# Complex Number Formulas

Simplify any complex expression easily by having a glance at the Complex Number Formulas. The Formulae list provided for Complex Numbers can be of extreme help during your calculations. You can arrive at the solutions easily with simple steps instead of lengthy calculations.

## List of Complex Numbers Formulae

To meet your requirements we have curated the list of complex number formulae below. You can solve the toughest problems too easily and quickly using the simpe formulas for Complex Numbers. These will defnitely help you cut through the hassle of doing lengthy calculations of your math problems.

1. Real number system

- Natural Number (N): N = {1,2,3, …………..}
- Whole Number (W): W = {0, 1, 2, …………..} = {N} + {0}
- Integers (Z or I): Z or I = {………-3, -2, -1, 0, 1, 2, 3, ………}
- Rational Numbers (Q): The numbers which are in the form of p/q (Where p, q ∈ I, q ≠ 0)
- Irrational Numbers: The numbers which are not rational i.e. which can not be expressed in p/q form or whose decimal part is non terminating non repeating but which may represent magnitude of physical quantities. e.g. \(\sqrt{2}\), 5
^{1/3}, π, e, ………etc. - Real Numbers (R): The set of Rational and Irrational Number is called as set of Real Numbers i.e. N ⊂ W ⊂ Z ⊂ Q ⊂ R

2. Imaginary Number

x = ± \(\sqrt{-1}\) is imaginary and \(\sqrt{-1}\) = i (iota)

3. Integral power of iota

i = \(\sqrt{-1}\) so i^{2} = -1; i^{3} = -i and i^{4} = 1

Hence i^{4n+1} = i; i^{4n+2} = -1; i^{4n+3} = -i; i^{4n} or i^{4n+4} = 1

4. Complex Number

A number of the form z = x + iy where x, y ∈ R and i = \(\sqrt{-1}\) is called a complex number where x is called as real part and y is called imaginary part of complex number and they are expressed as

Re (z) = x, Im (z) = y, | z | = \(\sqrt{x^{2}+y^{2}}\); amp (z) = arg (z) = θ = tan^{-1} = \(\frac{y}{x}\)

- Polar representation: x = r cos θ, y = r sin θ & r = \(\sqrt{x^{2}+y^{2}}\) = |z|
- Exponential Form : z = re
^{iθ} - Vector representation: P (x, y) then its vector representation is z = \(\overrightarrow{\mathrm{OP}}\)

5. Properties of Conjugate Complex Number

If z = a + ib

- \(\overline{(\bar{z})}\) = z
- z + \(\bar{z}\) = 2a = 2 Re (z) = purely real
- z – \(\bar{z}\) = 2ib = 2i Im (z) = purely imaginary
- z \(\bar{z}\) = a
^{2}+ b^{2}= | z |^{2} - z + \(\bar{z}\) =0 or z = – \(\bar{z}\) ⇒ z = 0 or z is purely imaginary
- z = \(\bar{z}\) ⇒ z is purely real

6. Properties of modulus of a Complex Number

- If z = x + iy then |z| = \(\sqrt{x^{2}+y^{2}}\)
- “|z|” is distance of any point “z” on argand plane from origin
- |z
_{1}z_{2}…….z_{n}| = |z_{1}|. |z_{2}|. |z_{3}| ……… |z_{n}|, if z_{1}= z_{2}= …. = z_{n}then |z^{n}| = |z|^{n} - \(\left|\frac{z_{1}}{z_{2}}\right|=\frac{\left|z_{1}\right|}{\left|z_{2}\right|}\), where |z
_{2}| ≠ 0 - ||z
_{1}| – |z_{2}|| ≤ |z_{1}– z_{2}| ≤ |z_{1}| + |z_{2}| - z \(\bar{z}\) = |z|
^{2} - z
^{-1}= \(\frac{\bar{z}}{|z|^{2}}\) - |z
_{1}± z_{2}|^{2}= |z_{1}|^{2}+ |z_{2}|^{2}± 2Re (z_{1}\(\bar{z}_{2}\)) - |z
_{1}+ z_{2}|^{2}+ |z_{1}– z_{2}|^{2}= 2[|z_{1}|^{2}+ |z_{2}|^{2}]

