Conversions between Number Systems in Computing
In computing, conversions between binary, decimal, octal, and hexadecimal systems are essential. Here’s a breakdown of conversion methods between these number systems:
Binary to Decimal Conversion
To convert binary to decimal, multiply each binary digit by 222 raised to the power of its position from right to left (starting from 0) and sum the results.
Example:
- Binary: 1011
- Decimal: 1×23+0×22+1×21+1×20=8+0+2+1=111 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 111×23+0×22+1×21+1×20=8+0+2+1=11
Decimal to Binary Conversion
To convert decimal to binary, divide the decimal number by 2 and record the remainder. Continue dividing the quotient by 2 until you reach 0, then read the remainders in reverse order.
Example:
- Decimal: 13
- Binary: 13÷2=613 \div 2 = 613÷2=6 remainder 1; 6÷2=36 \div 2 = 36÷2=3 remainder 0; 3÷2=13 \div 2 = 13÷2=1 remainder 1; 1÷2=01 \div 2 = 01÷2=0 remainder 1
- Binary result: 1101
Binary to Hexadecimal Conversion
Group binary digits into sets of four (from the right) and convert each set to its hexadecimal equivalent.
Example:
- Binary: 101110
- Grouped as: 0010 1110
- Hexadecimal: 2E
Hexadecimal to Binary Conversion
Convert each hexadecimal digit to its 4-bit binary equivalent.
Example:
- Hexadecimal: 4F
- Binary: 0100 1111
Binary Arithmetic: Addition, Subtraction, and Multiplication
1. Binary Addition
Binary addition follows simple rules:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (binary 2), which results in 0 and carries over 1 to the next higher bit.
Example:
markdownCopy code 1011
+ 1101
--------
11000
Explanation:
- Rightmost bits: 1+1=101 + 1 = 101+1=10 (0 and carry 1)
- Next column: 1+1+1=111 + 1 + 1 = 111+1+1=11 (1 and carry 1)
- And so on…
2. Binary Subtraction
Binary subtraction uses “borrow” concepts, similar to decimal subtraction:
- 0 – 0 = 0
- 1 – 0 = 1
- 1 – 1 = 0
- 0 – 1 = 1 (borrow 1 from the next higher bit)
Example:
markdownCopy code 1010
- 0011
--------
0111
Explanation:
- Rightmost bits: 0−10 – 10−1 requires a borrow, making it 10 – 1 = 1
- Next column proceeds similarly.
3. Binary Multiplication
Binary multiplication is straightforward and similar to decimal multiplication:
- 0 × 0 = 0
- 1 × 0 = 0
- 0 × 1 = 0
- 1 × 1 = 1
Steps:
- Multiply each digit of the first binary number by each digit of the second binary number.
- Add the results, aligning them by their position.
Example:
diffCopy code 101
× 110
---------
0000 (101 × 0)
1010 (101 × 1, shift one position)
+ 10100 (101 × 1, shift two positions)
---------
11110
Explanation:
- Step-by-step, you multiply each digit in 110 by 101 and align according to position.
Summary
- Conversions between binary, decimal, octal, and hexadecimal are critical for understanding data representation in computing.
- Basic arithmetic in binary includes addition, subtraction, and multiplication, which follow specific rules related to bitwise operations.