Complex Numbers
Complex Numbers DEFINITION: Complex numbers are definited as expressions of the form a + ib where a, b ∈ R & i = \(\sqrt { -1 } \) . It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as
imaginary part of z (Im z).
Every Complex Number Can Be Regarded As
Purely real Purely imaginary Imaginary
If b = 0 If a = 0 If b ≠ 0
Note:
- The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
- Zero is both purely real as well as purely imaginary but not imaginary.
- i = \(\sqrt { -1 } \) is called the imaginary unit. Also i² = −1 ; i3 = −i ; i4 = 1 etc.
- \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) only if atleast one of either a or b is non-negative.
Conjugate Complex | Complex Numbers
If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z. i.e. \(\bar { z } \) = a − ib.
Note:
- z + \(\bar { z } \)= 2 Re(z)
- z − \(\bar { z } \)= 2i Im(z)
- z \(\bar { z } \) = a² + b² which is real
- If z lies in the 1st quadrant then \(\bar { z } \) lies in the 4th quadrant and \(\bar {-z } \) lies in the 2nd quadrant.
Algebraic Operations | Complex Numbers
The algebraic operations on complex numbers are similar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.
e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless .
However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers,
z12 + z22 = 0 does not imply z1 = z2 = 0.
Equality in Complex Number
Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real & imaginary parts coincide.
Representation of Complex Number in Various Forms
- Cartesian Form (Geometric Representation):
Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y). length OP is called modulus of the complex number denoted by |z| & θ is called the argument or amplitude
e.g. |z| = \(\sqrt {x^{2} + Y^{2}}\)
θ = tan−1\(\frac {y}{x}\)
(angle made by OP with positive x−axis)
Note:
1. |z| is always non-negative. Unlike real numbers \(\left| z \right| = \left[ \begin{array}{ccc}{\mathbf{z}} & {\text { if }} & {\mathrm{z}>0} \\ {-\mathbf{z}} & {\text { if }} & {\mathbf{z}<0}\end{array}\right.\) is not correct
2. Argument of a complex number is a many valued function . If θ is the argument of a complex number then 2 nπ + θ ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ.
3. The unique value of θ such that – π < θ ≤ π is called the principal value of the argument.
4. Unless otherwise stated, amp z implies principal value of the argument.
5. By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.
6. There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers. - Trignometric / Polar Representation:
z = r (cos θ + i sin θ) where | z | = r ; arg z = θ ; \(\bar {z}\) = r (cos θ − i sin θ)
Note: cos θ + i sin θ is also written as CiS θ.
Also \(\cos x=\frac{e^{i x}+e^{-i x}}{2} \& \sin x=\frac{e^{i x}-e^{-i x}}{2}\) - Exponential Representation:
z = reiθ ; | z | = r ; arg z = θ ; \(\bar {z}\) = re-iθ
Important Properties of Conjugate/ Moduli/ Amplitude | Complex Numbers
If z , z1 , z2 ∈ C then ;
- z + \(\bar {z}\) = 2 Re (z) ; z − \(\bar {z}\) = 2 i Im (z) ; \(\overline{(\overline{z})}=\mathbf{z}\) ; \(\overline{z_{1}+z_{2}}=\overline{z}_{1}+\overline{z}_{2}\) ;
\(\overline{z_{1}-z_{2}}=\overline{z}_{1}-\overline{z}_{2}\) ; \(\overline{z_{1} z_{2}}=\overline{z}_{1} \cdot \overline{z}_{2}\) ; \(\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\overline{z}_{1}}{\overline{z}_{2}}\) ; z2 ≠ 0 - |z1 + z2|2 + |z1 – z2|2 = 2 [|z1|2 + |z2|2]
||z1| − |z2|| ≤ |z1 + z2| ≤ |z1| + |z2| - (i) amp (z1 . z2) = amp z1 + amp z2 + 2 kπ. k ∈ I
(ii) amp \(\frac {z_{1}}{z_{2}}\) = amp z1 – amp z2 + 2kπ; k ∈ I
(iii) amp(zn) = n amp(z) + 2kπ .
where proper value of k must be chosen so that RHS lies in (− π , π ].
