Complex Numbers

Complex Numbers are the numbers of the form (a + i b) where a & b are the real numbers and i 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.

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 (x² = a) is not solvable where a > 0 and a ∈ Q i.e x² = 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 (x² = a) is not solvable where a < 0 and a ∈ R i.e x² + 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 (i² = -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 Form also called Standard 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: eⁱ⁽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 i

So, 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 z₁ = (2 + 2 i), z₂ = (-2 + 3 i), z₃ = (-1 – 3 i), z₄ = (1 – i) on the Complex plane.

Solution:

Given:

z₁ = (2 + 2 i)

z₂ = (-2 + 3 i)

z₃ = (-1 – 3 i)

z₄ = (1 – i)

So, the points are z₁ (2, 2), z₂(-2, 3), z₃(-1, -3), and z₄(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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