Number Systems and Encoding
Contents
Number Systems
All numbers can be represented as sums of the powers of their base/radix.
For example
- Base 10: 3597 = 3 * 10^3 + 5 * 10^2 + 9 * 10^1 + 7 * 10^0
- Base 16: F24B = 15 * 16^3 + 2 * 16^2 + 4 * 16^1 + 11 * 16^0
- Base 2: 1011 = 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0
Concept: Subtraction
Subtraction is the addition of a number’s additive inverse.
a - b = a + (-b)
Two’s Complement
In binary arithmetic, the two’s complement of a number is its additive inverse.
For an n-digit number, its two’s complement is: b* = 2^n - b
We can simplify this as ~b - 1 (Flip all the bits and subtract 1, and keep only n LSBs)
Overflow Detection
In a n-bit two’s complement system
- Positive Overflow occurs when: a + b > 2^(n-1) - 1
- Negative Overflow occurs when: -a -b < 2^n - 1