7. Properties of argument of a Complex Number

- For any complex number z = x + iy,

arg(z) or amp(z) = tan^{-1}\(\left(\frac{y}{x}\right)\) or amp(z) = tan^{-1}\(\left[ \frac { Im(z) }{ Re(z) } \right] \) - For any complex number.z, -π ≤ amp(z) ≤ π
- amp (any real positive number) = 0
- amp (any real negative number) = π
- amp (z – \(\bar{z}\)) = ± π/2
- amp (z
_{1}. z_{2}) = amp (z_{1}) + amp (z_{2}) - amp \(\left(\frac{z_{1}}{z_{2}}\right)\) = amp (z
_{1}) – amp (z_{2}) - amp (\(\bar{z}\)) = – amp (z) = amp (1/z)
- amp (- z) = amp (z) ± n
- amp (z
^{n}) = n amp (z) - amp (z) + amp (\(\bar{z}\)) = 0

8. Square root of a complex number

The square root of z = a + ib is

\(\sqrt{a+i b}\) = ± \(\left[\sqrt{\frac{|z|+a}{2}}+i \sqrt{\frac{|z|-a}{2}}\right]\) for b > 0 and

= ± \(\left[\sqrt{\frac{|z|+a}{2}}-i \sqrt{\frac{|z|-a}{2}}\right]\) for b < 0.

9. Triangle Inequalities

- |z
_{1}± z_{2}| ≤ |z_{1}| + |z_{2}| - |z
_{1}± z_{2}| ≥ |z_{1}| – |z_{2}|

Condition of equality, equality sign holds if z_{1}, z_{2} and origin are colinear.

10. Some important points

(i) If ABC is an equilateral triangle having vertices z_{1}, z_{2}, z_{3} then z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1} or \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}\) = 0

(ii) If z_{1}, z_{2}, z_{3}, z_{4} are vertices of parallelogram then z_{1} + z_{3} = z_{2} + z_{4}

(iii) Amplitude of a complex number:

The amplitude or argument of a complex number z is the inclination of the directed line segment representing z, with real axis. The amplitude of z is generally written as amp z or arg z, thus if x = x + iy then amp z = tan^{-1}(y/x).

(iv) While finding the solution of equation of form x^{2} + 1 = 0, x^{2} + x + 1 = 0, the set of real number was extended into set of complex numbers. First of all ‘Euler’ represented \(\sqrt{-1}\) by the symbol i and proved that the roots of every algebraic equation are number of the form a + ib where a, b ∈ R. A number of this form called complex Number.

(v) Distance formulae:

The distance between two points P(z_{1}) and Q(z_{2}) is given by

PQ = |z_{2} – z_{1}|

= |affix of Q – affix of P|

Section formula:

If Re(z) divides the line segment joining P(z_{1}) and Q(z_{2}) in the ratio m_{1} : m_{2} (m_{1}, m_{2} > 0)

Then,

(a) For internal division z = \(\frac{m_{1} z_{2}+m_{2} z_{1}}{m_{1}+m_{2}}\)

(b) For external division z = \(\frac{m_{1} z_{2}-m_{2} z_{1}}{m_{1}-m_{2}}\)

(vi) Some particular locus

(a) Equation of the line joining complex number z_{1} and z_{2} is z = tz_{1} + (1 – t)z_{1}, t ∈ R or \(\frac{z-z_{1}}{z_{2}-z_{1}}\) = \(\frac{\bar{z}-\bar{z}_{1}}{\bar{z}_{2}-\bar{z}_{1}}\) = \(\left|\begin{array}{lll}z & \bar{z} & 1 \\

z_{1} & \bar{z}_{1} & 1 \\z_{2} & \bar{z}_{2} & 1\end{array}\right|\) = 0

(b) General equation of a line in complex plane \(\bar{a} z+a \bar{z}+b\) = 0 where b ∈ R and a is a fixed non zero complex number.

