Number Systems
3
The value or magnitude of a given decimal number can be expressed as the sum of the various digits
multiplied by their place values or weights.
As an illustration, in the case of the decimal number 3586.265, the integer part (i.e. 3586) can be
expressed as
3586 = 6 × 10
0 + 8 × 10
1 + 5 × 10
2 + 3 × 10
3 = 6 + 80 + 500 + 3000 = 3586
and the fractional part can be expressed as
265 = 2 × 10
−1 + 6 × 10
−2 + 5 × 10
−3 = 02 + 006 + 0005 = 0265
We have seen that the place values are a function of the radix of the concerned number system and
the position of the digits. We will also discover in subsequent sections that the concept of each digit
having a place value depending upon the position of the digit and the radix of the number system is
equally valid for the other more relevant number systems.
1.4 Binary Number System
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 higherorder 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 2
0 , 2
1 , 2
2 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.
1.4.1 Advantages
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.
3
The value or magnitude of a given decimal number can be expressed as the sum of the various digits
multiplied by their place values or weights.
As an illustration, in the case of the decimal number 3586.265, the integer part (i.e. 3586) can be
expressed as
3586 = 6 × 10
0 + 8 × 10
1 + 5 × 10
2 + 3 × 10
3 = 6 + 80 + 500 + 3000 = 3586
and the fractional part can be expressed as
265 = 2 × 10
−1 + 6 × 10
−2 + 5 × 10
−3 = 02 + 006 + 0005 = 0265
We have seen that the place values are a function of the radix of the concerned number system and
the position of the digits. We will also discover in subsequent sections that the concept of each digit
having a place value depending upon the position of the digit and the radix of the number system is
equally valid for the other more relevant number systems.
1.4 Binary Number System
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 higherorder 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 2
0 , 2
1 , 2
2 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.
1.4.1 Advantages
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.