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# Number Systems

**Number systems** are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

In this article, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will learn the conversions between these number systems and solve examples for a better understanding of the concept.

## What are Number Systems?

A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

### Number Systems Definition

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

## Types of Number Systems

There are different types of number systems in which the four main types are as follows.

- Binary number system (Base – 2)
- Octal number system (Base – 8)
- Decimal number system (Base – 10)
- Hexadecimal number system (Base – 16)

We will study each of these systems one by one in detail after going through the following number system chart.

### Number System Chart

Given below is a chart of the main four types of number system that we use to represent numbers.

## Binary Number System

The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 10001_{2}, 111101_{2}, 1010101_{2} are some examples of numbers in the binary number system.

Octal Number System

The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just as the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example: 35_{8}, 23_{8}, 141_{8} are some examples of numbers in the octal number system.

Decimal Number System

The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example: 723_{10}, 32_{10}, 4257_{10} are some examples of numbers in the decimal number system.

Hexadecimal Number System

The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B3_{16}, 6F_{16}, 4B2A_{16} are some examples of numbers in the hexadecimal number system.

Conversion Rules of Number Systems

A number can be converted from one number system to another number system using number system formulas. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa, and so on. Let us see the steps required in converting number systems.

### Steps for Conversion of Binary to Decimal Number System

To convert a number from the binary to the decimal system, we use the following steps.

**Step 1:**Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.**Step 2:**The exponents should start with 0 and increase by 1 every time we move from right to left.**Step 3:**Simplify each of the above products and add them.

Let us understand the steps with the help of the following example in which we need to convert a number from binary to decimal number system.

**Example:** Convert 100111_{2} into the decimal system.

**Solution:**

**Step 1:** Identify the base of the given number. Here, the base of 100111_{2} is 2.

**Step 2:** Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 2^{0}, 2^{1}, 2^{2} , and so on from right to left.

**Step 3:** We just simplify each of the above products and add them.

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.

100111 = (1 × 2^{5}) + (0 × 2^{4}) + (0 × 2^{3}) + (1 × 2^{2}) + (1 × 2^{1}) + (1 × 2^{0})

= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)

= 32 + 0 + 0 + 4 + 2 + 1

= 39

Thus, 100111_{2} = 39_{10}.

### Conversion of Decimal Number System to Binary / Octal / Hexadecimal Number System

To convert a number from the decimal number system to a binary/octal/hexadecimal number system, we use the following steps. The steps are shown on how to convert a number from the decimal system to the octal system.

**Example:** Convert 4320_{10} into the octal system.

**Solution:**

**Step 1:** Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

**Step 2:** Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.

**Step 3: **The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Therefore, 4320_{10} = 10340_{8}

### Conversion from One Number System to Another Number System

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

**Example:** Convert 1010111100_{2} to the hexadecimal system.

**Solution:**

**Step 1: **Convert this number to the decimal number system as explained in the above process.

Thus, 1010111100_{2} = 700_{10} → (1)

**Step 2:** Convert the above number (which is in the decimal system), into the required number system (hexadecimal).

Here, we have to convert 700_{10} into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively.

Thus, 700_{10} = 2BC_{16} → (2)

From the equations (1) and (2), 1010111100_{2} = 2BC_{16}

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