What is the connection between binary and hexadecimal numbers in the programming world?

Answer 1

The connection between binary and hexadecimal in the programming world is exactly the same as the connection in the mathematical world. All numeric bases that are themselves a power of 2 (base-4, base-8, base-16, base-32, etc) are trivial to convert both to and from binary.

A single base-4 digit maps to exactly 2 binary digits:

0 = 00

1 = 01

2 = 10

3 = 11

A single base-8 (octal) digit maps to exactly 3 binary digits:

0 = 000

1 = 001

2 = 010

3 = 011

4 = 100

5 = 101

6 = 110

7 = 111

A single base-16 (hexadecimal) digit maps to exactly 4 binary digits (also known as a nybble):

0 = 0000 1 = 0001 2 = 0010 3 = 0011

4 = 0100

5 = 0101

6 = 0110

7 = 0111

8 = 1000

9 = 1001

a = 1010

b = 1011

c = 1100

d = 1101

e = 1110

f = 1111

And so on for base-32, base-64, etc.

Hexadecimal is the most useful notation because a byte is normally 8 binary digits in length and we can represent a byte as two nybbles using just two hexadecimal digits as opposed to 8 binary digits.

For longer binary values, such as 32-bit, 64-bit or 128-bit values, using a more concise notation makes the value much easier to read. Consider the following:

1011010010110100101101001011010010110100101101001011010010110100

1011010010110100101111001011010010110100101101001011010010110100

These two binary values look the same but they are not. With hexadecimal notation it becomes easier to see the difference because there are fewer digits to compare:

b4b4b4b4b4b4b4b4

b4b4bcb4b4b4b4b4

Answer 2

Binary is base-2 and hexadecimal is base-16, which is convenient because they are both powers of 2. This means that we can easily represent binary numbers using hex. For example, say we have 1 byte (8 bits) of binary data:

011010112

A single hex digit is from 0-15 (numbers above 9 are represented using A - F), which can represent 4 binary digits (2^4), so we can represent the above 8 digit binary number with 2 hex digits:

01102 = 610 = 616

10112 = 1110 = B16

Therefore 011010112 = 6B16 (also written as 0x6B)

This becomes very useful when attempting to make sense of large amounts of binary data.

Answer 3

All number bases that are themselves a power of 2 (base-4, base-8, base-16, base-32, etc) are directly related to binary. A single hexadecimal digit maps to exactly 4 bits. This is because 4 bits have 16 possible permutations, representing all the values in the range 0 to 15 decimal, which is represented by the digits 0 to F in hexadecimal. Since 4 bits is half-a-byte, we call them nybbles and two hex nybbles make a full byte.

This is really no different to the way a single octal digit (base-8) can represent any group of 3 bits, hence you will encounter both octal and hexadecimal notations in programming -- because they're easier to comprehend than decimal and easier to write than binary .

Answer 4

Sets of four binary digits are equivalent to one hexadecimal digit.