Number System

Table of Contents

In this article, we will try to understand the number system. All will be well acquainted with the decimal number system. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to represent any number in the decimal number system. The base of this system is 10. Radix is the number of digits used in a system. Any number is a set of digits. For example, 343.95 is a number in which the integer value is 343 and the fractional part is 0.95. The integer and fractional parts are separated from each other by a radix point (.). Radix point is also called decimal point. Apart from the decimal system, there are some other number systems as well which are as follows – Read this article in Hindi

Binary system
Octal number system
Hexadecimal number system

Decimal Number System

Decimal number system is widely used in our daily life. Each digit of every number written in the decimal system has its own permanent value. It is also called multiplier or multiplying factor. Multiples are thousands, hundreds, tens and ones etc. Similarly, to display numbers of value less than 1, digits are written to the right of the decimal point and its place value can be obtained by multiplying by 1/10, 1/100, 1/1000 respectively. For example, the decimal number 4343.95 can be represented as follows.

(4343.95)₁₀=4X10³+3X10²+4X10¹+3X10⁰+9X10^-1+5X10^-2

In the above example, 4 is multiplied by its place value one thousand (10³), 3 by hundred (10²), 4 by ten (10¹) and 3 by one (10⁰). Similarly, in the fractional part, 9 is multiplied by 1/10 i.e. 10^-1 and 5 is multiplied by 1/100 i.e. 10^-2.

In general, every number system has a set of symbols called digits. Every system has rules for adding, subtracting, multiplying and dividing. The group of these digits is called a number. A number has two parts :(I) Integer and (II) Fractional part. Integer and fractional parts are separated by a dot (.). This point is called radix point or decimal point. For example, if a number has n integers, m fractional numbers and radix b, then this number (N) can be written in the following way.

where N= a number
b=radix or base number of the system
n = number of digits in the integer part
m = number of digits in the fractional part
dn-1 = Most Significant digit (MSD)
d-m = Least significant digit (LSD)

Characteristics of various number systems

Radixes of different number systems and the symbols used in them are given below.

Number System	Radix or base Number	Symbol	Examples
Binary	2	0,1	1100110
Octal	8	0,1,2,3,4,5,6,7	235.23
Decimal	10	0,1,2,3,4,5,6,7,8,9	896.95
Hexadecimal	16	0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F	3FA9.56

Binary Number System

In this system only two digits 0 and 1 are used. Hence the base number (radix) of this system is 2. Binary digits 0 and 1 are called bits. For example, 10101 is a binary number because only the digits 0 and 1 have been used in it. Similarly 10101.1101 is also a binary number. The dot (.) used after 10101 in the number is called binary point. The place values to the left and right of the binary point are converted to increasing and decreasing powers of 2.

The decimal value of a binary number can be found as follows:

Binary(101011)₂=1×2⁵+0x2⁴+1×2³+0x2²+1×2¹+1×2⁰

=32+08+8+0+2+1

==(43)₁₀ in decimal system

Similarly, to convert a fractional binary number to decimal, the digits to the right of the binary point are multiplied by 2^-1, 2^-2, 2^-3, etc., respectively.

for example (0.10101)₂=1×2^-1+0x2^-2+1×2^-3+0x2^-4+1×2^-5

=1/2+0+1/8+0+1/32

=(0.65625)₁₀in decimal system

so binary number 10101.1101 is equal to decimal number 43.65625

Octal Number System

The base or radix of any number system is equal to the number of digits used in it. The digits 0 to 7 in the octal number system are like the decimal system. Numbers 0 to 7 are used in octal number system. The maximum number in the octal system is 7. It does not have the digits 8 and 9. Hence, the value of the radix in the octal system is 8.

To convert binary into octal system, groups of 3-3 bits are made and after writing the value of each group in decimal, octal number is obtained. for example

Binary Number- 11111011110101

Groups of 3-3 bits 11, 111, 011, 110, 101

Writing the value of each group in decimal is 3 7 3 6 5 (octal)

So (37365)₈ is the octal equivalent of the binary number (11111011110101)₂. In the octal system, two decimal digits are used together to write numbers greater than 7. For example, 10 in octal represents 8 in the decimal system.

Hexadecimal Number System

The hexadecimal system is widely used in microprocessor functions. The base number of the hexadecimal system is 16. In this system, the digits from 0 to 9 are just like the decimal system. In hexadecimal system, 10 is represented by A, 11 by B, 12 by C, 13 by D, 14 by E, 15 by F and decimal number 16 by 10 in hexadecimal system. Decimal numbers represent 17 by 11, 18 by 12 and 32 by 20. The binary and hexadecimal equivalents of some decimal numbers are shown in the table below.In the hexadecimal system, the place value of each digit of a number is represented by a power of 16. The place values of the digits of a hexadecimal number are as follows.

16⁴ 16³ 16² 16¹ 16⁰ . 16^-1 16^-2 16^-3

Thus, to convert a hexadecimal number into a decimal number, each digit is multiplied by its place value and all the products are added. For example, the decimal equivalent of the hexadecimal number 5A9 can be found as follows-

(5A9)₁₆= 5×16²+Ax16¹+9×16⁰

=1280+10×16+9

=(1449)₁₀(decimal system

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