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In every step of life, we definitely find the use of numbers. If you need to buy something, you will have to pay a certain amount of money for which you will have to count the money. Likewise, the shopkeeper will also count the goods to give you and same for your changes. Therefore, number system can simply be defined as a way to represent numbers.
For example, a number system can be used to represent the number of players in a certain game like 11 players in a football team or for the number of audience for a concert like 25000 concert-goers, etc. So number system can also be viewed as a set of values that is used to represent different quantities.
1) Decimal Number System:
The decimal number system is the most commonly used number system in our daily life. This generally used number system is also known as the base 10 number system because it uses just the 10 symbols i.e. 0 to 9. It is also known as the denary number system because any numeric value there is, these system’s digits can easily represent them.The decimal system is specially used in the computer interface. The weight and position of the digit dictate the value represented by it.
In this system, each number consists of the digits that are located at different positions. The positions of the 1^{st} and the 2^{nd} digits towards the right side of the decimal point are ^{-1} and ^{-2.} Similarly, the positions of the 1^{st} and 2^{nd} digits towards the left side of the decimal point are 0 and 1 respectively.
The value of the number is determined by adding the results out of the multiplication of the digits with the weight of their position. This method is called the expansion method. Under this method, the rightmost digit of the number is called the Least Significant Digit (LSD), as it has the lowest weight. Likewise, the leftmost digit of the number is called the Most Significant Digit (MSD), as it has the highest weight.
Examples:
(a) The weights and positions of each digit of the number 796 are as follows:
Positions | 2 | 1 | 0 |
Weights | 10^{2} | 10^{1} | 10^{0} |
Face value | 7 | 9 | 6 |
The above table indicates that:
The value of digit 7 = 7x10^{2}=700
The value of digit 9 = 9x10^{1}=90
The value of digit 6 = 6x10^{0}=6
The actual number can be formed by adding the values obtained by the digits as follows:
700+90+6=796
Here, the digit 7 in the number 796 is the most significant digit and 6 is the least significant digit.
(b) The weights and positions of each digit of the number 125.64 are as follows:
Positions | 2 | 1 | 0 | -1 | -2 |
Weights | 10^{2} | 10^{1} | 10^{0} | 10^{-1} | 10^{-2} |
Face value | 1 | 2 | 5 | 6 | 4 |
The above table indicates that:
The value of digit 1 = 1x10^{2} = 100
The value of digit 2 = 2x10^{1} = 20
The value of digit 5 = 5x10^{0}= 5
The value of digit 6 = 6x10^{-1}= 0.6
The value of digit 4 = 4x10^{-2}= 0.4
The actual number can be formed by adding the values obtained by the digits as follows:
100+20+5+0.6+0.4=125.64
Binary Number System refers to the number system that uses only two symbols i.e. 0 and 1, that's 0why it is called a base 2 number system. It is also known as Binary Digit (BIT). This number system is especially used in the internal processing of computer system. When we count up from 0 in binary, symbols are much more frequently run out as this system only uses 0 and 1 and 2 do not exist. Therefore, we use 10 in this system because 10 is equal to 2 in decimal. The combination of binary numbers can be used to represent different quantities like 1001. In Binary, each digit’s positional value is twice the face value or place value of the digit of its right side. Each position’s weight is a power of 2.
According to the position and weight, the place value of the digits is as follows:
Positions | 3 | 2 | 1 | 0 |
Weights | 2^{3} | 2^{2} | 2^{1} | 2^{0} |
Example: Convert 101.11
Positions | 2 | 1 | 0 | -1 | -2 |
Face value | 1 | 0 | 1 | 1 | 1 |
Weight | 2^{3} | 2¹ | 2^{0} | 2^{-1} | 2^{-2} |
The above table indicates that:
101.101 = 1x2²+0x2¹+1x2^{0}+1x2^{-1}+1x2^{-2}
= 1x4+0+1x1+1/2+1/4
= 4+0+1+0.5+0.25
= 5.75
Octal Number System is the base 8 system. Like the decimal number system, this system is also used in the internal processing of computer system. It is the system that consists of eight digits i.e. {0, 1, 2, 3, 4, 5, 6, 7} which is used for the representation of long binary numbers short-handedly. In this system, each digit position represents a power of 8. The number 708 will not be valid in this system as 8 is not a valid digit.
According to the position and weight, the place value of the digits is as follows:
Example: Convert 128 to a decimal number.
Positions | 4 | 3 | 2 | 1 | 0 |
Weights | 8^{4} | 8³ | 8² | 8¹ | 8^{0} |
128 = 1x8¹+2x8^{0}
= 1x8+2x1
= 8+2
= 10
Hexadecimal Number System is the number system that represents long binary numbers in shortcut method. It is a base 16 system as it consists of 16 digits i.e. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } where the alphabets represent the decimal numbers 10 to 15. This system is also used in the computer system, mainly in the memory management. As the name suggests, each digit's position represents a power of 16 in this system.
According to the position and weight, the place value of the digits is as follows:
Example: Convert 2A16 to a decimal number.
Positions | 4 | 3 | 2 | 1 | 0 |
Weights | 16^{4} | 16³ | 16² | 16¹ | 16^{0} |
2A16 = 2x16¹+Ax16^{0}
= 2x16+10x1
= 32+10
= 42
(Karn & Pudasaini, 2015)
Karn, M. K., & Pudasaini, D. (2015). Computer Science I. Anamnagar, Kathmandu: Buddha Publication.
ASK ANY QUESTION ON Number System: Introduction and Types
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Samar khanal
what is positional and non positional no system
Jan 03, 2017
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