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A number system is a representation of numbers that are used for counting or measuring objects according.It is only the mathematical representation of numbers. Numbers are used to performing arithmetic calculations. We can represent various numbers such as natural numbers, complex numbers, whole numbers etc. in various forms.

**Types of number system:- **There are various types of number system as under:

As the name indicates, decimal means 10. So a decimal system is a number system having base-10. It only uses 10 digits which are 0,1,2,3,4,5,6,7,8,9. These are a total 10 digits. In a decimal system the successive positions to the left of the decimal represents ones, tens, hundreds, thousands, one thousands, and so on. In the decimal number system, every position is represented as a power of the base 10. For example,

In number system, 247 is represented as:-

- 247 = (2×10^2)+(4×10^1)+(7×10^0)

** ** = 200+40+7

** **** ** =247

- In number system 3838, can be represented as :-

** **3838 = (3×10^3)+(8×10^2)+(3×10^1)+(8×10^0)

** ** = 3000+800+30+8

** ** = 3838

As the name indicates binary means 2. So a number is said to be represented in the binary number system if it has base 2. The figures represented using this number system are known as binary numbers. For example, in binary number system 14 is represented as:-

** **

14 = (1110)^2

As the name indicates, octal means 8. So the octal number system is a system which has base 8 and it uses only 8 digits which are 0,1,2,3,4,5,6,7 i.e. 0 to 7. For example:

- 235^8
- 346^8
- 560^8

In these examples, all the digits are from 0 to 7 so these numbers are represented in the octal number system.

In the hexadecimal system, the numbers are represented with the base 16. The hexadecimal system uses 10 digits and 6 alphabets to represent a number with base 16. The 10 digits used in this system are 0,1,2,3,4,5,6,7,8,9 and the alphabets used are A,B,C,D,E,F. So in total these are 16.

**Que 1:- How 10 is represented in the binary number system?**

(1101)^2

(1010)^2

(1111)^2

(1100)^2

**Ans **Option 2 is the correct option

**Que 2:- How can 340 be represented in the decimal number system?**

(3×10^2)+(0×10^1)+(4×10^0)

(4×10^2)+(4×10^1)+(0×10^0)

(3×10^2)+(4×10^1)+(0×10^0)

(0×10^2)+(4×10^1)+(3×10^0)

**Ans **Option 3 is the correct option

**Que 3 Convert (110)****^2 into the decimal number system.**

14

12

18

22

**Ans **Option 2 is the correct option.

Numbers and different number systems play a very important role in our daily life. Everything we do from looking at the wall clock first thing in the morning to having only 2 toasts in the morning breakfast to writing complex computer software programs depends on the numbers.

We learn the concepts of numbers starting with whole numbers and numerical operations with base ten as early as elementary school. Students typically in 3rd and 4th grade may learn fractions as part of their Math curriculum. Also middle school students start their journey with a number system and understand different types of numbers & their applications such as plotting a positive and negative number on a number line, and differentiating between a rational number and irrational numbers. As students advance their grades, they also continue to learn new concepts and by the time they reach high school, students start to deal with math problems related to imaginary numbers & complex numbers.

Not only this binary system is used widely by college level students in their study of computer science and engineering. Therefore, it is crucial for one to have a strong understanding & clarity on the concept of number system.

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**Question 1: ****What is the formula of the number system?**

Formulas are used for solving number systems. There are several type of formulas below:

**Basic algebraic formula:**

**Formula for number series:**

**Divisibility of Number:**

- If the unit digit of a number is any of 0, 2, 4, 6, or 8, the number is divisible by 2.
- If the sum of a number's digits is divisible by 3, the number is divisible by 3.
- If the last two digits of a number are divisible by 4, the number is divisible by 4.
- If the unit digit of a number is either 0 or 5, it is divisible by 5.
- If a number can be divided by both 2 and 3, it is divisible by 6.
- If a last three digit of the number is divisible by 8, it is divisible by 8
- If the sum of a number's digits is divisible by 9, the number is divisible by 9.
- If the number ends with zero, it is divisible by 10.
- If the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a value divisible by 11, the number is divisible by 11.

**Question 2: ****What is a number system with examples?**

A number system is the system in which numbers are expressed based on their relationship and the laws govern them. Hindu-Arabic numerals 0 to 9 digits are used in the number system.

There are various type of number system:

- Decimal Number system. (Base 10)
- Binary Number system. (Base 2)
- Octal Number system. (Base 8)
- Hexadecimal Number system. (Base 16)

For Example:

**Question 3: ****What is the base 5 number system?**

The base 5 number system is the numeral system with five as a base. In this number system, numbers from 0 to 4 are used to represent any real number.

In this case, place values are powers of 5: 5**^**0 = 1, 5**^**1 = 5, 5**^**2 = 25, 5**^**3 = 125, 5**^**4=625 representing first, second, third and fourth place.

Example: represent 63 in base 5.

First we divide 63/25=2, 13 as a remainder

Now, we divide 13/5=2, 3 as a remainder

Again, 3/1=3

Thus, 223 is the 5 base number system of 63.

Check:

= 2(5**^**2) + 2(5**^**1) + 3(5**^**0)

** ** = 2(25) + 2(5) + 3(1)

** ** =50+10+3

** ** = 63

**Question 4: ****Who invented the number system?**

There have been many number systems created. In modern days we use the hindu-arabic number system, which was developed in India, then extended to Arabic-speaking populations in the Middle East, then to Europe and the rest of the globe. The number system was complete after the discovery of zero. Aryabhata, who invented place value in the 5th century, has played the most important role in the formation of the number system with base 10. So, he is known as the father of the number system.

**Question 5: ****Why are rules required for a number system to be useful?**

The number system is a system of representing numbers in the form of digits. Most commonly used number system is the decimal number system. Rules are required in order to understand the number in the system and without rules we are unable to understand the mathematical computation used in the system to represent the number.

For example:

If we write 29, we will not understand until we know the rule used in it.

We considered it as 2 & 9 two digits only, but it is 29 (twenty-nine) by using the decimal number system (base 10).

**Question 6: **** What is the real number system?**

Real number is the set of numbers that contains both rational and irrational numbers. All the numbers present on the number line is a real number and all the mathematical operations are done on a real number. All the numbers are the subset of the real numbers. Real number is denoted by ‘R’.

Example:

2,-3,2/3,0.002,√3 all are real numbers.

**Question 7: ****What is an octal number system?**

The Octal number system is a numeral system with a base eight number system. It uses 0 to 7 eight digits to represent the decimal number. Octal number systems are mostly used in computer applications.

Example:

Convert in octal number system

**Question 8: ****What is a binary number system?**

Binary number system is a number system with base two. 0 and 1 digits are used to represent the decimal number system. Computers use binary systems to perform tasks.

Example:

Convert 141 in binary system:

141/2 = 70, remainder = 1

70/2=35, remainder = 0

35/2=17, remainder = 1

17/2=8, remainder = 1

8/2=4, remainder = 0

4/2=2, remainder = 0

2/2=1, remainder = 0

1/2= 0, remainder = 1

So, Binary number for 141 is 10001101** **