Detailed Notes: Complex Numbers | Mathematics (Maths) for JEE Main & Advanced PDF Download
If 'a', and 'b' are two real numbers, then a number of the form a + ib is called a complex number.
Set of complex Numbers: The set of all complex numbers is denoted by C.
i.e. C = {a + ib | a, b ∈ ℝ}
Equality of Complex Numbers: Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and b1 = b2, i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2).
Fundamental Operations on Complex Numbers
1. Addition
Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers.
Then their sum z1 + z2 is defined as the complex number (a1 + a2) + i (b1 + b2).
Properties of addition of complex numbers
(i) Addition is commutative: For any two complex numbers z1 and z2, we have
z1 + z2 = z2 + z1
(ii) Addition is associative: For any three complex numbers z1, z2, z3, we have
(z1 + z2) + z3 = z1 + (z2 + z3)
(iii) Existence of additive identity: The complex number 0 = 0 + i0 is the identity element for addition, i.e.,
z + 0 = z = 0 + z for all z ∈ C
(iv) Existence of additive inverse: For every complex number z, there exists -z such that
z + (-z) = 0 = (-z) + z
The complex number -z is called the additive inverse of z.
2. Subtraction
Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then the subtraction of z2 from z1 is denoted by z1 - z2 and is defined as the addition of z1 and -z2.
Thus,
z1 - z2 = (a1 - a2) + i (b1 - b2)
3. Multiplication
Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then, the multiplication of z1 with z2 is denoted by z1z2 and is defined as the complex number.
(a1a2 - b1b2) + i (a1b2 + a2b1)
Properties of Multiplication
(i) Multiplication is commutative: For any two complex numbers z1 and z2, we have
z1 z2 = z2 z1
(ii) Multiplication is associative: For any three complex numbers z1, z2, z3, we have
(z1 z2) z3 = z1 (z2 z3)
(iii) Existence of identity element for multiplication: The complex number 1 = 1 + i0 is the identity element for multiplication, i.e., for every complex number z, we have
z ⋅ 1 = z
(iv) Existence of multiplicative inverse: Corresponding to every non-zero complex number z = a + ib, there exists a complex number z1 = x + iy such that
z ⋅ z1 = 1 ⇒ z1 = 1 / z
The complex number z1 is called the multiplicative inverse or reciprocal of z and is given by:
z1 = (a / (a² + b²)) + i(-b / (a² + b²))
Did You Kmow
Multiplication of Complex Numbers is Distributive Over Addition
For any three complex numbers z1, z2, z3, we have:
(i) z1(z2 + z3) = z1z2 + z1z3 (Left distributivity)
(ii) (z2 + z3)z1 = z2z1 + z3z1 (Right distributivity)
4. Division
The division of a complex number z1 by a non-zero complex number z2 is defined as the multiplication of z1 by the multiplicative inverse of z2 and is denoted by (z1 / z2).
Thus,
z1 / z2 = z1 z2-1 = z1 (1 / z2)
Conjugate , Modulus and Argument of a Complex Number
1. Conjugate
Let z = a + ib be a complex number. Then the conjugate of z is denoted by z̅ and is equal to a - ib.
