KembaraXtra – Computer Science – Unsigned Numbers

1. The Need for Unsigned Numbers

Signed vs. Unsigned:

• Signed integers (using two's complement) represent both positive and negative numbers.
• Unsigned integers represent only non-negative numbers (positive and zero).

Why Use Unsigned?

• When negative values are unnecessary, using signed integers wastes half the representable range.
• Unsigned integers allow representing larger positive values with the same number of bits.

2. Understanding Unsigned Representation

Interpretation Matters: A binary sequence can represent different values depending on whether it's interpreted as signed or unsigned.

Example (4-bit):

Generalization (n-bit unsigned number):

• Maximum value: (2ⁿ) − 1
• Minimum value: 0
• Count of unique values: 2ⁿ

3. Operations and Interpretation

Adder Circuit's Perspective: The adder circuit performs the same binary addition regardless of whether the numbers are signed or unsigned.

Interpretation is Key: After calculation, the program decides how to interpret the result (signed or unsigned).

Example: Adding 1011 and 0010 results in 1101.

• Signed: −5 + 2 = −3
• Unsigned: 11 + 2 = 13

4. Integer Overflow

Carry-out Bit (Unsigned): A carry-out of 1 indicates an integer overflow. The result is too large to be represented with the given number of bits.

Carry-in/Carry-out (Signed):

• Overflow: Most significant carry-in bit not equal to most significant carry-out bit.
• No Overflow: Most significant carry-in bit equal to most significant carry-out bit (carry-out can be ignored).

Importance of Overflow Detection: Overflows can lead to incorrect results and unexpected program behavior if not handled properly.

Real-world Example: Pac-Man level 256 glitch is caused by an integer overflow in the level counter.