Digital Number Systems: Binary, Hexadecimal, and BCD

Posted on Jan 20, 2026 in Systems Computer Engineering

Understanding Number Systems

Number systems are fundamental to computing and digital electronics. They are essentially methods for representing numbers using a specific set of symbols (digits). The key concept differentiating them is the base or radix, which is the total number of unique digits available in that system.

Common Number Systems Breakdown

1. Decimal (Base-10)

This is the number system we use every day.

Base (Radix): 10
Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Concept: Each digit’s position represents a power of 10.
Example: 123₁₀ = (1 × 10²) + (2 × 10¹) + (3 × 10⁰)

2. Binary (Base-2)

This is the native language of computers because digital circuits use two distinct states: ON/OFF or High/Low voltage, which can be represented by two digits.

Base (Radix): 2
Digits Used: 0, 1
Concept: Each position represents a power of 2. A single binary digit is called a bit.
Example: 1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀

3. Octal (Base-8)

Octal is often used as a compact way to represent binary numbers, as three binary digits can be perfectly represented by one octal digit (2³ = 8).

Base (Radix): 8
Digits Used: 0, 1, 2, 3, 4, 5, 6, 7
Concept: Each position represents a power of 8.
Example: 23₈ = (2 × 8¹) + (3 × 8⁰) = 16 + 3 = 19₁₀

4. Hexadecimal (Base-16)

Hexadecimal is the most common compact representation for binary in computing (e.g., in memory addresses, color codes, MAC addresses, etc.). It is especially useful because four binary digits (a nibble) fit exactly into one hexadecimal digit (2⁴ = 16).

Base (Radix): 16
Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Note: Where A=10, B=11, C=12, D=13, E=14, F=15 in decimal.
Concept: Each position represents a power of 16.
Example: 2A₁₆ = (2 × 16¹) + (A × 16⁰) = (2 × 16) + (10 × 1) = 32 + 10 = 42₁₀

Key Takeaway: Conversions

The most important aspect of these systems is the ability to convert between them.

Conversion Type	Method
Any Base to Decimal	Multiply each digit by its positional weight (the base raised to the power of the position) and sum the results.
Decimal to Any Base	Use the repeated division method: Divide the decimal number by the new base, keep track of the remainders, and read the remainders from bottom to top.
Binary to Octal	Group binary digits in sets of three starting from the right. Convert each group to its octal equivalent.
Binary to Hexadecimal	Group binary digits in sets of four starting from the right. Convert each group to its hexadecimal equivalent.

Number System Conversions and BCD

Understanding number system conversions is crucial for working in digital electronics and computer science. The main conversion methods depend on whether you are converting to or from Decimal (Base-10), or converting between binary-related bases (Binary, Octal, Hexadecimal).

1. Conversion Methods

A. To/From Decimal (Base-10)

Conversion Type	Method	Example
Any Base (B) → Decimal (10)	Positional Weight: Multiply each digit by its positional weight (Bⁿ) and sum the results.	1011₂ → 10: (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀
Decimal (10) → Any Base (B)	Repeated Division: Divide the decimal number by the new base (B). Collect the remainders; the result is the remainders read from bottom (MSB) to top (LSB).	13₁₀ → 2: 13 ÷ 2 = 6 remainder 1; 6 ÷ 2 = 3 remainder 0; 3 ÷ 2 = 1 remainder 1; 1 ÷ 2 = 0 remainder 1 → 1101₂

B. Conversions Between Binary-Related Bases

Since Octal is 2³ and Hexadecimal is 2⁴, conversion to and from Binary is direct by grouping bits.