Thus, z = a + ib ⇒ z̅ = a - ib
Properties of Conjugate
If z, z1, z2 are complex numbers, then:
- z + z̅ = 2 Re(z)
- z - z̅ = 2 Im(z)
- z = z̅ ⇔ z is purely real
- z + z̅ = 0 ⇒ z is purely imaginary
- z z̅ = {Re(z)}² + {Im(z)}²
- z̅1 + z̅2 = z̅1 + z̅2
- z̅1 - z̅2 = z̅1 - z̅2
- z̅1 z2 = z̅1 z̅2
- (z1 / z2) z̅ = z̅1 / z̅2, z2 ≠ 0
- z̅(z) = z
2. Modulus
- The modulus of a complex number z = a + i b is denoted by |z| and is defined as
- The multiplicative inverse of a non-zero complex number z is same as its reciprocal and is given by
- If b is positive then
- If b is negative then
Properties of Modulus of z
- |z1 + z2|² = |z1|² + |z2|² + 2|z1||z2| cos(θ1 - θ2)
or
|z1 + z2|² = |z1|² + |z2|² + 2 Re(z1z̅2) - |z1 - z2|² = |z1|² + |z2|² - 2 |z1||z2| cos(θ1 - θ2)
or
|z1 - z2|² = |z1|² + |z2|² - 2 Re(z1z̅2) - |z1 + z2|² + |z1 - z2|² = 2 (|z1|² + |z2|²)
- |z1 + z2| = |z1 - z2| ⇒ arg(z1) - arg(z2) = π/2
- |z1 + z2| = |z1| + |z2| ⇔ arg(z1) = arg(z2)
- |z1 + z2|² = |z1|² + |z2|² ⇔ (z1 / z2) is purely imaginary.
- |z1 + z2| ≤ |z1| + |z2|
- |z1 - z2| ≤ |z1| + |z2|
- |z1 + z2| ≥ ||z1| - |z2||
- |z1 - z2| ≥ ||z1| - |z2||
- |z1 z2| = |z1| |z2|
- |zn| = |z|n
- |z|² = z z̅
- |z| = |z̅| = |-z| = |-z̅|
3. Argument
The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane.
It is denoted by “θ” or “φ”. It is measured in the standard unit called “radians”.
Argument or (amplitude) of a Complex Number
- If x and y both are positive, then the argument of z = x + iy is the acute angle given by: tan-1 |y/x|
- If x < 0 and y > 0, then the argument of z = x + iy is π - α, where α is the acute angle given by: tan-1 |y/x|
- If x < 0 and y < 0, then the argument of z = x + iy is α - π, where α is the acute angle given by: tan α = |y/x|
- If x > 0 and y < 0, then the argument of z = x + iy is -α, where α is the acute angle given by: tan α = |y/x|
Properties of Argument of z
- arg(z̅) = -arg(z)
- arg(z1z2) = arg(z1) + arg(z2)
- arg(z1z̅2) = arg(z1) - arg(z2)
- arg(z1 / z2) = arg(z1) - arg(z2)
- arg(zn) = n arg(z)
- |z1 + z2| = |z1 - z2| ⇒ arg z1 - arg z2 = π/2
- |z1 + z2| = |z1| + |z2| ⇒ arg z1 = arg z2
- If arg z = 0, then z is purely real
- If arg z = ± π/2, then z is purely imaginary
Polar Form of a Complex Number
- Let z = x + iy be a complex number represented by a point P (x, y) in the Argand plane. Then, by the geometrical representation of z = x + iy, we have
⇒ z = r(cosθ + i sinθ), where r = |z| and θ = arg(z) - This form of z is called a polar form of z.
- The Euclearian form of a complex number is eiθ = cosθ + i sinθ and e-iθ = cosθ - i sinθ
1. Distance Between Two Points
If z1 and z2 are the affixes of points P and Q respectively in the Argand plane, then:
PQ = |z2 - z1|
2. Section Formula
Let z1 and z2 be the affixes of two points P and Q respectively in the Argand plane. Then, the affix of a point R dividing PQ internally in the ratio m : n is:
(m z2 + n z1) / (m + n)
but if R is an external point, then the affix of R is:
(m z2 - n z1) / (m - n)
3. Mid Point Formula
- If R be the mid-point, then the affix of R is: (z1 + z2) / 2
- If z1, z2, z3 are affixes of the vertices of a triangle, then the affix of its centroid is:
(z1 + z2 + z3) / 3 - The equation of the perpendicular bisector of the line segment joining points having affixes z1 and z2 is:
z(z̅1 - z̅2) + z̅(z1 - z2) = |z1|² - |z2|² - The equation of a circle whose centre is at point having affix z0 and radius R is:
|z - z0| = R
Note
- If the centre of the circle is at the origin and radius R, then its equation is |z| = R.
