Converting between different bases and representing signed integers are fundamental concepts in computer science and digital arithmetic. Here’s an overview of both:
Converting Between Bases:
Decimal to Binary:
- Divide and Remainder: Divide the decimal number by the base (2 for binary). Keep track of the remainders.
- Binary Digits: Write down the remainders from step 1 in reverse order to obtain the binary representation.
Binary to Decimal:
- Place Values: Write down the binary digits, assigning place values (1, 2, 4, 8, 16, etc.) from right to left.
- Multiply and Sum: Multiply each binary digit by its corresponding place value and sum the results to obtain the decimal representation.
Decimal to Hexadecimal/Octal:
- Divide and Remainder: Similar to converting to binary, but divide by 16 for hexadecimal or 8 for octal.
- Hexadecimal/Octal Digits: Write down the remainders from step 1, using hexadecimal digits (0-9, A-F) for hexadecimal and octal digits (0-7) for octal.
Hexadecimal/Octal to Decimal:
- Place Values: Write down the hexadecimal or octal digits, assigning place values (1, 16, 256, etc. for hexadecimal and 1, 8, 64, etc. for octal) from right to left.
- Multiply and Sum: Multiply each digit by its corresponding place value and sum the results to obtain the decimal representation.
Signed Integer Representation:
Signed Magnitude:
- Sign Bit: Use one bit to represent the sign (0 for positive, 1 for negative). The remaining bits represent the magnitude of the number.
- Example: In a 4-bit signed magnitude representation, 1101 represents -5 (1 for negative, 101 for magnitude).
One’s Complement:
- Negation: To negate a number, flip all its bits (change 0s to 1s and vice versa).
- Example: In a 4-bit one’s complement representation, 0110 represents 6, and its negation is 1001, representing -6.
Two’s Complement:
- Negation: To negate a number, invert all its bits and add 1.
- Example: In a 4-bit two’s complement representation, 0110 represents 6, and its negation is obtained by inverting bits (1001) and adding 1 (1010), representing -6.
Sign Extension:
- Extending Width: When converting a shorter signed number to a longer one, sign extension involves copying the sign bit of the shorter number to fill the additional bits.
- Example: Extending a 4-bit signed number 1101 to an 8-bit representation involves copying the sign bit (1) to fill the additional bits (11111101).
Importance:
- Flexibility: Understanding base conversion allows for representing numbers in different contexts efficiently.
- Data Representation: Signed integer representations are crucial for representing positive and negative numbers in computer systems accurately.
- Arithmetic Operations: Different representations affect arithmetic operations such as addition, subtraction, multiplication, and division, requiring appropriate handling for correct results.