Bits and Dits

Binary digits and binary strings 🧮

The most basic form of computational language is binary code.

• A bit (binary digit) is the basic unit of computational information.

• Each bit can either take the value of 0 or 1.

• Since there are only two possible options (\(2\) degrees of freedom), it is called binary.

• A bit string is a sequence of bits.

\[\begin{split} \text{Bitstring} = x_1x_2x_3... x_n \\ \text{ with }x_i \in \{0,1\} \end{split}\]

 

Tertiary, quaternary and higher ‘ary’s 🔢

• A dit is a unit of information with \(d\) possible options.

• Each dit can either take the value of 0, 1, …, up to d-1 and has \(d\) degrees of freedom.

• A dit string is a sequence of dits.

\[\begin{split} \text{Ditstring} = y_1y_2y_3... y_n \\ \text{ with }y_i \in \{0,1,..,d-1\} \end{split}\]

 

For a string of length \(n=4\), give the formula for the total possible number of (i) bit strings and (ii) dit strings. 🧠

Answer

Each bit represents a variable and each variable has only 2 options, i.e. 0 or 1.

For a bit string of length \(n=4\):

\[ \text{Total combinations of bitstrings } (T_{bit}) = 2\times 2\times 2\times 2 = 2^4 \]

For a dit string of length \(n=4\) :

\[ \text{Total combinations of ditstrings } (T_{dit}) = d\times d\times d\times d = d^4 \]

 

For a string of any length \(n\), give the formula for the total possible number of (i) bit strings and (ii) dit strings. 🧠

Answer

For any length \(n\), the total possible # of bitstrings:

\[ \text{Total combinations of bitstrings } (T_{bit}) = 2^n \]

For any length \(n\), the total possible # of ditstrings:

\[ \text{Total combinations of ditstrings } (T_{dit}) = d^ n \]