How do you convert the hexadecimal number ff to binary?

Answer 1

Any base that is itself a power of 2 is easily converted to and from binary. With base 4, each digit represents 2 bits. With base 8 (octal), each digit represents 3 bits. And with base 16 (hexadecimal), each digit represents 4 bits. Thus two hexadecimal digits represent an 8-bit binary value. This is convenient because we typically refer to a unit of computer memory as an 8-bit byte, thus every byte value can be represented using just 2 hex digits. If we had a system with a 9-bit byte we'd use 3 octal digits instead. A 24-bit value can either be represented using 6 hex digits or 8 octal digits.

To convert a hexadecimal value to binary, we simply consult the following table (note that 0x is the conventional prefix for a hexadecimal value):

hex = binary

0x0 = 0000

0x1 = 0001

0x2 = 0010

0x3 = 0011

0x4 = 0100

0x5 = 0101

0x6 = 0110

0x7 = 0111

0x8 = 1000

0x9 = 1001

0xA = 1010

0xB = 1011

0xC = 1100

0xD = 1101

0xE = 1110

0xF = 1111

Here, hexadecimal digit 0xF has the binary value 1111, thus 0xFF would be 11111111. Note that the bit patterns are in the same order as the hexadecimal digits. Thus 0x0F becomes 00001111 and 0xF0 becomes 11110000.

Knowing this, we can easily convert binary values into hexadecimal, we simply divide the binary value into groups of 4 bits and convert each group to the corresponding hex digit. Thus 101101001100 becomes B4C (1011=B, 0100=4 and 1100=C). If there aren't enough bits, we simply pad the first group with leading zeroes.

We can use a similar technique to convert between octal and binary, we simply divide the bits into groups of 3:

octal = binary

00 = 000

01 = 001

02 = 010

03 = 011

04 = 100

05 = 101

06 = 110

07 = 111

Note that a leading 0 is the conventional prefix for octal values. Thus binary value 100010 would be written 042 in octal to avoid confusion with 42 decimal.