A number system is a system to express or representing a number by using a set of mathematical notation or digits or symbols in a specific manner. Number system represents a unique notation or code for every number and it also allows to operation of mathematical operations like addition, subtraction, and division.

The number of systems are used to quantify the magnitude of something. In other words, the number systems are called position weighted system, since the weight of each digit depends on Its relative position within the number.

**Key points:**

If the base of the number system is ‘r’ the no of different symbols used in the system is ‘r’. the largest value of digits in the base ‘r’ system is r-1.

**Types of number systems**

1. Binary number system (base – 2)

2. Octal number system (Base – 8)

3. Decimal number system (base – 10)

4. Hexadecimal number system (base-16)

The base of the binary number system is ‘2’

The maximum digit in binary system =

r-1

2-1 = 1

The base 2 number system is also known as the binary number system. The binary number system uses only two digits 0 and 1.

Each position of the digit in a binary number represents a 0 power of the base (2). An example of a binary number is **001011.**

**The conversion process of Decimal to binary**

**For the Integer part:** – For the Integer part we will divide the given integer with base ‘r’ and we will collect the reminder then again quotient will be divided by base ‘r’ and we will collect the remainder.

We will repeat this process until the quotient become zero

**For Decimal part:**

For Decimal part, we will multiply the given decimal with base r and we will collect the integer part

Again we will multiply the remaining decimal with base ‘r’ and collect the integer part.

We will repeat this process until the decimal part becomes zero.

In the case of the Octal number system, its **base is 8** which means it represents 8 unique states. The word OCT means 8. A number system is called an octal number if this is expressed with **base-8 (0, 1, 2, 3, 4, 5, 6, 7).** There is the value of the base is 8 in octal number system.

The octal number system is used in shift resistor. The maximum decimal number in octal number system is** (8-1) = 7**

As we know 8 = (2)3

So each digit in octal number system will be replaced by its equivalent binary using three bits. Hence direct conversion of octal to binary is possible.

(12.5)8 = (001010.101)2

For binary to octal conversion, we will form a group of three bits starting from the origin and then we will write the decimal equivalent of each group.

(001010.101)2 = (12.5)1

The numbers are represented or expressed with **base 10** in the decimal number system. In this system, 10 digits (0 – 9) uses to represent a number. This is also called the base -10 number system. Each digit has a value in decimal number according to its value, this is called place value. The place value increased 10 times the position of number when moving right to left.

Hexadecimal number is also called the **base-16** number or sometimes called Hex. This is the combination of Hexa (6) and decimal (10). Hexadecimal number is the numbering technique to represent any number in the form of base 16, in the hexadecimal uses 0-9 and then uses A to F . there are only 16 symbols or digits used to represent the hexadecimal number. These 16 symbols and digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A is equal to 10, B is equal to 11 and so on till F is equal to 16.

Hexadecimal number system used to represent large value binary numbers. The hex number is much easier to read or express than binary and decimal.

1. Make groups of 4 digits from right to left (In fractional-left to right after the point). If numbers are not in groups of 4 digits then add extra 0 left end (also right end in fractional number) to make groups of 4

2. Find the equivalent hexadecimal number for each group

3. Write the hexadecimal of each group as per groups order

1. Place the value of binary from octal number of each digits using binary to octal table.

2. Now, make pairs of 4 of binary digits. If required more digits then place extra 0 on right end of numbers to make pair of 4.

3. Place the value of Hexa of each pair using the binary to hexadecimal Table.

• Many error codes and other values of codes are represented by hexadecimal on the computer. Error and any warning codes on blue screen in computers represented generally in hex format. Hex codes are shorter than binary and decimal so the programmer uses this.

• In the programming of web designing a specific Color-code is also represented in HTML format using the Hex codes. For example – hex value FF0000 to define the color red.

• Programmers use Hex codes in the coding of a program because of the value of Hex codes are shorter than the decimal or binary.

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