Number System-Definition, Types, And Examples

Number systems are a way of representing numbers using different symbols and are understood by computers. A number is a mathematical value used for counting, measuring objects, and performing arithmetic calculations.

There are various categories of numbers like natural numbers, whole numbers, rational and irrational numbers, etc. Similarly, there are various types of number systems that have different properties. Some examples include the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

We will explore different types of number systems that we use in this article, such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

What is Number System in Math?

A number system is a way to write numbers using symbols.  In mathematics, set notation is used to represent a given set of numbers. It provides a unique representation for every number in the set, allowing us to perform arithmetic operations such as addition, subtraction, and division. The set notation also reveals the algebraic structure of the numbers, making it a valuable tool for solving mathematical problems.

The value of any given digit in a number can be determined by three factors:

• The digit itself
• Its position in the number
• The base of the number system

But before discussing the different types of number system examples, let’s first define what a number is.

What is a Number?

A number is a mathematical value used for counting, measuring, or labeling objects. Numbers are used to performing arithmetic calculations. Examples of numbers include natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value.

A number can be classified in several ways, such as even and odd, or prime and composite. Even and odd terms are used to describe whether a number is divisible by 2. Prime numbers are those that have only two factors, while composite numbers have more than two factors.

There are different types of number systems that use digits to represent numbers. The most common number system is the binary system, which uses 0 and 1 as digits. Other number systems use 0 to 9 digits.

For example, the common base-10 system uses the ten digits from 0 through 9. In contrast, binary representations use two digits (0 and 1). Other number systems exist that use a different number of digits.

Types of Number System

There are four main types of number systems:

• The decimal system, which is used in most everyday situations

• The binary system, which is used in computer programming

• The hexadecimal system, which is used in some types of data storage

• The octal system, which is used in some fields of mathematics

Decimal Number System (Base- 10 Number system)

The decimal number system is the most common one used to represent numbers in everyday life. It uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with the base number being 10. If any number is represented without a base specified, it is assumed to have a base of 10.

For example, 10285 has place values as

(1 × 10⁴) + (0 × 10³) + (2 × 10²) + (8 × 10¹) + (5 × 10⁰)

1 × 10000 + 0 × 1000 + 2 × 100 + 8 × 10 + 5 × 1

10000 + 0 + 200 + 80 + 5

10285

Binary Number System (Base 2 Number System)

The binary number system is a numbering system that uses two digits, 0 and 1, to represent numbers. Binary numbers can be used in electronic devices and computer systems because they can be easily represented using just two states, ON and OFF, which correspond to 0 and1. Decimal Numbers 0-9 are represented in binary as: 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001

For example:

14 can be written as 1110

19 can be written as 10011

50 can be written as 110010

Hexadecimal Number System

The hexadecimal system is a way of writing or representing numbers with base 16. In the hex system, the numbers are first represented just like in the decimal system, from 0 to 9. Then, the numbers are represented using the letters A to F. The table below shows the representation of numbers in the hexadecimal system:

Hexadecimal	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

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 fewer digits than several other systems, hence there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system.

The octal number system is used in minicomputers with digits from 0 to 7. For example, 35₈, 23₈, and 141₈ are numbers in the octal number system.

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