0.4.6 Connecting Binary and Hexadecimal

Four Binary Bits Match One Hexadecimal Digit

Four bits can form sixteen different patterns:

through:

One hexadecimal digit also has sixteen possible values:

through:

That creates a direct correspondence.

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

This table is the key connection between binary and hexadecimal.

Convert Binary to Hex by Grouping Four Bits

Consider:

10101100

Split the binary value into groups of four bits from the right:

1010 1100

Convert each group separately.

From the table:

1010 = A
1100 = C

Therefore:

10101100₂ = AC₁₆

You do not need to convert the entire binary number to decimal first.

The four-bit groups map directly to hexadecimal digits.

Another Example

Convert:

00111110

Group the bits:

0011 1110

Convert each group:

0011 = 3
1110 = E

So:

00111110₂ = 3E₁₆

Convert Hex to Binary One Digit at a Time

The process works in reverse.

Suppose the hexadecimal value is:

7B

Convert each hex digit to four bits.

7 = 0111
B = 1011

Combine the groups:

01111011

Therefore:

7B₁₆ = 01111011₂

Leading Zeros Can Preserve the Four-Bit Group

The hexadecimal digit:

corresponds to:

not merely:

Both binary strings represent the same numeric quantity when interpreted as unsigned values, but 0011 preserves the full four-bit group.

That makes hex-to-binary conversion easier to organize consistently.

Hexadecimal Compresses the Written Representation

Compare the same pattern:

Binary:      1110101011010010
Hexadecimal: EAD2

Each hexadecimal digit replaces one group of four binary bits:

1110 1010 1101 0010
  E    A    D    2

The hexadecimal form is shorter for a person to read while preserving a direct connection to the underlying bit pattern.

Decimal Is Still Useful for Quantity

Binary and hexadecimal often make low-level representation easier to see.

Decimal remains convenient when people are thinking about ordinary numeric quantity.

For example, the same value can be represented as:

Binary:      00101010
Hexadecimal: 2A
Decimal:     42

Each form represents the same quantity.

The best representation depends on what you are trying to understand.

Number Systems Are Different Views of the Same Value

The module has now connected three positional number systems:

Decimal

Base 10.

Digits:

0–9

Place values grow by powers of 10.

Binary

Base 2.

Digits:

0–1

Place values grow by powers of 2.

Hexadecimal

Base 16.

Digits:

0–9 and A–F

Place values grow by powers of 16.

The systems use different symbols and bases, but the underlying positional idea is the same.

The Binary–Hex Relationship Is Especially Direct

The most useful connection to remember is:

4 binary bits ↔ 1 hexadecimal digit

That relationship explains why hexadecimal appears so often near binary information.

Hexadecimal gives people a shorter way to represent bit patterns without losing the clean four-bit structure underneath.