Number systems

Review these topic from S1:

• Number systems
• Data representation
• Integers

Signed integers

To denote signed integers, there are 3 approaches.

• Sign-and-magnitude
• One’s complement
• Two’s complement

Sign-and-magnitude

MSB denotes the sign. Remaining bits denote the magnitude.

2 problems with this representation:

• Two zeros ($0000$ and $1000$)
• Arithmetic is cumbersome

Note

One’s complement fixes the arithmetic. Two’s complement fixes both of the problems. Two’s complement is the widely used one.

Special formats

Binary Coded Decimal

Aka. BCD. Each digit is represented by a fixed set of bits. Usually in length 4 or 8. Only 10 of the available representations are used.

Gray codes

Aka. reflected binary. An ordering of the binary numeral system such that two successive values differ only by 1 bit. Named after Frank Gray. Useful in minimizing errors in digital systems, especially in state transitions.

If two adjacent states have more than 1 bit changed (eg: 3 and 4), then the value transition might take some noticable time and could lead to issues.

A gray code is said to be cyclic, if the first and last numbers also differ by only a bit.

Generating gray code for n bits

To generate the Gray code for n bits:

• Start with the Gray code for 1 bit: 0 and 1.
• For each additional bit:
  • Reflect the current sequence (write it in reverse order).
  • Prefix the original sequence with 0.
  • Prefix the reflected sequence with 1.

Example

For 2-bit Gray codes:

• 00
• 01
• 11
• 10

For 3-bit Gray codes:

• 000
• 001
• 011
• 010
• 110
• 111
• 101
• 100