01. Introduction
Dated: 14-08-2025
Number System
In our daily life, we use \(10\) symbols (\(0, 1, 2, \ldots, 9\)) to do calculations. This number \(10\) is called the base of the system we are using. Therefore, with a base \(N\), we need \(0, 1, \ldots, (N - 1)\) symbols to represent any number.1
Following systems are used in computers usually
| Base \(N\) | Number |
|---|---|
| \(2\) | Binary |
| \(8\) | Octal |
| \(10\) | Decimal |
| \(16\) | Hexadecimal |
Any arbitrary number1 can be represented as
\[
a = a_m N^m + a_{m-1} N^{m-1} + \dots + a_1 N^1 + a_0 + a_{-1}N^{-1} + \dots + a_{-m}N^{-m}
\]
The decimal number1 \(1729\) is represented as
\[
(1729)_{10} = 1 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0
\]
Computers use bits (\(0\) or \(1\) - base 2) to represent a number.1
Conversion Example
Convert \((47)_{10}\) to binary
| Base | Post Division | Remainder | |
|---|---|---|---|
| Starting: | \(2\) | \(47\) | |
| \(2\) | \(23\) | \(1\) | |
| \(2\) | \(11\) | \(1\) | |
| \(2\) | \(5\) | \(1\) | |
| \(2\) | \(2\) | \(1\) | |
| \(2\) | \(1\) | \(0\) | |
Most Significant Bit |
\(0\) | \(1\) |
\[(47)_{10} = (101111)_2\]
Conversion Example
Convert \((0.7625)_{10}\) to binary
| Fractional Part | Operation | Product | Integer |
|---|---|---|---|
| \(0.7625\) | \(\times2\) | \(1.5250\) | \(1\) |
| \(0.5250\) | \(\times2\) | \(1.0500\) | \(1\) |
| \(0.0500\) | \(\times2\) | \(0.1000\) | \(0\) |
| \(0.1000\) | \(\times2\) | \(0.2000\) | \(0\) |
| \(0.2000\) | \(\times2\) | \(0.4000\) | \(0\) |
| \(0.4000\) | \(\times2\) | \(0.8000\) | \(0\) |
| \(0.8000\) | \(\times2\) | \(1.6000\) | \(1\) |
| \(0.6000\) | \(\times2\) | \(1.2000\) | \(1\) |
| \(0.1000\) | \(\times2\) | \(0.2000\) | \(0\) |
\[(0.7625)_{10} = (0.110 \overline{00011})_2\]
The number starts from \(110\) but the sequence \(00011\) is represented indefinitely.
Conversion Example
Convert \((59)_{10}\) to binary and than octal.
| Base | Post Division | Remainder | |
|---|---|---|---|
| Starting: | \(2\) | \(59\) | |
| \(2\) | \(29\) | \(1\) | |
| \(2\) | \(14\) | \(1\) | |
| \(2\) | \(7\) | \(0\) | |
| \(2\) | \(3\) | \(1\) | |
| \(2\) | \(1\) | \(1\) | |
Most Significant Bit |
\(0\) | \(1\) |
\[(59)_{10} = (111011)_2\]
\[(111011)_2 = (73)_8\]
References
Read more about notations and symbols.