Hexadecimal Numbers

• Hexadecimal (hex) is another number system used regularly in programming
• Hex uses a combination of digits and letters in order to represent a number
• Hexadecimal numbers are shorter than binary
• Hexadecimal (or base-16) uses sixteen different digits
• A single hex character can represent any denary number from 0 - 15
• To represent 0-15 in binary would require 4 bits (a nibble), so each hex character equals to a nibble in binary
• Programmers often prefer hex when coding because:

  • It's simpler to remember large numbers in hex - they're far shorter than binary numbers
  • Due to hex numbers being shorter, there's less chance of input errors
  • It's easier to convert between binary and hex than binary and denary
• Computers themselves don't use hex - they still have to convert everything to binary to process it
• Denary - hex - binary

  0 - 0 - 0000
• Denary - hex - binary

  1 - 1 - 0001
• Denary - hex - binary

  2 - 2 - 0010
• Denary - hex - binary

  3 - 3 - 0011
• Denary - hex - binary

  4 - 4 - 0100
• Denary - hex - binary

  5 - 5 - 0101
• Denary - hex - binary

  6 - 6 - 0110
• Denary - hex - binary

  7 - 7 - 0111
• Denary - hex - binary

  8 - 8 - 1000
• Denary - hex - binary

  9 - 9 - 1001
• Denary - hex - binary

  10 - A - 1010
• Denary - hex - binary

  11 - B - 1011
• Denary - hex - binary

  12 - C - 1100
• Denary - hex - binary

  13 - D - 1101
• Denary - hex - binary

  14 - E - 1110
• Denary - hex - binary

  15 - F - 1111
• Convert hex to denary by multiplying each character
• In hex, moving right to left, place values increase in powers of 16: 4096 - 256 - 16 - 1
• Example

  Convert the hexadecimal 87 into denary

  • First draw a table with 2 columns and 2 rows (16 and 1 across the top / hex on the bottom)

  16 1

  8 7

  • Then write the hex number and multiply the numbers in each column

  > 8 x 16 = 128 7 x 1 = 7

  • Add up the results

  > 128 + 7 = 135

  = So the hex number 87 is 135 in denary
• Example

  Convert the denary number 106 into hexadecimal

  • Draw this table

  16 1

  6 A

  • Start at the left. Divide 106 by 16, then hold onto the remainder

  > 106 / 16 = 6 r 10

  • Divide the remainder from the last calculation by 1

  > 10 / 1 = 10 = A

  = So the denary number 106 is 6A in hex
• Convert binary in hex by splitting it into nibbles
• Each hex character is equal to a nibble in binary, so it is possible to convert from binary to hex by splitting the binary code in 4-bit chunks
• Binary to hex conversions can be much easier than converting from binary to denary, as you only have to deal with the nibbles one at a time
• Example

  Convert the binary number 10111001 to hex

  • Firstly split the binary number into nibbles: 1011 1001
  • Draw a table with columns labelled 8, 4, 2, 1, then repreat the values for as many nibbles as you require

  8 4 2 1 / 8 4 2 1

  • Fill in the table with your binary number

  8 4 2 1 / 8 4 2 1

  1 0 1 1 / 1 0 0 1

  • For each nibble, add up the numbers with 1 in the column, then convert the value to hex and put the values together

  > The binary number 10111001 is B9 in hex
• When converting binary to hex, if the number can't be split into nibbles, you'll have to put some zeros on the front
• Example

  Convert the binary number 11,110 to hex

  • Add 0s to the front of the binary number, so that you can split it into nibbles

  0 0 1 1 / 1 1 1 0

  • Draw a repeating table of 8, 4, 2, 1

  8 4 2 1 / 8 4 2 1

  • Write your binary number in the table

  8 4 2 1 / 8 4 2 1

  0 0 1 1 / 1 1 1 0

  • Add up each nibble and convert each value to hex

  > 2 + 1 = 3 8 + 4 + 2 = 14 = E

  > The binary number 11,110 in 3E in hex
• Example

  Convert the hexadecimal number 8C to binary

  • Find the denary value of each character

  8 4 2 1 / 8 4 2 1

  1 0 0 0 / 1 1 0 0

  • Find the binary value of each number
  • Put the nibbles together to get the equivalent binary number

  > The hexidecimal number 8C is 10001100 in binary