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## Introduction

A means to express and represent numbers is called a number system. The decimal system, which employs ten digits, is the most widely used number system. (0-9). Binary, octal, and hexadecimal are additional number systems that are often used in digital electronics and computers. These systems may be translated to and from decimal and have bases of 2, 8, and 16 correspondingly. Understanding various number systems is crucial for computer science and engineering as well as for comprehending the fundamental ideas of mathematics and computers.

## Decimal number system

The most popular number system, which employs 10 digits, is the decimal one. (0-9). It is a positional notation system, which means that each digit's value is determined by where it appears in the number. Units are represented by the first digit on the right, tens by the second, hundreds by the third, and so on. With a decimal point dividing the total from the fractional components, decimals may represent both integers and non-integers. It is simple to execute arithmetic operations and computations because decimal arithmetic adheres to well-established rules and conventions.

Each place of numerals indicates a respective degree of the base (10). For example, 4563 where 3 exemplify units’ place, 6 exemplify tens place, 5 exemplify hundreds place, and 4 exemplify thousands place. Its value can be written as

## Binary number system

In computers and digital electronics, a base-2 number system called the binary number system is employed. It is used to represent numbers, characters, and data and only has two digits, 0 and 1. A binary number's digits each represent a power of 2, with the rightmost digit standing in for 20 (1), the next for 21 (2), and so on. Since binary numbers are easily implemented in electrical circuits, they are essential to computers and digital electronics. Binary numbers may be translated to and from decimal and other number systems.

### Example

Write (160)10 as a binary number.

To convert a decimal number to binary, we can use the process of successive division by 2.

## Octal number system

In computers, the octal number system, which has a base of 8, is also widely used. It represents the numbers 0 through 7 using the digits 0 through 7. Each digit's place value is defined by its placement while counting from right to left. Divide the decimal value by 8, then write down the remainder in octal form until the quotient equals zero to convert it to octal. It is simple to switch between the two systems since each octal digit may be represented by three binary digits. However, because hexadecimal has become so popular, octal numerals are now less widely utilized.

The base-16 hexadecimal number system is widely used in computing. It represents the numbers 0–15 with the numerals 0–9 and the letters A–F, respectively. Each digit's place value is defined by its placement while counting from right to left. Divide the decimal value by 16 and then put the remainder in hexadecimal form until the quotient equals zero to convert a decimal number to hexadecimal. Four binary digits may be used to represent each hexadecimal digit, making it simple to convert between the two systems.

## Solved Example

1. Convert binary number (101101) into equivalent decimal number.

2. Convert the octal number 67 into an equivalent decimal number.

Q3. Convert the octal number 235 into an equivalent binary  number.

To convert an octal number to a binary number, we need to convert each octal digit to a binary digit.

The octal number 235 can be broken down as follows:

• 2 in octal is equivalent to 010 in binary

• 3 in octal is equivalent to 011 in binary

• 5 in octal is equivalent to 101 in binary

Therefore, the octal number 235 in binary is 010011101.

## Conclusion

The number system methods were primarily developed to prevent illicit use and misuse of crucial data.  All different numeral systems can be converted into one another using specific methods and forms. Simply put, the number system is a way to describe or symbolize numbers. There are many different kinds of number systems, but the decimal, binary, octal, and hexadecimal systems are the ones that are most frequently used.

Q1. Why is the Number System Important?

A number system helps in representing numbers with a limited collection of symbols. Computers typically use the binary digits 0 and 1 to simplify the calculation and reduce the amount of circuitry needed, which uses the least amount of room, energy, and money.

Q2. What is the name of the Base 1 Number System?

The simplest numeral system to describe natural numbers is the base-1 system, also known as the unary numeral system.

Q3.Why are numbers with a base of 10 referred to as decimal numbers?

Ans. This number system is also known as an arabic number system or a hindu arabic number system in mathematics.  The latin term decimus, derived from the base word ‘decem’ or 10, is where the word decimal first appeared.