Number Systems
We typically use the denary number system, where everything is based on 10, but computers work in binary, where everything is base 2.
So denary looks like:
1 → 10 → 100 → 1000
And binary digits represent:
1 → 2 → 4 → 8 → 16 → 32 → 64 → 128 → 256
The binary number 10100 is equivalent to denary 20.
And 01111111 is 127
Conversion Practice
64 → Binary
=01000000
130 → Binary
= 10000010
255 → Binary
= 11111111
196 → Binary
= 0110000100
Handling negatives
BCD is a primitive method for handling negatives, but it is unused because it is slow and inefficient in terms of resource utilization.
So instead we use a sign bit, where we make the largest column in the binary pattern an indicator for whether or not the number is positive or negative.
So you would replace the heading:
128 - 64 - 32 - 16 - 8 - 4 - 2 - 1
With:
Sign - 64 - 32 - 16 - 8 - 4 - 2 - 1
This limits the range of numbers that can be stored, but can be applied to larger binary patterns too.
The main method used is Two’s Complement
Binary Task
Converting binary to denary
14 → Bin
64 → 32 → 16 → 8 → 4 → 2 → 1 0 → 0 → 0 → 1 → 1 → 1 → 0
= 0001110
60 → Bin
64 → 32 → 16 → 8 → 4 → 2 → 1 0 → 1 → 1 → 1 → 1 → 0 → 0
= 0111100
200 → Bin
128 → 64 → 32 → 16 → 8 → 4 → 2 → 1 1 → 1 → 0 → 0 → 1 → 0 → 0 → 0
= 11001000
Binary Arithmetic
56 + 32 = 88
64 → 32 → 16 → 8 → 4 → 2 → 1 0 → 1 → 1 → 1 → 0 → 0 → 0
56 = 0111000
64 → 32 → 16 → 8 → 4 → 2 → 1 0100000
0111000 0100000
1111000
Advanced numbers and mathematical systems are explained here