In the decimal number system, we use the digits \(0,1,2,\dots,9\) to write numbers. In the binary number system (from bi, meaning two), we use only the digits \(0\) and \(1\).
Just like decimal numbers, binary numbers can be added, subtracted, multiplied, and divided. But why use binary instead of decimal?
Electronic devices are built from switches that have only two possible states: ON and OFF. These states are naturally represented by the digits \(1\) and \(0\), which makes the binary system ideal for electronic calculators and computers.
In expanded form, the decimal number \(253\) is written as:
\[ 253 = 2\times10^{2} + 5\times10^{1} + 3\times10^{0} \]This is why the decimal system is called the base-10 system.
Using the base-2 system, the same number can be written as:
\[ 253 = 1\times2^{7} + 1\times2^{6} + 1\times2^{5} + 1\times2^{4} + 1\times2^{3} + 1\times2^{2} + 0\times2^{1} + 1\times2^{0} \]Therefore, the binary representation of \(253\) is:
\[ 253_{10} = 11111101_{2} \]You can verify your conversions using this online decimal-to-binary converter.
Binary addition follows rules very similar to decimal addition:
\[ \begin{aligned} 0 + 0 &= 0 \\ 1 + 0 &= 1 \\ 0 + 1 &= 1 \\ 1 + 1 &= 10 \quad (\text{which is } 2 \text{ in decimal}) \end{aligned} \]Now let us add the binary numbers \(1101\) and \(1111\).
Important notes:
To check the result, convert each number to decimal:
\[ 1101_2 = 13_{10}, \quad 1111_2 = 15_{10} \] \[ 13 + 15 = 28 = 11100_2 \]The decimal and binary results agree exactly.
Binary numbers can be added, subtracted, multiplied, and divided just like decimal numbers. Their key advantage is simplicity: using only two digits makes it easier to design reliable digital systems such as computers.