2’s complement

In the binary system, a positive number is simply represented as a simple binary number (with the most significant bit as 0). Meanwhile, a negative number needs to be represented using one of the complements: 1’s complement and 2’s complement.

The most significant bit is the sign bit. For the remaining part,

– with 1’s complement, simply invert each digit of the given number.

– with 2’s complement, add 1 to the result of 1’s complement.

2’s complement is widely used, because “inversion and add one” works well for arithmetic operation: https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html

Fixed point and floating point numbers

Here, we focus on Binary number representation only.

To store a real number (a number with fractional component), there are two major approaches: fixed point number representation and floating point number representation.

First, what is a binary point?

A binary point is like the decimal point in a decimal system. It acts as a divider between the integer and the fractional part of a number.

A binary point represents the coefficient of the term 2⁰ = 1. All digits (or bits) to the left of the binary point carries a weight of 2⁰, 2¹, 2², and so on. Digits (or bits) on the right of binary point carries a weight of 2^-1, 2^-2, 2^-3, and so on. For example, the number

For example, the number 11010.1₂ =

1 * 2⁴ + 1 * 2³ + 0 * 2² + 1 * 2¹ + 0* 2⁰ + 1 * 2^-1

= 16 + 8 + 2 + 0.5

= 26.5

Fixed point

This representation implicitly fixes the binary point to be at some position of a numeral. In this case, both the integer part and the fraction part is with fixed number of bits.

A negative fixed-point number is not hard to represent as long as we use 2’s compliments.

Floating point

This representation reserves a certain number of bits for the number (called the mantissa or significand) and a certain number of bits to indicates where within that number the decimal place sits (called the exponent).

For 32-bit system, a floating point number contains 3 parts:

– Sign (1 bit)

– Exponent (8 bit) 
– Mantissa (23 bit) it can be fraction or an integer.