The Binary Numeral System

In ancient time, when man first created the numeric system, they came up with the idea of counting with their fingers. Humans have only 10 fingers, therefore initially they were limited by this. So the decimal numeral system was born, which counts from 0 to 10, and continues after every 10's.

When electronics started to surface, human realized that decimal is too much for machines to decipher. Human is logical and rational; machines are logical but not rational. We understand what is wrong and what is right, what are the limits for certain situations. Machines know only two things: true or false.

Hence, a new numeral system must be created for both machine and human to understand and communicate with each other. The binary numeral system was the solution.



In 1854, British mathematician George Boole presented an algebraic system of logic that would later become known as the Boolean algebra. His logical calculus was the foundation in the design of digital electronic circuitry. Boolean algebra is based on the binary numeral system.

The decimal numeral system uses 10 distinct digits, i.e. from '0' to '9'. Every 1's that is added to a number contribute to a higher number, until number '9'. After '9', there is no more distinct digit to represent the number. Therefore, the 1 that is added will revert the first digit to '0' and a second digit '1' is added. Then the trend repeats, from 10 to 19 and changes to 20. This is what we learnt in kindergarten. This is our fundamentals in mathematics.

But in binary numeral system, things are a bit different. Binary only uses two distinct digits: '0' and '1'. When a numeral system is limited only by two digits, the calculation changes. The trend is still the same: every 1's that is added to a number contribute to a higher number. But in binary, when the number reaches '1', it cannot change to '2', because '2' is not a binary number. Therefore, the added number '1' will instead revert the first digit back to '0' and a second digit '1' is added, i.e. '10'.

Continuing the trend, decimal number '3' is binary number '11'. Again, binary does not contain the number '4'. So the first bit '1' will revert back to '0', second bit '1' will also revert back to '0', and a third bit '1' is added. This is the binary numeral system.

The following is an example list of binary numbers and the corresponding decimal numbers for easier observation and understanding. A 3-bits system is used.

000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7


That is all for today. Happy learning! :)