write a case study of various number system in detail
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Understanding Number Systems
Number systems use different number bases. A number base indicates how many different digits are available when using a particular numbering system. For example, decimal is number base 10, which means it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Binary is number base 2, which means that is uses two digits: 0 and 1. Different number bases are needed for different purposes. Humans use number base 10 whereas computers use binary.
The number base determines how many digits are needed to represent a number. For example, the number 78 in decimal (base 10) requires two digits. The binary (base 2) equivalent is 1001110 which requires seven digits. As a consequence of this, there are many occasions in computing when very long binary numbers are needed. To solve this problem, other number bases can be used, which require fewer digits to represent numbers. For example, some aspects of computing use number base 16 which is referred to as hexadecimal.
The accepted method of representing different number bases (in technical documentation) is to show the number with the base in subscript. For example:
3710 is decimal
11012 is binary
12FF16 is hexadecimal
You will often see hexadecimal numbers represented using the C programming syntax where the prefix 0x is added to the value: 0x12FF.
Hexadecimal
Hexadecimal (or Hex) is particularly useful for representing large numbers as fewer digits are required. Hex is used in a number of ways. Memory addresses are shown in hex format, as are colour codes. The main advantage of hex is that two digits represent one byte.
Consider the number 100010012. This is an 8-bit number which when converted to decimal equals 13710. The same number in hex is 8916. This basic example shows that an 8-bit number in binary can be represented as a two-digit number in hex.
As it is number base 16, hex uses 16 different digits: 0 to 9 and A to F. The image below shows a hex viewer displaying all byte values from 0 to 255 in decimal, or 0 to FF in hex.
Hex viewer showing all values from 0 to FF
Working with Number Bases
When performing calculations, humans use number base 10, probably because we have 10 digits on our hands. Commonly, this system is known as decimal and uses 10 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. When we get to 9 we add an extra digit to the left and start again. When we get to 99 we add an extra digit to the left and so on. Each digit we add is worth ten times the previous digit. The weighted value for each position is as follows:
Decimal Numbering System
The number 3265 is easy to understand in decimal terms. It is made up as follows:
(3 x 1000) + (2 x 100) + (6 x 10) + (5 x 1)
When creating a number, we start with the units and add further digits as needed to create the number we want.
Binary numbers use number base 2 and works on exactly the same principal. This time, we only have two digits: 0 and 1. It has to be binary because computers work with either a zero or a one (off and on). Therefore, 1 is the largest value we can have with one bit. To increase the size of the number, we add more bits. Each bit is worth two times the previous bit because we are using number base 2. The table below shows an 8-bit binary number 10010111. The value of each new bit doubles in value as binary is base 2.
Binary Numbering System
Using the same principal to work out the number, we have:
(1 x 128) + (1 x 16) + (1 x 4) + (1 x)
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