Number Systems

Last updated May 26, 2023 Edit Source

• Compsci
• Theory
• Datarep

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

64 -> Binary

=01000000

130 -> Binary

= 10000010

255 -> Binary

= 11111111

196 -> Binary

= 0110000100

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

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

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

Theory