The binary number system is a __radix-2 number system__ with ‘0’ and ‘1’ as the two independent digits. All larger binary numbers are represented in terms of ‘0’ and ‘1’. The procedure for writing higher order binary numbers after ‘1’ is similar to the one explained in the case of the decimal number system.

For example, the first 16 numbers in the binary number system would be 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110 and 1111. The next number after 1111 is 10000, which is the lowest binary number with five digits.

This also proves the point made earlier that a maximum of only 16 (=2^{4}) numbers could be written with four digits. Starting from the binary point, the place values of different digits in a mixed binary number are 20, 21, 22 and so on (for the integer part) and 2^{−}^{1}, 2^{−}^{2}, 2^{−}^{3} and so on (for the fractional part).

**Example of Binary Number System**

*Consider an arbitrary number system with the independent digits as 0, 1 and X. What is the radix of this number system? List the first 10 numbers in this number system.*

*Solution*

*Solution*

- The radix of the proposed number system is 3.
- The first 10 numbers in this number system would be 0, 1, X, 10, 11, 1X, X0, X1, XX and 100.

**Advantages of ****Binary Number System**

Logic operations are the backbone of any digital computer, although solving a problem on computer could involve an arithmetic operation too. The introduction of the mathematics of logic by George Boole laid the foundation for the modern digital computer. He reduced the mathematics of logic to a binary notation of ‘0’ and ‘1’.

As the mathematics of logic was well established and had proved itself to be quite useful in solving all kinds of logical problem, and also as the mathematics of logic (also known as Boolean algebra) had been reduced to a binary notation, the binary number system had a clear edge over other number systems for use in computer systems.

Yet another significant advantage of this number system was that all kinds of data could be conveniently represented in terms of 0s and 1s. Also, basic electronic devices used for hardware implementation could be conveniently and efficiently operated in two distinctly different modes.

For example, a bipolar transistor could be operated either in cut-off or in saturation very efficiently. Lastly, the circuits required for performing arithmetic operations such as addition, subtraction, multiplication, division, etc., become a simple affair when the data involved are represented in the form of 0s and 1s.