# Complex Numbers

**Complex Numbers** are the numbers of the form **(a + i b)** where * a* &

*are the real numbers and*

**b***is an imaginary unit called iota that represents √-1. For example, 2 + 3i, is a complex number in which 2 is a real number and 3i is an imaginary number.*

**i**### Need for Complex Numbers

Initially, we only have the set of natural numbers (**N**) which is extended to form a set of integers (**I**) as the equation (x + a = b) is not solvable for a>b where a, b ∈ N.

Then this set of integers (I) is extended to a set of rational numbers (**Q**) as every equation of the form (x.a = b) is not uniquely solvable where a ≠ 0 and a, b ∈ I.

Again this set of integers (Q) is extended to a set of real numbers (rational & irrational) (**R**) as every equation of the form (x^{2} = a) is not solvable where a > 0 and a ∈ Q i.e x^{2} = 5 because there is not any rational number whose square is 5. Hence, the notion of irrational numbers is given to numbers like √2, √3, √5, etc.

Finally, we extended this set of real numbers (R) to a set of complex numbers (**C**) as the equation of the form (x^{2} = a) is not solvable where a < 0 and a ∈ R i.e x^{2} + 5 = 0 because there is not any real number whose square is -5. Hence, the notion of complex (imaginary) numbers is given to numbers like √-1, √-2, √-3, √-5, etc. where √-1 is called imaginary unit (iota) and represented by the symbol * i *(i

^{2}= -1).

### Classification of Complex Numbers

As we know the standard form of a complex number is * z = (a + i b)* where a, b ∈ R, and

*i*is iota (an imaginary unit). So depending on the values of a (

*called real part*) and b (

*called imaginary part*), the complex numbers are classified into four types:

1. Zero Complex Number(a = 0 & b = 0)Example: 0 (zero)

2. Purely Real Numbers(a ≠ 0 & b = 0)Examples: 2, 3, 7, etc.

3. Purely Imaginary Numbers(a = 0 & b ≠ 0)Examples: -7i, -5i, -i, i, 5i, 7i, etc.

4. Imaginary Numbers(a ≠ 0 & b ≠ 0)Examples: (-1 – i), (1 + i), (1 – i), (2 + 3i), etc.

### Different Forms of Complex Numbers

1. Rectangular Formalso calledStandard Form =(a + i b)Examples: (5 + 5i), (-7i), (-3 – 4i), etc.

2. Polar Form =r [cos(θ) + i sin(θ)]Examples: [cos(π/2) + i sin(π/2)], 5[cos(π/6) + i sin(π/6)], etc.

3. Exponential Form =Examples: e

^{i(0)}, e^{i(π/2)}, 5.e^{i(π/6)}, etc.

**NOTE: **All three forms of the complex numbers discussed above are interconvertible.

### The Complex Plane

The plane on which the complex numbers are uniquely represented is called the *Complex plane or Argand plane or Gaussian plane*.

The Complex plane has two axes:

1. X-axis

- All the purely real complex numbers are uniquely represented by a point on it.
- Real part Re(z) of all complex numbers are plotted with respect to it.
- That’s why X-axis is also called
Real axis.

2. Y-axis

- All the purely imaginary complex numbers are uniquely represented by a point on it.
- Imaginary part Im(z) of all complex numbers are plotted with respect to it.
- That’s why Y-axis is also called
Imaginary axis.

### Geometrical Representation of Complex Numbers

As we know that every complex number *(z = a + i b)* is represented by a unique point *p(a, b) *on the complex plane and every point on the complex plane represents a unique complex number.

To represent any complex number z = (a + i b) on the complex plane follow these conventions:

- Real part of z (Re(z) = a) becomes the X-coordinate of the point p
- Imaginary part of z (Im(z) = b) becomes the Y-coordinate of the point p
And finally z (a + i b) ⇒ p (a, b) which is a point on the complex plane.

**Example 1: Plot these complex numbers z _{ }= 3 + 2 i on the Complex plane.**

**Solution:**

Given:

z

_{ }= 3 + 2 iSo, the point is z(3, 2). Now we plot this point on the below graph, here in this graph x-axis represents the real part and y-axis represents the imaginary part.

**Example 2: Plot these complex numbers z _{1 }= (2 + 2 i), z_{2 }= (-2 + 3 i), z_{3} = (-1 – 3 i), z_{4} = (1 – i) on the Complex plane.**

**Solution:**

Given:

z

_{1 }= (2 + 2 i)z

_{2 }= (-2 + 3 i)z

_{3}= (-1 – 3 i)z

_{4}= (1 – i)So, the points are z

_{1 }(2, 2), z_{2}(-2, 3), z_{3}(-1, -3), and z_{4}(1, -1). Now we plot these points on the below graph, here in this graph x-axis represents the real part and y-axis represents the imaginary part.

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