Conversion Type	Grouping Method	Example
Binary → Octal	Group the binary digits into sets of 3, starting from the right (pad with leading zeros if necessary). Convert each group to its octal digit.	1101011₂ → 8: (001)(101)(011) → 153₈
Binary → Hexadecimal	Group the binary digits into sets of 4, starting from the right. Convert each group to its Hex digit.	1101011₂ → 16: (0110)(1011) → 6B₁₆
Octal → Binary	Convert each octal digit into its 3-bit binary equivalent.	72₈ → 2: (7)(2) → (111)(010) → 111010₂
Hexadecimal → Binary	Convert each Hex digit into its 4-bit binary equivalent.	A5₁₆ → 2: (A)(5) → (1010)(0101) → 10100101₂

Tip: To convert Octal ↔ Hexadecimal, always go through Binary as an intermediate step.

2. Binary Coded Decimal (BCD)

The Binary Coded Decimal (BCD) system is a special type of binary encoding used to represent decimal numbers, often found in digital displays like digital clocks, calculators, and meters.

What is BCD?

Principle: Instead of converting the entire decimal number into its pure binary equivalent, BCD converts each individual decimal digit into its 4-bit binary representation.
Encoding: The most common form is 8421 BCD (Natural BCD), where the four bits are weighted by 2³=8, 2²=4, 2¹=2, and 2⁰=1.

Decimal Digit	4-bit BCD Code (8421)
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001

BCD vs. Pure Binary

Feature	BCD (Binary Coded Decimal)	Pure Binary
Encoding	Each decimal digit (0-9) is converted separately.	The entire decimal number is converted.
Efficiency	Less efficient (wastes 6 of the 16 possible 4-bit combinations: 1010 to 1111 are invalid BCD codes).	More efficient (uses fewer bits to represent larger numbers).
Conversion	Easy to convert to and from decimal (human-readable).	More complex calculation for human-readable decimal conversion.
Use	Applications requiring easy decimal display (clocks, pocket calculators, financial systems).	General computing, memory, and high-speed processing.

BCD Conversion Examples

1. Decimal → BCD

Take the decimal number 485₁₀. Convert each digit individually:

Decimal Digit	4	8	5
BCD Code	0100	1000	0101

2. BCD → Decimal

Take the BCD number 1001 0011_BCD. Group into 4-bit blocks and convert each block:

4-bit Block	1001	0011
Decimal Equivalent	9	3

BCD is a method of representing each decimal digit (0 through 9) using its own four-bit binary code. Here is a breakdown of the types mentioned:

Types of BCD and Digital Codes

1. Natural Binary Code (NBCD / 8421 Code)

Definition: This is the most common form of BCD. Each decimal digit is represented by its natural, 4-bit, unweighted binary equivalent.
Weighted Code Status: It is a Weighted Code.
Weights: The bit positions have the weights 8, 4, 2, and 1, from most significant bit (MSB) to least significant bit (LSB).
Example: Decimal 5 is 0101 (which is 0 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 5).

2. Weighted Codes

Definition: In a weighted code, each bit position in the four-bit group is assigned a specific numerical weight. The decimal value is the sum of the products of each bit and its weight.
Examples:
- 8421 (Natural BCD)
- 2421
- 7421
- 84-2-1 (Codes like 2421 and 84-2-1 are often designed to be self-complementing).

3. Self-Complementing Codes

Definition: A code is self-complementing if the 1’s complement of the binary code for a decimal digit D is equal to the code for the digit’s 9’s complement (9 – D). This property simplifies subtraction operations.
Condition: A necessary condition for a weighted code to be self-complementing is that the sum of the weights must equal 9.
Examples: 2421, 84-2-1 (weighted), and Excess-3 (non-weighted).
Excess-3: This is a non-weighted code derived by adding 3 (0011₂) to the Natural BCD (8421) code of the digit.

4. Cyclic Codes (Unit Distance Codes)

Definition: A code is called a cyclic code (or unit distance code) if successive code words for consecutive decimal digits differ in only one bit position.
Application: These codes are useful in applications where a change in an input value must not cause a momentary flash of incorrect codes, such as in rotary encoders.
Example: The most common example is the Gray Code, which is a non-weighted, cyclic code. While BCD itself is a category for decimal encoding, Gray code is the most relevant example of a cyclic code in the context of digital codes. Note: The BCD codes (like 8421) themselves are not cyclic.