Revision Notes on Complex Numbers

• z₁ = a + ib and z₂ = c + id then z₁ = z₂ implies that a = c and b = d.

• i² = –1, i³ = – i, i⁴ = 1, i⁴ⁿ⁺¹ = i, i⁴ⁿ⁺² = – 1, i⁴ⁿ⁺³ = – i, i⁴ⁿ = 1

• z₁ + z₂ = (Re(z₁) + Re (z₂)) + i (Im (z₁) + Im (z₂))

• z₁ – z₂ = (Re(z₁) – Re(z₂)) + i (Im (z₁) - Im (z₂))

• Re (z₁. z₂) = Re (z₁). Re(z₂) – Im(z₁). Im (z₂)

  Im (z₁. z₂) = Im(z₁). Re(z₂) + Im(z₂). Re(z₁)

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• Square root of a complex number:

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• Argument of a Complex Number:

1. Argument of a complex number p(z) is defined by the angle which OP makes with the positive direction of x-axis.

2. Argument of z generally refers to the principal argument of z (i.e. the argument lying in (–π, π) unless the context requires otherwise.

• General argument: General value of argument = 2nπ + θ; where θ is one of the argument of z.

• Amplitude (Principal value of argument):

z = a + ib, then calculate ∝ = tan^-1 |b/a|. 

Then consider the following cases:

Case I: I quadrant

amp (z) = θ  = ∝

Case II: II quadrant

amp (z) = θ = (π – ∝)

Case III: III quadrant

amp (z) = θ = – (π – ∝)

Case IV: IV quadrant

amp (z) = θ = – ∝

• ?Least  Positive argument: Value of θ such that 0 < θ ≤ 2π

• If OP = |z| and arg (z) = θ, then obviously z = r (cos θ + i sin θ) and is called the polar form of complex number z.  

• |(z-z₁) / (z-z₂)| =1 the locus of point representing z is the perpendicular bisector of line joining z₁ and z₂.

• -|z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|

• If a and b are real numbers and z₁ and z₂ are complex numbers then |az₁ + bz₂|² + |bz₁ - az₂|² = (a² + b²) (|z₁|² + (|z₂|²)

• The distance between the complex numbers z₁ and z₂ is given by |z₁ - z₂|.

• In parametric form, the equation of line joining z₁ and z₂ is given by z = tz₁ + (1 – t)z₂.

• If A(z₁) and B(z₂) are two points in the argand plane, then the complex slope μ of the straight line AB is given by μ, where μ =

• Two lines having complex slopes μ₁ and μ₂ are:

1. Parallel iff μ₁ = μ₂

2. Perpendicular iff μ₁ = - μ₂ or μ₁ + μ₂ = 0

• If A(z₁), B(z₂), C(z₃) and D(z₄) are four points in the argand plane, then the angle θ between the lines AB and CD is given by θ = arg{(z₁ - z₂)/ (z₃ – z₄)}

• Some basic properties of complex numbers:

I. ||z₁| - |z₂|| = |z₁ + z₂| and |z₁ - z₂| = |z₁| + |z₂| iff origin, z₁, and z₂ are collinear and origin lies between z₁ and z₂.

II. |z₁ + z₂| = |z₁| + |z₂| and ||z₁| - |z₂|| = |z₁ - z₂| iff origin, z₁ and z₂ are collinear and z₁ and z₂ lie on the same side of origin.

III.amp (z₁ z₂) = amp (z₁) + amp (z₂) + 2kπ, k ∈ I

IV. amp(zⁿ) = n amp z

V. The least value of |z - a| + |z - b|  is |a - b|.

• Demoivre's Theorem: The theorem can be stated in two forms:

Case I: If n is any integer, then

(i) (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

(ii) (cos θ₁ + i sin θ₁) . (cos θ₂ + i sin θ₂) ......... (cos θ_n + i sin θ_n) = cos (θ₁ + θ₂ + θ₃ ^{.................. +} θ_n) + i sin (θ₁ + θ₂ + ..............  θ_n)

Case II: For p and q such that q ≠ 0, we have

(cos θ + i sin θ)^p/q = cos((2kπ + pθ)/q) + i sin((2kπ + pθ/q)  where k = 0,1,2,3,....., q-1

• Demoivre’s formula does not hold for non-integer powers.

• Main application of Demoivre’s formula is in finding the n^th roots of unity. So, if we write the complex number z in the polar form then,

z = r(cos x + isin x)

Then z^1/n = [r (cos x + i sin x)]^1/n

              = r ^1/n [ cos (x + 2kπ/n) + i sin (x + 2kπ/n)]

Here k is an integer. To get the n different roots of z one only needs to consider values of k from 0 to n – 1.

• Cube roots of unity

  • 1, ω, ω ² are the 3 cube roots of unity

  • 1 + ω + ω ² = 0 & ω ³ = 1

  • 1 + ω ^r + ω ^2r = 0 if r is not multiplier of 3

                                  or 3 if r is multiple of 3

• n, nth roots of unity 1, α, α² . . . . αⁿ are n nth roots of unity, then 1 + α + α² + . . . . . + α^{n – 1} = 0.

• Continued product of the roots of a complex quantity should be determined using theory of equations.

• The only complex number with modulus zero is the number (0, 0).

• The following figures illustrate geometrically the meaning of addition and subtraction of complex numbers:

• Different forms of equation of circle:

  • (i) |z| = r, r ∈ R⁺ then locus of z represent a circle whose centre is the origin and radius is equal to r.

  • (ii) |z – z₀| = r, r ∈ R⁺ then locus of z represents a circle whose centre is z₀ and radius is equal to r. 

  • (iii) Equation   represents a circle whose centre is z0 and radius is equal to r.

• The formulae for centroid, incentre and orthocentre are as given below: