Complex numbers

Introduction

We already know about the real numbers. These are the numbers that can be represented on the number line. We also know that the real can be classified into broad categories as Rational and Irrational numbers. Rational numbers can be defined as the ratio of two integers with no common factor. On the other hand, irrational numbers cannot be expressed as the ratio of two integers and can be expressed in decimal notations. This is because they are non-terminating and non-recurring decimals.

This chapter will discuss the complex number definition, roots, and formulas.

Complex numbers Definition 

In the previous chapter, we learned about the square root of negative numbers by using the term “i” equal to the square of negative 1. These numbers are called imaginary numbers. According to the theoretical perspective, Complex numbers definition is the set of all numbers in the form a+ib where a is an actual number and bi is an imaginary number. 

Real numbers are simply positive, negative, whole, or decimal numbers. Therefore, any number which you think exists is an actual number. 

Imaginary numbers are those that do not exist, and we only assume them to solve mathematical problems like the square of “i.”

Examples of complex numbers are -2+4i, where -2 is an actual number and 4i is an imaginary number.

Difference Between Real and Imaginary Numbers

Any number contained in a number system like positive, fractions, irrational, rational, integer, zero, harmful, etc., are numbers that are real. It’s represented as Re(). So, for example, twelve, 45, zero, 1/7, 2.8, √5, etc., tend to be accurate numbers.

Imaginary numbers solve mathematical problems of complex numbers and quadratic equations. The symbol x = ±√-1, denotes it. We denote √-1 with the symbol’ i’, known as iota.

Complex Number Formulas

We all know that we perform various mathematical operations on numbers like addition, subtraction, division, and multiplication of two natural numbers. However, there are some alternative ways to perform these operations on complex numbers, and we can’t use these operations directly on complex numbers. So, here are the ways to apply mathematical operations to complex numbers. 

•   Addition– if we add two complex numbers (a+bi) and (c+di), we will get one in real form and one in complex form

(a+ib)+(c+id)=(a+c)+i(b+d)

• Subtraction– if we do the subtraction between two complex numbers, we will get the same results as before but the sign will be minus in between the real and complex ones

(a+ib)-(c+id)=(a-c)+i(b-d)

• Equality– if two complex numbers are equal like (a+bi) =(c+di), then we can conclude that a=c and b=d
• Multiplication– The multiplication formula of complex numbers is 

(a+ib) × (c+id) = (ac-bd) + i(ad+bc)

• Multiplication of Conjugates- the formula of multiplication of conjugates is 

(a+bi)(a-bi) = a^2+b^2.

 It is essential among Complex Number Formulas.

• Division– The division formula of complex numbers is

a+bi/c+di  =(a+bi)(c-di)/(c+di)(c-di)   =(ac+bd)+(bc-ad)i/c^2+d^2   =ac+bd/c^2+d^2+(bc-ad)i/c^2+d^2

Polar Form to find Roots of Complex Numbers

Let’s quickly recap what we have studied in the previous chapter, a complex number A plus Bi can be plotted in two ways. First, A units to the right and B units up. Lets us assume that OB has R units and this angle from the positive x-axis is . So, the second way to represent point P is R, if you are looking at this right triangle , there are three length a,b and r. Also,angle POA is Θ and another angle is 90 degree. Now, let’s focus on the triangle, to determine the length of r in terms of a and b.

As this is a right triangle, Pythagoras Theorem can be applied in this:-

OP=r=|z|=a^2+b^2

The square of the hypotenuse is equal to the sum of the square of the base and square of the perpendicular.

Taking the square root of both side we can get the value:-

OP=|z|=a^2+b^2

So, here the above term is known as the modulus of the complex number z. If the complex number z is equal to a plus bi, then the modulus of this complex number set is the square root of a(square) plus b(square), it is also known as the absolute value of the complex number. 

Here, is known as the argument of the complex number. 

Conclusion

In this chapter we have studied the complex numbers, definition, roots of complex numbers and formulas. The arithmetical operation in complex numbers is different from the real numbers. In this chapter we have also learnt how to perform addition, subtraction, multiplication and division on the complex number. We understand that complex numbers are the set of all numbers in the form a+bi where a is a real number and bi is an imaginary number. This topic is very important for the examination, especially for the IIT JEE Entrance examination. This topic is correlated with Sets, relation & function and power set.  So, it is advised to not escape this topic and focus on making your fundamentals strong.