B | 1011 | 11 C | 1100 | 12 D | 1101 | 13 E | 1110 | 14 F | 1111 | 15

As you study this section, answer the following questions:

> What is the difference between a binary numbering system and a hexadecimal numbering system? > What are the possible values in a binary number? > In a hexadecimal number, how many possible characters can be used for each number space? > In a 3-bit binary number, how many possible combinations are there?

Octal Number

A base-8 number system that uses the digits 0-7.

Binary Number

A number system that only has two values, typically 0 (zero) and 1 (one).

Hexadecimal Number

A numbering system with 16 symbols, 0-9 and A-F.

Binary Numbers and Decimal Equivalents

Because computers rely heavily on binary numbers, you must convert decimal numbers to binary (and vice versa) to be an effective network administrator. The following table lists several binary values and their decimal equivalents: (Binary Value -Decimal Value) 10000000 - 128 01000000 - 64 00100000 - 32 00010000 - 16 00001000 - 8 00000100 - 4 00000010 - 2 00000001 - 1 To find the decimal value of a binary number, simply add the decimal values of the 1 bits in the number. For example, the decimal value of the binary number 10010101 is: 10000000 = 128 00010000 = 16 00000100 = 4 00000001 = 1 Total: 128 + 16 + 4 + 1 = 149 Because there are only two possible values in a binary number (0 and 1), you can express binary numbers in terms of powers of two: (# of bits - Exponent - Exponent value) 1 - 2^1 - 2 2 - 2^2 - 4 3 - 2^3 - 8 4 - 2^4 - 16 5 - 2^5 - 32 6 - 2^6 - 64 7 - 2^7 - 128 8 - 2^8 - 256 9 - 2^9 - 512 10 - 2^10 - 1024 11 - 2^11 - 2048 12 - 2^12 - 4096

Binary Numbers

Computers natively use the binary numbering system to represent and process data. In binary, there are only two possible numbers for every number place: 0 or 1. For example, a three-bit binary number contains a combination of 0s and 1s. Because there are only two possible numbers for each slot, there are eight possible combinations for a three-bit binary number: > 000 > 001 > 010 > 011 > 100 > 101 > 110 > 111

Hexadecimal Numbering System

The hexadecimal numbering system is also frequently used with computers and networking. Hexadecimal is a Base 16 numbering system, which means there are 16 different characters possible for each number place. These characters go from 0 to 9, as decimal does; however, hexadecimal uses the letter A to represent the decimal number 10. B represents 11 and so on up to F, which represents 15. The easiest way to convert between decimal, binary, and hexadecimal is to memorize the corresponding values for each hexadecimal number using the following table.

Octal Numbering System

The octal numeral system is a base eight number system and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for 74 is 1001010. You can add two zeros to the left: (00)1 001 101. This makes the corresponding the octal digits 1 1 2, creating the octal representation of 112.

1.5.2 Numbering System Facts

When managing networks, you will frequently work with number systems other than the decimal numbering system you are already familiar with. These additional numbering systems include: > Binary (Base 2) > Octal (Base 8) > Hexadecimal (Base 16)