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Negative Binary Numbers Simplified Revision Notes for A-Level OCR Computer Science
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Negative Binary Numbers
Overview
While binary numbers are often associated with positive values, representing negative numbers in binary is essential for calculations in computing. To handle negative numbers, we use techniques like Sign and Magnitude and Two's Complement. These methods allow computers to differentiate between positive and negative values and perform arithmetic accurately.
Sign and Magnitude Representation
Overview:
In the Sign and Magnitude method, the leftmost bit (most significant bit) is used as the "sign bit."
0in the sign bit indicates a positive number.1in the sign bit indicates a negative number.
Structure:
For an 8-bit binary number, the structure would look like this:
[Sign Bit] [Magnitude Bits]
For example, 10001010 in sign and magnitude notation represents 10.
Conversion from Denary to Sign and Magnitude:
- Identify the sign: If the number is negative, set the sign bit to
1; if positive, set it to0. - Convert the magnitude (absolute value) of the number to binary.
- Combine the sign bit with the magnitude bits.
Limitations:
Sign and Magnitude is limited in use because it complicates binary arithmetic (e.g., addition and subtraction).
Example:
Convert 13 to 8-bit binary using Sign and Magnitude.
- Absolute value of
13in binary:00001101 - Add the sign bit:
10001101(sign bit1indicates negative). - So,
13in Sign and Magnitude is10001101.
Two's Complement Representation
Overview:
Two's Complement is the most widely used method for representing negative numbers in binary. It simplifies binary arithmetic operations and eliminates the need for a separate "sign bit."
How It Works:
- Positive numbers are represented in standard binary form.
- Negative numbers are represented by inverting the bits of the absolute value and adding 1.
Conversion from Denary to Two's Complement:
- Convert the absolute value of the number to binary.
- Invert all the bits (change 1s to 0s and 0s to 1s).
- Add
1to the inverted binary number.
Example:
Convert 13 to 8-bit binary using Two's Complement.
- Absolute value of
13in binary:00001101 - Invert all bits:
11110010 - Add
1to the inverted bits:11110010 + 1 = 11110011 - So,
13in Two's Complement is11110011.
Conversion from Two's Complement to Denary:
- If the leftmost bit is
0, the number is positive, and you can read it directly in binary. - If the leftmost bit is
1, the number is negative. Invert the bits, add1, and interpret the result as a negative denary number.
Comparison of Sign and Magnitude vs. Two's Complement
| Feature | Sign and Magnitude | Two's Complement |
|---|---|---|
| Sign Bit | Uses leftmost bit for the sign | No explicit sign bit |
| Arithmetic Operations | Complex | Simplified |
| Range | Limits usable range of bits | Extends usable range |
| Representation | Two values for zero (positive and negative zero) | One value for zero |
Examples
Example 1: Denary to Sign and Magnitude
Convert -9 to 8-bit binary in Sign and Magnitude.
- The absolute value of
9in binary:00001001 - Add the sign bit for negative:
10001001 - So,
9in Sign and Magnitude is10001001.
Example 2: Denary to Two's Complement
Convert -9 to 8-bit binary in Two's Complement.
- The absolute value of
9in binary:00001001 - Invert the bits:
11110110 - Add
1:11110110 + 1 = 11110111 - So,
9in Two's Complement is11110111.
Example 3: Two's Complement to Denary
Convert 11110111 in Two's Complement to denary.
- The leftmost bit is
1, so it's negative. - Invert all bits:
00001000 - Add
1:00001000 + 1 = 00001001(which is9in decimal). - Therefore,
11110111in Two's Complement is9.
Note Summary
Common Mistakes
- Incorrect Bit Inversion in Two's Complement: Forgetting to add
1after inverting the bits when converting from a positive number. - Confusion Between Representations: Mistaking Sign and Magnitude for Two's Complement, which leads to incorrect calculations and interpretations.
- Double Counting the Sign Bit in Sign and Magnitude: Remember that the leftmost bit is not part of the magnitude and only indicates the sign.
Key Takeaways
- Sign and Magnitude: Uses a sign bit to indicate positive or negative; straightforward for storage but limited in arithmetic operations.
- Two's Complement: Most commonly used for binary arithmetic; simplifies operations and represents both positive and negative values with a single zero representation.
- Conversion Skills: Practice converting between denary, Sign and Magnitude, and Two's Complement to become confident in representing negative binary numbers.
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