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 (= 24 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 1.1**

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**

• 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.