Complex Number Formulas

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

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

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3. Integral power of iota
i = \(\sqrt{-1}\) so i² = -1; i³ = -i and i⁴ = 1
Hence i⁴ⁿ⁺¹ = i; i⁴ⁿ⁺² = -1; i⁴ⁿ⁺³ = -i; i⁴ⁿ or i⁴ⁿ⁺⁴ = 1

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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 = \(\frac{y}{x}\)

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5. Properties of Conjugate Complex Number
If z = a + ib

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6. Properties of modulus of a Complex Number

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7. Properties of argument of a Complex Number

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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.

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9. Triangle Inequalities

Condition of equality, equality sign holds if z₁, z₂ and origin are colinear.

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10. Some important points
(i) If ABC is an equilateral triangle having vertices z₁, z₂, z₃ then z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁ or \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}\) = 0

(ii) If z₁, z₂, z₃, z₄ are vertices of parallelogram then z₁ + z₃ = z₂ + z₄

(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 x² + 1 = 0, x² + 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(z₁) and Q(z₂) is given by
PQ = |z₂ – z₁|
= |affix of Q – affix of P|

Section formula:
If Re(z) divides the line segment joining P(z₁) and Q(z₂) in the ratio m₁ : m₂ (m₁, m₂ > 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 z₁ and z₂ is z = tz₁ + (1 – t)z₁, 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 – z₀| = r where z₀ is the center and r is the radius of the circle further on squaring, we get
|z – z₀|² = r² ⇒ (z – z₀) \(\left(\bar{z}-\bar{z}_{0}\right)\) = r²

⇒ \(z \bar{z}+z_{0} \bar{z}_{0}-z \bar{z}_{0}-\bar{z} z_{0}\) = r²

(d) General equation of a circle:
\(z \bar{z}+a \bar{z}_{0}-z \bar{z}_{0}-\bar{z} z_{0}\) = r² where b ∈ R and a is fixed complex number. For this circle, centre is the points a and radius = \(\sqrt{|a|^{2}-b}\)

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11. Let z₁, z₂ are two complex no.’s then at complex plane

(i) |z₁ – z₂| 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 – z₁| + |z – z₂| = 2a where |z₁ – z₂| < 2a then locus of z is an ellipse, z₁, z₂ are two foci.

(iv) If |z – z₁| – |z – z₂| = 2a where |z₁ – z₂| > 2a then z describes a hyperbola where z₁, z₂ 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 – z₁|² + |z – z₂|² = k where \(\frac{1}{2}\)|z₁ – z₂|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

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

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.

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

You can get Collection of Complex Number Formulae from Onlinecalculator.guru

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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.

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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.

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