Hexadecimal
The hexadecimal numeral system, often shortened to "hex", is a numeral system made up of 16 symbols (base 16). The standard numeral system is called decimal (base 10) and uses ten symbols: 0,1,2,3,4,5,6,7,8,9. Hexadecimal uses the decimal numbers and six extra symbols. There are no numerical symbols that represent values greater than nine, so letters taken from the English alphabet are used, specifically A, B, C, D, E and F. Hexadecimal A = decimal 10, and hexadecimal F = decimal 15.
Humans mostly use the decimal system. This is probably because humans have ten fingers on their hands. Computers however, only have on and off, called a binary digit (or bit, for short). A binary number is just a string of zeros and ones: 11011011, for example. For convenience, engineers working with computers tend to group bits together. In earlier days, such as the 1960's, they would group 3 bits at a time (much like large decimal numbers are grouped in threes, like the number 123,456,789). Three bits, each being on or off, can represent the eight numbers from 0 to 7: 000 = 0; 001 = 1; 010 = 2; 011 = 3; 100 = 4; 101 = 5; 110 = 6 and 111 = 7. This is called octal.
As computers got bigger, it was more convenient to group bits by four instead of three. This doubles the numbers that the symbol would represent; it can have 16 values instead of eight. Hex = 6 and Decimal = 10, so it is called hexadecimal. In computer jargon four bits make a nibble (sometimes spelled nybble). A nibble is one hexadecimal digit, written using a symbol 09 or AF. Two nibbles make a byte (8 bits). Most computer operations use the byte, or a multiple of the byte (16 bits, 24, 32, 64, etc.). Hexadecimal makes it easier to write these large binary numbers.
To avoid confusion with decimal, octal or other numbering systems, hexadecimal numbers are sometimes written with a "h" after or "0x" before the number. For example, 63h and 0x63 mean 63 hexadecimal.
Hexadecimal values[change  change source]
Hexadecimal is similar to the octal numeral system (base 8) because each can be easily compared to the binary numeral system. Hexadecimal uses a fourbit binary coding. This means that each digit in hexadecimal is the same as four digits in binary. Octal uses a threebit binary system.
In the decimal system, the first digit is the one's place, the next digit to the left is the ten's place, the next is the hundred's place, etc. In hexadecimal, each digit can be 16 values, not 10. This means the digits have the one's place, the sixteen's place, and the next one is the 256's place. So 1h = 1 decimal, 10h = 16 decimal, and 100h = 256 in decimal.
Example values of hexadecimal numbers converted into binary, octal and decimal.

Conversion[change  change source]
Binary to hexadecimal[change  change source]
Changing a number from binary to hex uses a grouping method. The binary number is separated into groups of four digits starting from the right. These groups are then converted to hexadecimal digits as shown in the chart above for the hexadecimal numbers 0 through F. To change from hexadecimal, the reverse is done. The hex digits are each changed to binary and the grouping is usually removed.
Binary  Groupings  Hex  

01100101  0110  0101  65  
010010110110  0100  1011  0110  4B6  
1101011101011010  1101  0111  0101  1010  D75A 
When the quantity of bits in a binary numbers is not a multiple of 4, it is padded with zeros to make it so. Examples:
 binary 110 = 0110, which is 6 Hex.
 binary 010010 = 00010010, which is 12 Hex.
Hexadecimal to decimal[change  change source]
To convert a number from hexadecimal to decimal, there are two common ways.
The first method is more commonly done when converting it manually:
 Use the decimal value for each hexadecimal digit. For 09, it is the same, but A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
 Keep a sum of the numbers converted at each step below.
 Start with the least significant hexadecimal digit. That is the digit on the right end. This will be the first item in a sum.
 Take the secondleast significant digit. That is next to the digit on the right end. Multiply the decimal value of the digit by 16. Add this to the sum.
 Do the same for the thirdleast significant digit, but multiply it by 16^{2} (that is, 16 squared, or 256). Add it to the sum.
 Continue for each digit, multiplying each place by another power of 16. (4096, 65536, etc.)
Location  

6  5  4  3  2  1  
Value  1048576 (16^{5})  65536 (16^{4})  4096 (16^{3})  256 (16^{2})  16(16^{1})  1 (16^{0}) 
The next method is more commonly done when converting a number in software. It does not need to know how many digits the number has before it starts, and it never multiplies by more than 16, but it looks longer on paper.
 Use the decimal value for each hexadecimal digit. For 09, it is the same, but A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
 Keep a sum of the numbers converted at each step below.
 Start with the most significant digit (the digit on the far left). This is the first item in the sum.
 If another digit exists, multiply the sum by 16 and add the decimal value of the next digit.
 Repeat the above step until there are no more digits.
Example: 5Fh and 3425h to decimal, method 1


Example: 5Fh and 3425h to decimal, method 2

