Table of Contents

## There are three types of Number System

### 1.**Decimal Number System**

**2.Binary Number System**

**3.Octal Number System**

**4.Hexadecimal Number System**

So, let’s stat the article. In this article you will find an detailed explanation aon each type of the Number System-:

**Decimal Number System**

The decimal system of 10 numerals or symbols (Deca means 10, that is why this system is called **decimal system**). These 10 symbols are 0,1,2,3,4,5,6,7,8,9 ; using these symbols as digit of number, we can express any quantity. The decimal system, also called the base-10 system because it has 10 digits, has evolved naturally as a result of the fact that man has 10 fingers.

The decimal system is a positional-value system in which the value of a digit depends on its position. For example, consider the decimal number 729. We know that the digit 7 actually represents 7 **hundreds**, the 2 represents 2 **tens**, and 9 **units**. In essence, the 7 carries the most weight of three digits; it is referred to as the most significant digit (MSD). The 9 carries the least weight and is called the least significant digit (LSD).

Consider another example, 25.12. This number is actually equal to 2 tens plus 5 units plus 1 tenths plus 2 hundredths i.e., 2\times 10+5\times 1+1\times \frac { 1 }{ 10 } +2\times \frac { 1 }{ 100 }. The decimal point is used to separate the integer and fractional parts of the number.

More rigorously, the various positions relative to decimal point carry weights that can be expressed as power of . This is illistrated in the figure below where the number 2512.1971 is represented. The decimal point seperates the positive powers of 10 from the negetive powers. The number 2512.1971 is thus equal to

2\times { 10 }^{ 3 }+5\times { 10 }^{ 2 }+1\times { 10 }^{ 1 }+2\times { 10 }^{ 0 }+1\times { 10 }^{ -1 }+9\times { 10 }^{ -2 }+7\times { 10 }^{ -3 }+1\times { 10 }^{ -4 }.

In general, any number is simply the sum of the products of each digit value of its positional value.

The sequence of decimal numbers goes as 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18….

See after 9, each successive number is a combination of two (or more) (unique) symbols of this system.

**Binary Number System**

Unfortunately, the decimal number system does not lend itself to convenient implementation in digital system. For example, it is very difficult to design electronic equipments so that it can work with 10 different voltage levels (each one representing one decimal character, 0 through 9). On the other hand, it is very easy to design simple, accurate electronic circuits that operate with only two voltage levels. For this reason, almost every digital system uses the binary number system (base 2) as the basic number system of its operations, although other systems are often used in conjunction with binary.

In the binary system there are only two symbols or possible digit values, 0 and 1.

Even so, this base-2 system can be used to represent any quantity that can be represented in decimal or other number systems.

The binary system is also a positional-value system, wherein each binary digit has its own value or weight expressed as a power of 2. This is illustrated in the figure below:

Here, places to left of the binary point (counterpart of the decimal point) are positive powers of 2 and places to the right are negative powers of 2. The number 1010.0101 is shown represented in the figure.

To find the decimal equivalent of above shown binary number, we simply take the sum of the products of each digit value (0 or 1) and its positional value:

{ 1010.0101 }_{ 2 }=\left( 1\times { 2 }^{ 3 } \right) +\left( 0\times { 2 }^{ 2 } \right) +\left( 1\times { 2 }^{ 1 } \right) +\left( 0\times { 2 }^{ 0 } \right) +\left( 0\times { 2 }^{ -1 } \right) +\left( 1\times { 2 }^{ -2 } \right) +\left( 0\times { 2 }^{ -3 } \right) +\left( 1\times { 2 }^{ -4 } \right) .

= 8+0+2+0+0+0.25+0+0.0625

= { 10.3125 }_{ 10 }.

Notice in the preceding operation that subscripts (2 and 10) were used to indicate the base in which the particular number is expressed. This convention is used to avoid confusion whenever more than one number system is being employed.

In the binary system, the term Binary digit is often abbreviated to the term bit, which we’ll use henceforth. As you see in the above figure, there are 4 bits to the left of the binary point, representing the integer part of the number, and 4 bits to the right of the binary point, representing the fractional part. The leftmost bit carries the largest weight and hence, is called the most significant bit (MSB). The rightmost bit carries the smallest weight, and hence called least significant bit (LSB).

The sequence of binary numbers goes as 00,01,10,11,100,101,110,111,1000…….. The binary counting sequence has an important characteristic. The unit bit(LSB) changes either from 0 and 1 or 1 to 0 with each count. The second bit (two’s ({ 2 }^{ 1 }) position) stays at 0 for two counts, then at 1 for two counts, then at 0 for two counts, and so on. The third bit (four’s({ 2 }^{ 2 })position) stays at 0 for four counts, then at 1 for four counts, and so on. The fourth bit (eight’s ({ 2 }^{ 3 })position) stays 0 for eight counts, then at 1 for eight counts. If we wanted to count further we would add more places, and this pattern would continue with 0s and 1s alternating in groups of { 2 }^{ N-1 }.

## Octal Number System

The octal number system is very important in digital computer work. The octal number system has a base of **eight**, meaning that it has eight unique symbols:0,1,2,3,4,5,6 and 7. Thus, each digit of an octal number can have any value from 0 to 7.

The octal system is also a positional value system, wherein each octal digit has its own value or weight expressed as a power of 8 (see below figure). The places to the left of the octal point (counter-part of decimal point and binary point) are positive power of 8 and places to the right are negative powers of 8. The number 3721.2406 is shown represented in the figure.

To find the decimal equivalent of above sown octal number, simply take the sum of products of each digit value and its positional value:

{ 3721.2406 }_{ 8 }=\left( 3\times { 8 }^{ 3 } \right) +\left( 7\times { 8 }^{ 2 } \right) +\left( 2\times { 8 }^{ 1 } \right) +\left( 1\times { 8 }^{ 0 } \right) +\left( 2\times { 8 }^{ -1 } \right) +\left( 4\times { 8 }^{ -2 } \right) +\left( 0\times 2^{ -3 } \right) +\left( 6\times { 8 }^{ -4 } \right)= 3 \times 512 + 7 \times 64 + \times 8 + 1 \times 1 + 2 \times 0.0625 + 0 + 0.001464

= { 2001.313964 }_{ 10 }

The sequence of octal numbers goes as 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,22….. See each successive number after 7 is a combination of 2 or more unique symbols of octal system.

**Hexadecimal Number System**

The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It uses the digit 0 through 9 plus the letter A,B,C,D,E and F as the 16 digit symbols.

Just like above systems, hexadecimal system is also a positional- value system, wherein each hexadecimal digit has its own value or weight expressed as a power of 16. The digit positions in a hexadecimal number have weight as shown in the above figure.

Following table below shows the relationship between hexadecimal, octal, decimal and binary numbers.

Note that each hexadecimal digit represents a group of four binary digits. It is important to remember that hex (abbreviation for hexadecimal) digits A through F are equivalent to the decimal values 10 through 15.

**Final Wordings:**

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Source-: Sumita arora

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