We have dealt with calculating natural numbers in primary classes like addition, subtraction, division, and multiplication. The calculation becomes more accessible when the numbers are alike, but when the numbers are complex, then the methods of the calculator vary. The topic complex number is introduced in class XII for the first time. In this chapter, we have dealt with the numbers that do not exist and are imaginary. 

This chapter will learn about the complex number- definition, standard form, algebraic expression, conjugate, and polar form. A complex number is the combination of the real and the imaginary number.

Complex Numbers- Definition and formula

In the previous classes, we have learned that finding the square root of a negative number is possible, but it involves using a new term, ‘i’, which is equal to the square root of negative 1. 

For example, 3i is called an imaginary number and could be ‘bi’ where ‘b’ is an actual number. 

Here, ‘3’ is an actual number, and ‘i’ is an imaginary number and is equal to the square root of negative 1. 

Let us take another example; We have a number like 3+7i

Here 3 is the real number, and 7i is an imaginary number. The above number is an example of a complex number. 

Application of Complex Number

There are various applications of complex numbers used by us in daily life:-

• Engineers use it to study stresses on beams
• Phenomena of resonance
• Fluid flows around the objects, such as water around the pipe
• The flow of current through electronic circuits using resistance, inductance, and capacitors
• They are also used in electromagnetism

Complex Number Formulas

Arithmetical operations in complex numbers such as subtraction, division, multiplication, and addition are performed differently. As complex numbers are the combination of real and imaginary numbers, the methods differ from the operations of natural numbers. The different ways to perform procedures are given below:-

Addition

(x + iy) + (w + iz) = (x + w) + i(y+ z)

Subtraction

(x + iy) – (w + iz) = (x – w) + i(y- z)

Multiplication

In the multiplication of two complex numbers, the multiplication process might be the same as the two binomial numbers. The FOIL rule  (which means Distributive multiplication process) is used while performing the multiplication of these numbers.

(x + iy). (w + iz) = (xw – yz) + i(xz + yw)

Division

In the division of 2 complex numbers, the multiplication of the numerator and denominator is done by its conjugate value of the denominator, and after that FOIL method is applied.

(x + iy) / (w + iz) = (xw+yz)/ (w2+z2 )+ i(yw – xz) / (w2+z2 )

Integral powers of Iota

Above, we have understood the concept of an imaginary unit number “i.” It is the square root of -1. So if we square it, we will get -1. So the square of the unit imaginary number is negative 1. Well, let’s now try to work out more powers of “i”

Iota is an imaginary number denoted by the symbol “i,” and the value of Iota was √-1, which is i = √−1. When solving quadratic equations, you might face some of the problems where the value of the discriminant is negative. 

Let us take the example of a quadratic equation,

x2+ x + 1 = 0 

We apply the quadratic formula to determine this value and we get the resultant or discriminant negative value.

In those cases, we wrote √−3 as √−3 = (√−1 ×3). By this method, the answer of the aforementioned quadratic equation is x = (−1 ± √3i)/2.

Hence, the value of the Iota helps find square roots with the negative values.

Powers of Iota

As we know, the powers of the Iota, “i” repeat itself after a particular cycle pattern. So let’s begin with evaluating the value of powers of the Iota for primary cases & try to find out the way of it. For this, we will take  the value of Iota as, i= √−1.

Square of Iota 

We are well aware that the value of the Iota, i

is determined by, i = √−1. So, if we square both sides of the above equation, we get: i2 = -1, i.e., the value of the square of Iota is -1. Therefore, the square of Iota is i2= −1.

Algebraic Operations on Complex Numbers

Like natural numbers, we can also perform algebraic operations on complex numbers; the various functions are described below:

• In addition to the two Complex values 

             z1+z2 =( a1+a2 )+i( b1+b2 )

• Difference or Subtraction of two Complex values

    z1-z2 = (a1-a2)+i(b1-b2)

• Product or Multiplication of two Complex values

    z1 z2 = (a1 a2 – b1b2 ) + i(a1 b2 + a2 b1 )

• Division of 2 Complex values

Conjugate of Complex Number

In every complex number, another complex number is linked with it, known as the complex conjugate. It is another complex number that has the same real part as the original complex number. The imaginary part has the same magnitude and opposite sign or indication. The multiplication of a complex number and complex conjugate is a real number. 

Complex conjugate forms the mirror image of complex numbers on the real axis or horizontal axis in the Argand plane. In this chapter, we have explored the definition of the conjugate of the complex number, complex root theorem, its properties, some applications, and the use of complex conjugate.

Conclusion

Above, we have studied complex numbers. A complex number is different from real numbers, and the calculation method also differs. In this topic, we have read about the integral powers of Iota, algebraic operations and conjugate of complex numbers. Iota is an imaginary number denoted by the symbol “i,” and the value of Iota was √-1, which is i = √−1. In every complex number, another complex number is linked with it, known as the complex conjugate. It is another complex number that has the same real part as the original complex number. The imaginary part has the same magnitude and opposite sign or indication. 

It is advised to go through the complete notes of this chapter as it is considered the most important topic of mathematics for IIT JEE. 2020, around six questions of more than 18 marks are asked from this topic.