Number systems form the backbone of digital electronics and computing. A number system is a writing system for expressing numbers using consistent symbols and rules. In digital electronics, number systems are crucial because they allow information to be represented, processed, and transmitted efficiently. Digital circuits rely on discrete values rather than continuous ones, and understanding the relationship between different number systems is essential for designing, analyzing, and troubleshooting digital circuits.
Different number systems serve different purposes. The binary number system is used directly by digital circuits because electronic devices can easily recognize two states: high and low voltage, corresponding to 1 and 0. The hexadecimal system provides a compact representation of binary numbers, making it easier for engineers and programmers to read and interpret large binary values. The decimal system, familiar to humans, is often used for data interpretation, display, and interface with users.
This post explores the major number systems used in digital electronics—binary, decimal, and hexadecimal—as well as techniques for converting between them and their applications in modern digital systems.
Binary Number System
The binary number system, also known as base 2, uses only two digits: 0 and 1. It is the fundamental number system in digital electronics because digital circuits, such as transistors and logic gates, have two stable states: ON (1) and OFF (0).
Representation
Each digit in a binary number is called a bit (binary digit). Binary numbers are written from left to right, with each position representing a power of 2, starting from the rightmost bit (least significant bit, LSB) to the leftmost bit (most significant bit, MSB).
For example, the binary number 1011 can be interpreted as:
- 1×23=81 \times 2^3 = 81×23=8
- 0×22=00 \times 2^2 = 00×22=0
- 1×21=21 \times 2^1 = 21×21=2
- 1×20=11 \times 2^0 = 11×20=1
Adding these together: 8+0+2+1=118 + 0 + 2 + 1 = 118+0+2+1=11 in decimal.
Binary Arithmetic
Binary arithmetic follows simple rules similar to decimal arithmetic, but carries occur at base 2.
Addition Rules:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (carry 1 to the next higher bit)
Example of Binary Addition:
1011
+ 1101
--------
11000
Subtraction Rules:
Binary subtraction also follows rules similar to decimal subtraction, using borrowing when necessary.
- 0 – 0 = 0
- 1 – 0 = 1
- 1 – 1 = 0
- 0 – 1 = 1 (borrow 1 from the next higher bit)
Example of Binary Subtraction:
1010
- 0111
--------
0011
Conversion Between Binary and Decimal
To convert a binary number to decimal, multiply each bit by its corresponding power of 2 and sum the results.
To convert decimal to binary, repeatedly divide the number by 2, recording the remainder until the quotient is zero. Then read the remainders in reverse order to get the binary number.
Example: Convert decimal 13 to binary:
- 13 ÷ 2 = 6 remainder 1
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
Binary representation: 1101
Hexadecimal Number System
The hexadecimal system, or base 16, uses sixteen symbols: 0, 1, 2, …, 9, and A, B, C, D, E, F. Hexadecimal is widely used in digital electronics to represent binary numbers in a more compact and readable form, since each hexadecimal digit corresponds to four binary bits (a nibble).
Representation
- Binary 0000 → Hex 0
- Binary 0001 → Hex 1
- Binary 0010 → Hex 2
- …
- Binary 1111 → Hex F
Example: Convert binary 10111011 to hexadecimal:
- Divide into 4-bit groups from right: 1011 1011
- 1011 = B (hex), 1011 = B (hex)
- Result: BB (hex)
Conversions Between Binary, Decimal, and Hexadecimal
Binary to Hexadecimal: Group binary digits in sets of four starting from the LSB and convert each group to hex.
Hexadecimal to Binary: Convert each hexadecimal digit to its 4-bit binary equivalent.
Decimal to Hexadecimal: Divide the decimal number by 16 repeatedly, noting remainders. The remainders represent hexadecimal digits from LSB to MSB.
Hexadecimal to Decimal: Multiply each hex digit by 16 raised to the power of its position index and sum the results.
Decimal Number System
The decimal system or base 10 is the most familiar to humans, using ten symbols: 0 to 9. It is often used in digital electronics for display, interpretation, and user interface purposes. While digital circuits operate in binary, decimal numbers provide an intuitive representation for humans.
Representation
Each position in a decimal number represents a power of 10, starting from the rightmost digit (units place).
Example: 345 in decimal represents:
- 3×102=3003 \times 10^2 = 3003×102=300
- 4×101=404 \times 10^1 = 404×101=40
- 5×100=55 \times 10^0 = 55×100=5
Total = 300 + 40 + 5 = 345
Decimal in Digital Circuits
Decimal numbers are often converted to binary for processing by digital circuits. This conversion allows systems to perform calculations and logic operations efficiently, while the output may be converted back to decimal for display.
Conversion Techniques
Conversions between binary, decimal, and hexadecimal numbers are critical in digital electronics. Accurate conversions ensure proper data representation and processing.
Binary to Decimal
Multiply each binary bit by 2 raised to the power of its position index and sum:
Example: Binary 1101 → Decimal
- 1×23=81 \times 2^3 = 81×23=8
- 1×22=41 \times 2^2 = 41×22=4
- 0×21=00 \times 2^1 = 00×21=0
- 1×20=11 \times 2^0 = 11×20=1
Decimal = 8 + 4 + 0 + 1 = 13
Decimal to Binary
Divide the decimal number by 2 repeatedly, noting the remainder:
Example: Decimal 19 → Binary
- 19 ÷ 2 = 9 remainder 1
- 9 ÷ 2 = 4 remainder 1
- 4 ÷ 2 = 2 remainder 0
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Binary = 10011
Binary to Hexadecimal
Group binary digits in sets of 4 starting from LSB:
Example: Binary 111001 → 0011 1001 → Hex 39
Hexadecimal to Binary
Convert each hex digit to a 4-bit binary equivalent:
Example: Hex 7A → 0111 1010
Decimal to Hexadecimal
Divide the decimal number by 16 repeatedly:
Example: Decimal 254 → Hex
- 254 ÷ 16 = 15 remainder 14 → F E → Hex FE
Hexadecimal to Decimal
Multiply each hex digit by 16 raised to its position index:
Example: Hex 2F → Decimal
- 2 × 16^1 = 32
- F (15) × 16^0 = 15
- Decimal = 32 + 15 = 47
Applications of Number Systems
Number systems are essential for representing and processing information in digital electronics. Some key applications include:
1. Microprocessor Design
Microprocessors operate internally using binary numbers. Instruction sets, registers, memory addresses, and data are all represented in binary. Hexadecimal representation is often used for readability in programming and debugging.
2. Data Representation
Digital systems store numbers, characters, images, and audio using binary codes. Binary and hexadecimal systems allow efficient encoding and compression of data for storage and transmission.
3. Error Detection and Correction
Number systems facilitate error detection and correction in digital communication. Parity bits, checksums, and cyclic redundancy checks (CRC) rely on binary arithmetic to identify and correct errors in data transmission.
4. Programming and Assembly Language
Hexadecimal numbers are used in programming, particularly in assembly language and low-level hardware programming, because they provide a convenient way to represent binary instructions and memory addresses.
Leave a Reply