Vectorial Representation Of A Complex Number
Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then, \(\overrightarrow{\mathrm{OP}} = z\) & |\(\overrightarrow{\mathrm{OP}}\)| = |z|
Note:
- If \(\overrightarrow { OP }\)= z = reiθ then \(\overrightarrow { OQ }\) = z1 = rei(θ + φ) = z . eiφ. If \(\overrightarrow { OP }\) and \(\overrightarrow { OQ }\) are of unequal magnitude then φ \(\hat{\mathrm{OQ}}=\hat{\mathrm{OP}} \mathrm{e}^{\mathrm{i} \theta}\)
- If A, B, C & D are four points representing the complex numbers z1, z2, z3 & z4 then \(\mathrm{AB}| | \mathrm{CD} \quad \text { if } \quad \frac{\mathrm{z}_{4}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{1}}\) is purely real ;
\(A B \perp C D \text { if } \frac{z_{4}-z_{3}}{z_{2}-z_{1}}\) is purely imaginary ]
- If z1, z2, z3 are the vertices of an equilateral triangle where z0 is its circumcentre then (a) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{1} z_{2}-z_{2} z_{3}-z_{3} z_{1}=0\) (b) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\)
Demoivre’S Theorem
Statement: cos nθ + i sin nθ is the value or one of the values of (cos θ + i sin θ)n ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity
Note: Continued product of the roots of a complex quantity should be determined using theory of equations.
Cube Root Of Unity | Complex Numbers
- The cube roots of unity are 1, \(\frac{-1 + i\sqrt {3}}{2}, \frac{-1 – i\sqrt{3}}{2}\)
- If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general 1 + wr + w²r = 0 ; where r ∈ I but is not the multiple of 3.
- In polar form the cube roots of unity are:
\(\cos 0+i \sin 0 ; \cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}, \quad \cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\) - The three cube roots of unity when plotted on the Argand plane constitute the vertices of an equilateral triangle.
- The following factorisation should be remembered:
(a, b, c ∈ R & ω is the cube root of unity)
a3 − b3 = (a − b) (a − ωb) (a − ω²b); x2 + x + 1 = (x − ω) (x − ω2);
a3 + b3 = (a + b) (a + ωb) (a + ω2b);
a3 + b3 + c3 − 3abc = (a + b + c)(a + ωb + ω²c)(a + ω²b + ωc)
nth Roots Of Unity | Complex Numbers
If 1 ,1 ,α2 , α3 ….. αn − 1 are the n, nth root of unity then:
- They are in G.P. with common ratio ei(2π/n) &
- \(1^{\mathrm{p}}+\alpha_{1}^{\mathrm{p}}+\alpha_{2}^{\mathrm{p}}+\ldots\ldots+\alpha_{\mathrm{n}-1}^{\mathrm{p}}=0\) if p is not an integral multiple of n
= n if p is an integral multiple of n - (1 − α1) (1 − α2) …… (1 − αn – 1) = n &
(1 + α1) (1 + α2) ……. (1 + αn − 1) = 0 if n is even and 1 if n is odd. - 1 . α1 . α2 . α3 ……… αn − 1 = 1 or −1 according as n is odd or even.
The Sum Of The Following Series Should Be Remembered:
- \(\cos \theta+\cos 2 \theta+\cos 3 \theta+\ldots \ldots+\cos n \theta=\frac{\sin (n \theta / 2)}{\sin (\theta / 2)} \cos \left(\frac{n+1}{2}\right) \theta\)
- \(\sin \theta+\sin 2 \theta+\sin 3 \theta+\ldots \ldots+\sin n \theta=\frac{\sin (n \theta / 2)}{\sin (\theta / 2)} \sin \left(\frac{n+1}{2}\right) \theta\)
Straight Lines & Circles In Terms Of Complex Numbers:
- If z1 & z2 are two complex numbers then the complex number z =mn
divides the joins of z1 & z2 in the ratio m : n.
Note:
(i) If a , b , c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z1, z2 & z3 are collinear.
(ii) If the vertices A, B, C of a ∆ represent the complex nos. z1, z2, z3 respectively, then:
(a) Centroid of the ∆ ABC = \(\frac{z_{1}+z_{2}+z_{3}}{3}\)
(b) Orthocentre of the ∆ ABC = \(\frac{(a \sec A) z_{1}+(b \sec B) z_{2}+(c \sec C) z_{3}}{a \sec A+b \sec B+c \sec C}\) OR \(\frac{\mathrm{z}_{1} \tan \mathrm{A}+\mathrm{z}_{2} \tan \mathrm{B}+\mathrm{z}_{3} \tan \mathrm{C}}{\tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}}\)
(c) Incentre of the ∆ ABC = (az1 + bz2 + cz3) ÷ (a + b + c)
(d) Circumcentre of the ∆ ABC = :
(Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ÷ (sin 2A + sin 2B + sin 2C) - amp(z) = θ is a ray emanating from the origin inclined at an angle θ to the x− axis.
- |z − a| = |z − b| is the perpendicular bisector of the line joining a to b.
- The equation of a line joining z1 & z2 is given by;
z = z1 + t (z1 − z2) where t is a parameter. - z = z1 (1 + it) where t is a real parameter is a line through the point z1 & perpendicular to oz1.
- The equation of a line passing through z1 & z2 can be expressed in the determinant form as
\(\left| \begin{matrix} z & \bar { z } & 1 \\ { z }_{ 1 } & \bar { { z }_{ 1 } } & 1 \\ { z }_{ 2 } & \bar { { z }_{ 2 } } & 1 \end{matrix} \right| = 0\)
This is also the condition for three complex numbers to be collinear. - Complex equation of a straight line through two given points z1 & z2 can be written as
\(z\left(\overline{z}_{1}-\overline{z}_{2}\right)-\overline{z}\left(z_{1}-z_{2}\right)+\left(z_{1} \overline{z}_{2}-\overline{z}_{1} z_{2}\right)=0\) which on manipulating takes the form as \(\overline{\alpha} \mathrm{z}+\alpha \overline{\mathrm{z}}+\mathrm{r}=0\) where r is real and α is a non zero complex constant. - The equation of circle having center z0 & radius ρ is: |z − z0| = ρ or
\(z \overline{z}-z_{0} \overline{z}-\overline{z}_{0} z+\overline{z}_{0} z_{0}-\rho^{2}=0\) which is of the form \(\mathrm{zz}+\overline{\alpha} z+\alpha \overline{z}+r=0\) , r is real centre − α & radius
\(\sqrt{\alpha \overline{\alpha}-r}\). Circle will be real if \(\alpha \overline{\alpha}-r \geq 0\). - The equation of the circle described on the line segment joining z1 & z2 as diameter is:
\(\arg \frac{\mathrm{z}-\mathrm{z}_{2}}{\mathrm{z}-\mathrm{z}_{1}}=\pm \frac{\pi}{2} \quad \text { or }\left(\mathrm{z}-\mathrm{z}_{1}\right)\left(\overline{\mathrm{z}}-\overline{\mathrm{z}}_{2}\right)+\left(\mathrm{z}-\mathrm{z}_{2}\right)\left(\overline{\mathrm{z}}-\overline{\mathrm{z}}_{1}\right)=0\) - Condition for four given points z1, z2, z3 & z4 to be concyclic is, the number \(\frac{\mathrm{z}_{3}-\mathrm{z}_{1}}{\mathrm{z}_{3}-\mathrm{z}_{2}}\cdot\frac{\mathrm{z}_{4}-\mathrm{z}_{2}}{\mathrm{z}_{4}-\mathrm{z}_{1}}\). is real. Hence the equation of a circle through 3 non collinear points z1, z2 & z3 can be taken as \(\frac {(\mathrm{z}-\mathrm{z}_{2})(\mathrm{z}_{3}-\mathrm{z}_{1})}{(\mathrm{z}-\mathrm{z}_{1})(\mathrm{z}_{3}-\mathrm{z}_{2})}\) is real ⇒ \(\frac {(\mathrm{z}-\mathrm{z}_{2})(\mathrm{z}_{3}-\mathrm{z}_{1})}{(\mathrm{z}-\mathrm{z}_{1})(\mathrm{z}{3}-\mathrm{z}_{2})} = \frac { \left( \bar { z } -\bar { { z }_{ 2 } } \right) \left( \bar { { z }_{ 3 } } -\bar { { z }_{ 1 } } \right) }{ \left( \bar { z } -\bar { { z }_{ 1 } } \right) \left( \bar { { z }_{ 3 } } -\bar { { z }_{ 2 } } \right) } \)
Reflection points for a straight line: Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z1 & z2 will be the reflection points for the straight line \(\overline{\alpha} z+\alpha \overline{z}+r=0\) if and only if; \(\overline{\alpha} z_{1}+\alpha \overline{z}_{2}+r=0\) where r is real and α is non zero complex constant.
Inverse points w.r.t. a circle:
Two points P & Q are said to be inverse w.r.t. a circle with center ‘O’ and radius ρ, if :
- the point O, P, Q are collinear and on the same side of O.
- OP. OQ = ρ2.
Note that the two points z1 & z2 will be the inverse points w.r.t. the circle
\(z \overline{z}+\overline{\alpha} z+\alpha \overline{z}+r=0\) if and only if \(z_{1} \overline{z}_{2}+\overline{\alpha} z_{1}+\alpha \overline{z}_{2}+r=0\).
Ptolemy’s Theorem:
It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides.
i.e. |z1 − z3| |z2 − z4| = |z1 − z2| |z3 − z4| + |z1 − z4| |z2 − z3|.
Logarithm Of A Complex Quantity
- Loge(α + iβ) = ½Loge(α² + β²) + i( 2nπ + tan−1\(\frac {\beta}{\alpha}\) where n ∈ I.
- ii represents a set of positive real numbers given by \(\mathrm{e}^{-\left(2 \mathrm{n} \pi+\frac{\pi}{2}\right)}\)