write a detailed study on the topic number system.
Answers
Answer:
The number system is simply a system to represent or express numbers. There are various types of number systems and the most commonly used ones are decimal number system, binary number system, octal number system, and hexadecimal number system.
Answer:
A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction and division.
The value of any digit in a number can be determined by:
The digit
Its position in the number
The base of the number system.
Types of Number System
There are various types of number system in mathematics. The four most common number system types are:
Decimal number system (Base- 10)
Binary number system (Base- 2)
Octal number system (Base-8)
Hexadecimal number system (Base- 16)
Decimal Number System (Base 10 Number System)
Decimal number system has base 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on. This system is expressed in decimal numbers.
Every position shows a particular power of the base (10). For example, the decimal number 1457 consists of the digit 7 in the units position, 5 in the tens place, 4 in the hundreds position, and 1 in the thousands place whose value can be written as
(1×103) + (4×102) + (5×101) + (7×100)
(1×1000) + (4×100) + (5×10) + (7×1)
1000 + 400 + 50 + 7
1457
Binary Number System (Base 2 Number System)
The base 2 number system is also known as the Binary number system wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.
We can convert any system into binary and vice versa.
Example
Write (14)10 as a binary number.
Solution:

Base 2 Number System Example
∴ (14)10 = 11102
Octal Number System (Base 8 Number System)
In the octal number system, the base is 8 and it uses numbers from 0 to 7 to represent numbers. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.
Example: Convert 2158 into decimal.
Solution:
2158 = 2 × 82 + 1 × 81 + 5 × 80
= 2 × 64 + 1 × 8 + 5 × 1
= 128 + 8 + 5
= 14110
Hexadecimal Number System (Base 16 Number System)
In the hexadecimal system, numbers are written or represented with base 16. In the hex system, the numbers are first represented just like in decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabets from A to F. The below-given table shows the representation of numbers in the hexadecimal number system.
Hexadecimal0123456789ABCDEFDecimal0123456789101112131415
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Number System Chart
In the number system chart, the base values and the digits of different number system can be found. Below is the chart of the numeral system.

Number System Chart
Number System Conversion
Numbers can be represented in any of the number system categories like binary, decimal, hex, etc. Also, any number which is represented in any of the number system types can be easily converted to other. Check the detailed lesson on the conversions of number systems to learn how to convert numbers in decimal to binary and vice versa, hexadecimal to binary and vice versa, and octal to binary and vice versa using various examples.
Solved Examples
Q.1: Convert (1056)16 to octal number.
Solution: Given, 105616 is an hex number.
First we need to convert the given hexadecimal number into decimal number
(1056)16
= 1 x 163 + 0 x 162 + 5 x 161 + 6 x 160
= 4096 + 0 + 80 + 6
= (4182)10