- General Equation of a circle is:
z z̅ + a z̅ + az + b = 0, where b ∈ R and a is a complex number.
This represents a circle having centre at '-a' and radius:
√(|a|² - b) = √(a z̅ - b)
Complex Number as a Rotating Arrow in the Argand Plane
To obtain the point representing z eiα, we rotate OP through angle α in an anticlockwise sense. Thus, multiplication by eiα to z rotates the vector OP in an anticlockwise sense through an angle α.
Let z1 and z2 be two complex numbers represented by points P and Q in the Argand plane such that ∠POQ = θ.
Then, z1 eiθ is a vector of magnitude |z1| = OP along OQ and (z1 eiθ) / |z1| is a unit vector along OQ.
Some Important Results
1. I f z1, z2, z3 are the affixes o f the points A, B and C in the Argand plane, then
(i) ∠BAC = arg ( (z3 - z1) / (z2 - z1) )
(ii) ∠BAC = arg ( (z3 - z1) / (z2 - z1) ) = |(z3 - z1) / (z2 - z1)| (cos α + i sin α), where α = ∠BAC.
If z1, z2, z3, and z4 are the affixes of the points A, B, C, and D respectively in the Argand plane, then AB is inclined to CD at the angle: arg ( (z2 - z1) / (z4 - z3) )
(iii) The equation of the circle having z1 and z2 as the endpoints of a diameter is:
(z - z1)(z̅ - z̅1) + (z̅ - z̅1)(z - z2) = 0
De-Moivre’s Theorem
Statement
(i) If n ∈ Z (the set of integers), then:
(cosθ + i sinθ)n = cos(nθ) + i sin(nθ)
(ii) If n ∈ Q (the set of rational numbers), then cos(nθ) + i sin(nθ) is one of the values of (cosθ + i sinθ)n.
(iii) 1 / (cosθ + i sinθ) = cosθ - i sinθ
(iv) (cosθ1 + i sinθ2)(cosθ2 + i sinθ2) = cos(θ1 + θ2) + i sin(θ1 + θ2)
nth Roots of Unity
nth roots of unity are: α0 = 1, α, α2, α3, ...... αn-1 where α = ei2π/n = cos(2π/n) + i sin(2π/n)
Properties of nth Roots of Unity
- Property 1: nth roots of unity form a G.P. with common ratio ei2π/n
- Property 2: Sum of the nth roots of unity is always zero.
- Property 3: Sum of pth powers of nth roots of unity is zero, if p is not a multiple of n.
- Property 4: Sum of pth powers of nth roots of unity is n, if p is a multiple of n.
- Property 5: Product of nth roots of unity is (-1)n-1
- Property 6: nth roots of unity lie on the unit circle |z| = 1 and divide its circumference into n equal parts.
Properties of Cube Roots of Unity and Some Useful Results Related To Them
(i) Cube roots of unity are 1, ω, ω2 where:
ω = -1/2 ± i(√3/2), ω2 = -1/2 - i(√3/2)
(ii) arg(ω) = 2π/3
(iii) Cube roots of -1 are -1, -ω, -ω2
(iv) 1 + ω + ω2 = 0
(v) ω3 = 1
- Four fourth roots of unity are -1, 1, -i, i
- log(α + iβ) = (1/2) log(α2 + β2) + i tan-1(β/α)
Condition for points A(z1), B(z2), C(z3), D(z4) to be concyclic:
α + β
Condition(s) for four points A(z1), B(z2), C(z3), and D(z4) to represent vertices of a
1. Parallelogram:
(i) The diagonals AC and BD must bisect each other
⇔ (1/2) (z1 + z3) = (1/2) (z2 + z4)
(ii) Rhombus:
(a) The diagonals AC and BD bisect each other.
⇔ z1 + z3 = z2 + z4
(b) A pair of two adjacent sides are equal, i.e., AD = AB.
⇔ |z4 - z1| = |z2 - z1|
(iii) Square:
(a) The diagonals AC and BD bisect each other.
⇔ z1 + z3 = z2 + z4
(b) A pair of adjacent sides are equal, AD = AB.
⇔ |z4 - z1| = |z2 - z1|
(c) The two diagonals are equal, AC = BD.
⇔ |z3 - z1| = |z4 - z2|
(iv) Rectangle:
(a) The diagonals AC and BD bisect each other.
z1 + z3 = z2 + z4
(b) The diagonals AC and BD are equal.
⇔ |z3 - z1| = |z4 - z2|
(v) Incentre: I (z) of the ΔABC is given by
(vi) Circumcentre (z) of the ΔABC is given by
(v) Orthocentre (z) of the ΔABC is given by
(vi) Area of triangle ABC with vertices A(z1), B(z2), C(z3) is given by
(vi) Equation of line passing through A(z1) and B(z2) is
(vii) General equation of a line is:
a̅z + az̅ + b = 0, where a is a complex number and b is a real number.
(viii) Complex slope of a line joining points A(z1) and B(z2) is given by:
w = (z1 - z2) / (z̅1 - z̅2)
(ix) Two lines with complex slopes w1 and w2 are parallel if w1 = w2 and perpendicular if w1 + w2 = w
Length of Perpendicular from a Point to a Line
Length of perpendicular from a point A(ω) to the line a̅z + az̅ + b = 0 is:
p = |aω + a̅ω̅ + b| / (2 |a|)
Recognizing Some Loci by Inspection
(i) If z1 and z2 are two fixed points, then:
|z - z1| = |z - z2| represents the perpendicular bisector of the line segment joining A(z1) and B(z2).
(ii) If z1 and z2 are two fixed points and k > 0, k ≠ 1 is a real number, then:
|z - z1| / |z - z2| = k represents a circle. For k = 1, it represents the perpendicular bisector of the segment joining A(z1) and B(z2).
(iii) Let z1 and z2 be two fixed points and k be a positive real number.
(a) If k > |z1 - z2|, then:
|z - z1| + |z - z2| = k represents an ellipse with foci at A(z1) and B(z2) and length of major axis = k = CD.
(b) If k = |z1 - z2|, it represents the line segment joining z1 and z2.
(c) If k < |z1 - z2|, then |z - z1| + |z - z2| = k does not represent any curve in the Argand plane.
(iv) Loci Related to Fixed Points
(a) If k < |z1 - z2|, then |z - z1| - |z - z2| = k represents a hyperbola with foci at A(z1) and B(z2).
(b) If k = (z1 - z2), then:
(z - z1) - (z - z2) = k
represents the straight line joining A(z1) and B(z2) excluding the segment AB.
(v) Special Circle Property
If z1 and z2 are two fixed points, then:
|z - z1|² + |z - z2|² = |z1 - z2|²
represents a circle with z1 and z2 as extremities of a diameter.
(vi) Argument Condition for a Circle Segment
Let z1 and z2 be two fixed points and α be a real number such that 0 ≤ α ≤ π, then:
(a) If 0 < α < π and α ≠ π/2, then: arg((z - z1) / (z - z2)) = α represents a segment of the circle passing through A(z1) and B(z2).
(b) If α2 = π/2, then: arg((z - z1) / (z - z2)) = α = π/2 represents a circle with diameter as the segment joining A(z1) and B(z2).
(c) If α = π, then: arg((z - z1) / (z - z2)) = α represents the straight line joining A(z1) and B(z2) but excluding the segment AB.
(d) If α = 0, then: arg((z - z1) / (z - z2)) - α ( = 0 ) 
represents the segment joining A(z1) and B(z2).
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