Number System in Complex Numbers

The system of naming or otherwise portraying numbers is referred to as the number system or the numeral system. It is common knowledge that a number is a mathematical value that can be used to count or measure things, in addition to assisting in the execution of a variety of mathematical operations. In mathematics, there are many distinct kinds of number systems, such as the binary number system, the octal number system, the hexadecimal number system, and the decimal number system. 

Number system in maths: 

A system of writing that is used to express numbers is what is referred to as a number system. It is the notation used in mathematics to represent numbers that belong to a particular set in a standardised way by employing digits or other symbols in place of those numbers. It not only depicts the mathematical and algebraic structure of the figures, but it also offers a unique representation of each and every number. In addition to that, we are able to do arithmetic operations such as addition, subtraction, and division because of it.

The following factors can be used to assess the significance of any digit in a number: 

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

What is a number? 

A mathematical value that can be used for counting, measuring, or labelling things is referred to as a number. Arithmetic operations are carried out by using numbers in various capacities. Natural numbers, whole numbers, rational and irrational numbers, and other types of numbers are all examples of numbers. A null value can alternatively be represented by the integer 0, which stands for nothing. 

A number can exist in a variety of other forms, including prime and composite numbers, even and odd numbers, and even and odd numbers. If a number is divisible by two or not, it is referred to as even, and if it is not, it is referred to as odd. Prime and composite are terms that are used to differentiate between numbers that have only two factors and numbers that have more than two factors, respectively. 

Types of number system: 

In the field of mathematics, there are many distinct kinds of number systems. The following are the four most popular forms of number systems: 

1. Decimal number system (Base- 10)
2. Binary number system (Base- 2)
3. Octal number system (Base-8)
4. Hexadecimal number system (Base- 16) 

Decimal Number System (Base 10 Number System): 

Because it employs ten digits, ranging from 0 to 9, the decimal number system is said to have a base of 10. The locations that are successively moved to the left of the decimal point in the decimal number system denote different amounts of units, tens, hundreds, thousands, and so on. The numbers used in this system are given in decimal form. Every position illustrates a different facet of the base’s power (10). 

Binary Number System (Base 2 Number System): 

The number system with a base of 2 is also referred to as the Binary number system. In this system, there are only two binary digits, which are represented by the numbers 0 and 1. To be more specific, the typical base-2 system uses a radix of 2. The numbers that are described using this approach are referred to as binary numbers, and they are formed by combining the digits 0 and 1. One example of a binary number is the sequence 110101.

Any system can be converted into binary, and binary may be converted into any system. 

Octal Number System (Base 8 Number System): 

In the octal number system, eight is the basis, and the digits 0 through 7 are used to denote numbers in this system. Applications that run on computers frequently make use of octal numbers. Performing an octal conversion on a number is identical to performing a decimal conversion. 

Hexadecimal Number System (Base 16 Number System): 

Hexadecimal is a numbering system that uses base 16 as the foundation for written or represented integers. First, the numbers are represented in the hexadecimal system in the same manner as they are in the decimal system, that is, from 0 to 9. After then, the numbers are written using the letters from A through F of the alphabet. 

Complex number: 

A real number and an imaginary number make up a complex number, which is essentially the combination of the two. The formula for the complex number is a+ib, where a represents a real number and ib represents an imaginary number. Additionally, a and b are both real numbers, and the value of i = √-1. 

A complex number is an easy way to represent the addition of two numbers, namely a real number and an imaginary number. This is because a complex number is simply the addition of a real number and an imaginary number. It can be broken down into two categories: the purely real and the purely imagined parts. 

Real numbers: 

Real numbers are all numbers that are present in a number system, whether they be positive, negative, zero, integer, rational, irrational, fractions, or any other type of number. It is denoted by the symbol Re (). Real numbers include the following: 12, -45, 0; 1/7; 2.8; √5; and so on. 

Imaginary numbers: 

Imaginary numbers are any numbers that do not exist in the real world. When we square an imaginary number, the result that we get is a negative value. It is denoted by the symbol Im (). As an illustration, the numbers √-2, √-7, and √-11 are all examples of imaginary numbers.

We represent the value √-1 using the symbol i which stands for the letter iota (Imaginary number). 

Rational numbers: 

In mathematics, the term “rational number” refers to any number that can be expressed in the form p/q for situations in which q is greater than 0. Also, we can argue that any fraction belongs to the category of rational numbers, which are numbers in which both the denominator and the numerator are integers, and the denominator does not equal zero. When a rational number, also known as a fraction, is divided, the result will be in decimal form. The decimal form that is produced can either be terminating or repeating, depending on the circumstances. 

Conclusion: 

Mathematical number systems are the structures that allow for the expression of numbers in a variety of different formats that are comprehensible to computers. A mathematical value that may be used for counting and measuring things, as well as for conducting computations in arithmetic, is referred to as a number.