(c) Equation of circle in central form is |z – z_{0}| = r where z_{0} is the center and r is the radius of the circle further on squaring, we get

|z – z_{0}|^{2} = r^{2} ⇒ (z – z_{0}) \(\left(\bar{z}-\bar{z}_{0}\right)\) = r^{2}

⇒ \(z \bar{z}+z_{0} \bar{z}_{0}-z \bar{z}_{0}-\bar{z} z_{0}\) = r^{2}

(d) General equation of a circle:

\(z \bar{z}+a \bar{z}_{0}-z \bar{z}_{0}-\bar{z} z_{0}\) = r^{2} where b ∈ R and a is fixed complex number. For this circle, centre is the points a and radius = \(\sqrt{|a|^{2}-b}\)

11. Let z_{1}, z_{2} are two complex no.’s then at complex plane

(i) |z_{1} – z_{2}| is distance between two complex no.’s

(ii) then z = \(\frac{m\left(z_{2}\right) \pm n\left(z_{1}\right)}{m \pm n}\) “+” for internal division, “-” for external division

(iii) Let “z” is any variable point then |z – z_{1}| + |z – z_{2}| = 2a where |z_{1} – z_{2}| < 2a then locus of z is an ellipse, z_{1}, z_{2} are two foci.

(iv) If |z – z_{1}| – |z – z_{2}| = 2a where |z_{1} – z_{2}| > 2a then z describes a hyperbola where z_{1}, z_{2} are two foci.

(v) \(\left|\frac{z-z_{1}}{z-z_{2}}\right|\) = k is a circle if k ≠ 1 and is a line if k = 1

(vi) z discribes a circle if |z – z_{1}|^{2} + |z – z_{2}|^{2} = k where \(\frac{1}{2}\)|z_{1} – z_{2}|^{2} ≤ k

(vii) Let a is any complex number then for z = x + iy

\(\overline{\mathrm{a}} z+\mathrm{a} \bar{z}\) = k, where k is any real number represents equation of straight line. Real slope is \(\left(-\frac{a}{\bar{a}}\right)\) and complex slope is \(\left[ -\frac { Re(a) }{ Im(a) } \right] \).

(viii) Circle may given in any one of following manner

- |z – z
_{0}| = k where z_{0}is centre and k(real no.) is radius - \(z \bar{z}+a \bar{z}+\bar{a} z+k\) = 0, (kis real) with centre “-a” and radius \(\sqrt{|a|^{2}-k}\)
- (z – z
_{1}) \(\left(\bar{z}-\bar{z}_{2}\right)\) + (z – z_{2}) \(\left(\bar{z}-\bar{z}_{1}\right)\) = 0 where z_{1}& z_{2}are ends of diameter

(ix) \(\frac{z_{1}-z_{3}}{z_{1}-z_{2}}=\frac{\left|z_{1}-z_{3}\right|}{\left|z_{1}-z_{2}\right|}\) e^{iα}.

(x) Greatest and least value of |z| if \(\left|z+\frac{1}{z}\right|\) = a is

\(\frac{a+\sqrt{a^{2}+4}}{2}\) and \(\frac{-a+\sqrt{a^{2}+4}}{2}\)

### FAQs on Complex Number Formulas

**1. How do you solve Complex Number Expressions?**

You can solve Complex Number Expressions easily by taking the help of Fomulas and then simplify accordingly.

**2. Where do I get Collection of Complex Number Formulae?**

You can get Collection of Complex Number Formulae

**3. How do Complex Number Formulas help you?**

Complex Numbers Formulas help you solve difficult problems of Complex Numbers too easily and makes your job easy.

**4. How to memorize Complex Number Formulas?**

Best way to memorize Complex Number Formulas is through consistent practice as it is the only way to get good grip of them.