[Maths Class Notes] on Complex Numbers Pdf for Exam

The fundamental theorem of algebra tells us that every single variable equation of nth degree has n roots. That’s true! If we consider, it gives us the values 2 repeatedly. It means the curve crosses the point (2,0) two times. 

We have been solving many quadratic and cubic equations in our highschools. Do you remember what we did when we came across an equation like x² + 1 = 0 or like x³ + 1 = 0? According to the fundamental theorem, mentioned quadratic equation has 2 root and cubic equation have 3 roots. But where are they? If they would have been on the real line then we would have managed to get those values but they don’t. This problem of not getting all roots in real numbers leads to its extension.

Define Complex Numbers

As we discussed in the earlier section, to get all those roots that are not present in real numbers, we need to extend the real numbers. This extension is called complex numbers. If someone will ask you, what is complex numbers then simply it is an extension of real numbers that contains all the roots of a polynomial of degree n.

If we define i as the solution of the equation x² = -1 then the complex numbers are the set of numbers of the form a+ib. This set is represented as 

{a + ib la, b ∈ R}

We often use the variable z to represent complex numbers so z=a+ib. Here the number a represents the real part of the complex number and number b represents the imaginary part of the complex number. For example, consider z=7+3i. Here, 7 is independent of i so it’ll be the real part and 3 is the imaginary part. In any complex number, i is called iota. And its value is [sqrt{-1}].

i = [sqrt{-1}]

Definitions in Complex Numbers

• Equality of Complex Numbers

Two complex numbers are equal if and only if real and imaginary parts are equal. Consider complex numbers z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂. We say z₁ and z₂ are equal to each other if and only of a₁ = a₂ and b₁ = b₂.

• Purely Real Complex Numbers

A complex number is purely real if it’s imaginary part is zero. Consider z = a + ib as a complex number. z will be purely real if it’s imaginary part b=0.

• Purely Imaginary Complex Number

A complex number is purely imaginary if it’s real part is zero. Consider z = a + ib as a complex number. z will be purely imaginary if it’s real part a=0.

A complex number is called zero complex number if and only if its real and imaginary part simultaneously is zero. Consider z = a + ib as a zero complex number then a=0 and b=0.

This definition leads to an important point that needs to get address here. Remember that, every real number is a complex number because complex numbers are just an extension of real number but the converse is not true.

Graphical Representation of Complex Numbers

Graphically we represent on argand plane, which is also known as argand diagram or complex plane. It is similar to our conventional coordinate plane but there we have the x-axis here it’s the real axis. There we have the y-axis and here we have the imaginary axis. A point a+ib in the argand plane is represented as an ordered pair (a,b).

Mis-conception of Complex Number

Many students have the misconception that it is complicated which is why its name is complex. What are complex numbers actually then? If we define complex numbers in our own words then it means two different types of number, real and imaginary club together to form a complex number. Its just like a building complex where buildings joined together.

Powers of Iota (i)

Earlier we have discussed that the negative root of unity is called iota. That is i = [sqrt{-1}] 

Now let’s see some properties of it.

• i² = i x i = [sqrt{-1}] x [sqrt{-1}] = -1    

• i³ = i x i x i = [sqrt{-1}] x [sqrt{-1}] x [sqrt{-1}] = -1 x [sqrt{-1}] = -1 x i = -i

• i⁴ = i x i x i x i = [sqrt{-1}] x [sqrt{-1}] x [sqrt{-1}] x [sqrt{-1}] = (-1) x (-1) = 1  

It means, 

• i[^{n}] = i[^{4k}]  for somen n and k then i[^{n}] = 1

• i[^{n}] = i[^{4k+1}] for some n and k then i[^{n}] = i

• i[^{n}] = i[^{4k+2}]  for some n and k then i[^{n}] = -1

• i[^{n}] = i[^{4k+3}] for some n and k then i[^{n}] = -i

Addition of Complex Numbers

Let us consider z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are two complex numbers then the addition of two complex numbers can be defined as

z₁ + z₂ = (a₁ + ib₁) + (a₂ + ib₂)

⇒ z₁ + z₂ = a₁ + ib₁ + a₂ + ib₂

⇒ z₁ + z₂ = a₁ + a₂ + ib₁ + ib₂

⇒ z₁ + z₂ = (a₁ + a₂) + i(b₁ + b₂)

 Hence, 

z₁ + z₂ = (a₁ + a₂) + i(b₁ + b₂).

For example, consider two complex numbers as z₁ = 5 + i7 and z₂ = 2 + i3. On addition, they’ll give z₁ + z₂ = (5 + 2) + i(7 + 3) = 7 + i10 

Subtraction of Complex Numbers

Let us consider z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are two complex numbers then subtraction of two complex numbers can be defined as

z₁ – z₂ = (a₁ + ib₁) – (a₂ + ib₂)

⇒ z₁ – z₂ = a₁ + ib₁ – a₂ – ib₂

⇒ z₁ – z₂ = a₁ – a₂ + ib₁ – ib₂

⇒ z₁ – z₂ = (a₁ – a₂) + i(b₁ – b₂)

 Hence, 

z₁ – z₂ = (a₁ – a₂) + i(b₁ – b₂).

For example, consider two complex numbers as z₁ = 5 + i7 and z₂ = 2 + i3. On subtraction, they’ll give z₁ – z₂ = (5 – 2) + i(7 – 3) = 3 + i4

Product of Complex Numbers

Let u
s consider z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are two complex numbers then multiplication of two complex numbers can be defined as

z₁z₂ = (a₁ + ib₁) x (a₂ + ib₂) 

⇒ z₁z₂ = a₁a₂ + ia₁b₂ + ib₁a₂ + i²b₁b₂

⇒ z₁z₂ = a₁a₂ + ia₁b₂ + ib₁a₂ – b₁b₂ 

⇒ z₁z₂ = a₁a₂ – b₁b₂ + ia₁b₂ + ib₁a₂

⇒ z₁z₂ = a₁a₂ – b₁b₂ + i(a₁b₂ + b₁a₂)

 Hence, 

⇒ z₁z₂ = (a₁a₂ – b₁b₂) + i(a₁b₂ + b₁a₂)

For example, consider two complex numbers as z₁ = 5 + i7 and z₂ = 2 + i3. Their product will give us 

z₁z₂ = (5 x 2 – 7 x 3) + i(5 x 3 + 7 x 2)

⇒ z₁z₂ = (10 – 21) + i(15 + 14)

⇒ z₁z₂ = (-11) + i(29)

⇒ z₁z₂ = -11 + i29

Conjugate of a Complex Number

The conjugate of a complex number is also a complex number in the opposite imaginary direction on the argand plane. Consider the complex number z = a + ib. Its complex conjugate can be defined as z = a – ib. Arithmetically, we can get the complex conjugate of any complex number by just changing the sign of iota.

For example, consider the complex number z = 12 + i5. It complex conjugate will by z = 12 – i5.

Observe that the complex conjugate of a purely real number will be itself. Graphical representation of complex conjugates of -3 + 5i and 3 + 2i are mentioned below.

Division of Complex Numbers

Let us consider z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are two complex numbers then the division of two complex number can be computed by just rationalizing the complex number. In other words to get the division we just need to multiply and divide by the conjugate of the complex number.

Consider, [frac{z_{1}}{z_{2}}] = [frac{a_{1}+ib_{1}}{a_{2}+ib_{2}}]

On multiplying and divide by complex conjugate of denominator we get

[frac{z_{1}}{z_{2}}] = [frac{a_{1}+ib_{1}}{a_{2}+ib_{2}}] x [frac{a_{2}-ib_{2}}{a_{2}-ib_{2}}]

⇒ [frac{z_{1}}{z_{2}}] = [frac{(a_{1}a_{2} – b_{1}b_{2})+i(a_{1}b_{2}+b_{1}a_{2})}{a_{2}^{2}+ib_{2}^{2}}]

For example, consider 

[frac{z_{1}}{z_{2}}] = [frac{3+2i}{5-2i}]

On multiplying and divide by complex conjugate of denominator we get

[frac{z_{1}}{z_{2}}] = [frac{3+2i}{5-2i}] x [frac{5+2i}{5+2i}]

⇒ [frac{z_{1}}{z_{2}}] = [frac{3+2i}{5-2i}] x [frac{5+2i}{5+2i}]

⇒ [frac{z_{1}}{z_{2}}] = [frac{11+16i}{29}]

⇒ [frac{z_{1}}{z_{2}}] = [frac{11}{29}] + [frac{16}{29}]i