Representing raw data

In computer science, number systems are crucial for representing and processing data in digital systems. Different number systems are used depending on the context.

The most relevant of which would be Binary, Hexadecimal and Decimal.

Converting from one to another

There are two main steps we can use to easily convert from one number system to any other number system.

Converting to decimal

The following formula can be used to convert to base 10; b being the base of the number: w(z) = \sum_{i=0}^{n-1}z_{i} \cdot b^{i}

Example: 1010 0010_{2} = 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 164

Division method

Just perform the euclidean algorithm on a decimal number and read the rest from bottom to top:

Example: \begin{align} 164 \div 7 = 23 \; | \; r: 3 \\ 23 \div 7 = 3 \; | \; r: 2 \\ 3 \div 7 = 0 \; | \; r: 3 \\ \\ 323 \end{align}

IEEE 754

Single Precision: 1 bit sign, 8 bit exponent (-127); 23 bit mantissa.

\begin{align} x & = v \cdot m \cdot b^e \\ \\ v & = \text{sign} \\ m & = \text{mantisse} \\ b & = \text{base} \\ e & = \text{exponent} \\ \end{align}

TODO! This should be a separat note: