Complex Numbers DEFINITION: Complex numbers are definited as expressions of the form a + ib where a, b ∈ R & i = . 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
- 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 = is called the imaginary unit. Also i² = −1 ; i3 = −i ; i4 = 1 etc.
- 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. = a − ib.
- z + = 2 Re(z)
- z − = 2i Im(z)
- z = a² + b² which is real
- If z lies in the 1st quadrant then lies in the 4th quadrant and 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| =
θ = tan−1
(angle made by OP with positive x−axis)
1. |z| is always non-negative. Unlike real numbers 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 ; = r (cos θ − i sin θ)
Note: cos θ + i sin θ is also written as CiS θ.
- Exponential Representation:
z = reiθ ; | z | = r ; arg z = θ ; = re-iθ
Important Properties of Conjugate/ Moduli/ Amplitude | Complex Numbers
If z , z1 , z2 ∈ C then ;
- z + = 2 Re (z) ; z − = 2 i Im (z) ; ; ;
; ; ; 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 = 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, & || = |z|
- If = z = reiθ then = z1 = rei(θ + φ) = z . eiφ. If and are of unequal magnitude then φ
- If A, B, C & D are four points representing the complex numbers z1, z2, z3 & z4 then is purely real ;
is purely imaginary ]
- If z1, z2, z3 are the vertices of an equilateral triangle where z0 is its circumcentre then (a) (b)
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,
- 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:
- 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) &
- 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:
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.
(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 =
(b) Orthocentre of the ∆ ABC = OR
(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
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
which on manipulating takes the form as 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
which is of the form , r is real centre − α & radius
. Circle will be real if .
- The equation of the circle described on the line segment joining z1 & z2 as diameter is:
- Condition for four given points z1, z2, z3 & z4 to be concyclic is, the number . is real. Hence the equation of a circle through 3 non collinear points z1, z2 & z3 can be taken as is real ⇒
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 if and only if; 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
if and only if .
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 where n ∈ I.
- ii represents a set of positive real numbers given by