Revision Notes on Complex Numbers

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

• If we have a complex number z where z = a+ib, the conjugate of the complex number is denoted by z* and is equal to a-ib. In fact, for any complex number z, its conjugate is given by z* = Re(z) – Im(z).

• Division of complex numbers: The numerator as well as denominator should first be multiplied by the conjugate of the denominator and then simplified.  

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.

3. Hence, the argument of the complex number z = a + ib = r (cos θ + i sin θ) is the value of θ satisfying r cos θ = a and r sin θ = b.

4. The angle θ is given by θ = tan^-1 |b/a|.

5. The value of argument in various quadrants is given below:

• 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 = (z₁ - z₂)/ (₁ - ₂).

• 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. The product of n^th roots of any complex number z is z(-1)^n-1.

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 + pq)/q) + isin((2kπ+pq/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 sinx )]^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.

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

• The modulus of a complex number is given by |z| = √x²+y².

• 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: