If ‘a’ and ‘b’ are two numbers in the form (a + ib) then it is known as Complex Number. Number ‘a’ is called real part and ‘b’ is called the imaginary part of the complex number (a + ib). The complex number is generally represented by z.
A Complex Number is a combination of a real number and an imaginary number
Complex Number has a real part and an imaginary part. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.
Imaginary Unit(i)
i2 = -1 or i = √1
for x > 0, √(-x) = √(-1).x = √i2x = i√x
Power of an imaginary unit
i0 = 1
i1 = i
i2 = -1
i3 = -i
i4 = +1
i4k+1 = i
i4k+2 = -1
i4k+3 = -i
i4k = +1
where k is positive integer.
Conjugate of Complex number
In any two complex numbers, if only the sign of the imaginary part differs then, they are known as a complex conjugate of each other. Its is denoted by .
If z = a + ib is complex number, then = a – ib is called conjugate.
For example,
(i) Conjugate of z1 = 5 + 4i is = 5 – 4i
(ii) Conjugate of z2 = – 8 – i is = – 8 + i
(iii) Conjugate of z3 = 9i is = – 9i.
(iv) = 6 – 5i, = 6 + 5i
Property of Conjugate of Complex number
If z, z1 and z2 are complex number, then
=> If z is real. z =
Real z= z + 2
Imaginary z= z – 2i
=> = z
=> || = |z|
=> z. = .z
=> z. = |z|2
=> = +
=> = –
=> = .
=> z1z2 = where z2 ≠ 0
Additional of Complex number
Let z1 = (a + bi) , z2 = (c + di) two complex number then sum of two complex number as follow:
z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i which is again is complex number.
Example: (4 – 3i) + (-1 + 8i) = (4+(-1)) + (-3+8)i = 3 + 5i
The addition of complex number satisfies the following property:
1. Commutative Law:
For any two complex number z1 and z2
z1 + z2 = z2 + z1
2. Closure Law:
Sum of two complex number z1 and z2 is also complex number. ie z1 + z2 is also complex number.
3. Associative Law:
For any three complex number z1, z2 and z3
(z1 + z2) + z3 = z1 + (z2 + z3)
4. Additive Identity:
For any complex number a+ib, we have
(a + b.i) + (0 + 0.i) = a + bi
5. Additive Inverse:
Complex number -z is additive inverse of z.
Difference of Complex number
Let z1 = (a + bi) , z2 = (c + di) two complex number then difference of two complex number as follow:
z1 – z2 = z1 + (-z2)
= (a + bi) – (c + di) = (a – c) + (b – d)i
Example: (4 – 3i) – (-1 + 8i) = 4 – 3i + 1 – 8i = 5 – 11i
Multiplication of Complex number
Let z1 = (a + bi) , z2 = (c + di) two complex number then product of two complex number as follow:
z1z2 = (a + bi) × (c + di) = (ac – bd) + (ad + bc)i
Instead of memorizing that formula, just think “FOIL” multiplication, and remember that i2 = -1
Example: (4 – 3i)(-1 + 8i) = (4)(-1) + (4)(8i) + (-3i)(-1) + (-3i)(8i)
= -4 + 32i + 3i + (-24)i2
= -4 + 35i + (-24)(-1)
= -4 + 35i + 24
= 20 + 35i
The product of complex number satisfies the following property:
1. Commutative Law:
For any two complex number z1 and z2
z1z2 = z2z1
2. Closure Law:
Product of two complex number z1 and z2 is also complex number. ie z1z2 is also complex number.
3. Associative Law:
For any three complex number z1, z2 and z3
(z1z2)z3 = z1(z2z3)
4. Multiplication Identity:
i + 0i or 1 is the multiplication identity
(a + ib)(1 + 0.i) = a +ib
5. Multiplication Inverse:
Complex number 1/z is multiplication inverse of z.
6. Distributive Law:
For any three complex number z1, z2, z3
a) z1(z2 + z3) = z1z2 + z1z3
b) (z1 + z2)z3 = z1z3 +z2z3
Division of Complex number
For any three complex number z1, z2,
where z2 ≠ 0, the quotient z1z2 is defined by
z1z2 = z1 1z2
Here are the step require to divide complex numbers:
Step 1: To divide complex numbers, you must multiply by the conjugate. To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator.
Step 2: Distribute (or FOIL) in both the numerator and denominator to remove the parenthesis.
Step 3: Simplify the powers of i, specifically remember that i2 = –1.
Step 4: Combine like terms in both the numerator and denominator, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.
Step 5: Write you answer in the form a + bi.
Step 6: Reduce your answer if you can.
Module of a complex number
Point P(x,y) represents the complex number z = x + iy in the complex plane. OP is known as the moduus of complex number z. |z| = /a2 + b2
θ, the angle between OP and x-axis, is known as argument or amplitude of the complex number z. The set of complex numbers is denoted by C.
Polar representation of a complex number
Point P is uniquely determined by the ordered pair of a real number(r,θ), called the polar coordinates of point P.
x = r cosθ, y = rsinθ
therefore, z=r(cosθ + isinθ)
where r =√a2 + b2 and θ =tan-1 =b/a
The latter is said to be polar form of complex number.
Here r = √x2 + y2 = |z| is the modus of z and θ is called argument(or amplitude) of z is denoted by arg z.
If z1,z2,——zn are the complex numbers then
z1.z2. —–zn = r1.r2.r3——.rn{cos(θ1+θ2+—+θn) + iSin(θ1+θ2+—+θn)}
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