W1032 Negative Integers

Within these castle walls be forged Mavens of Computer Science ...

— Merlin, The Coder

Prerequisites

• W1031 Positive Integers

Background

What methods might we use to encode negative integers? As a first approach, perhaps we can use one of the bits of our word to represent the sign. Let's consider a single-byte word and the number $5_{10}$. This may be encoded as:

Positive 5

If we were to reserve the most-significant bit, bit seven, to indicate whether our number was positive or negative, we can use the remaining seven bits to store the absolute value of the number. In such a case, the number $-5_{10}$ would be encoded as:

Negative 5, Reserved Bit

Initially this may appear to be a good solution, but let's explore further.

Observe

Observe, Ponder, and Journal:

Section 1: Reserved Bit Encoding

1. How would $\color {White}+11_{10}$ be encoded?
2. How would $\color {White}-11_{10}$ be encoded?
3. How would you add these two numbers ($\color {White}+11_{10}+-11_{10}$) in binary?
4. How many alternative representations are there for the number 0?
5. Given the above, do you think that this method of encoding is ideal? Why or why not?

References

• Complements (Wikipedia)