Complex Number
What is Complex Number ?
The system of real numbers is not sufficient to solve all the algebraic equations. There are no real number which can satisfy the elementary quadratic equation X² + 1 = 0 or X²= -1 ( It follows from the properties of real numbers that the square of a real number is never negative). In order to solve such equations, that is, to find square roots of negative quantities, we extend the system of numbers and introduce a new class of numbers known as imaginary or complex numbers.
Now we can solve more general type of equations,
a₀xⁿ+ a₁x(n-1)+.........+aₙ=0, where a₀, a₁,........,aₙ are real numbers.
☛Definition:
A complex number z is defined to be an ordered pair of real numbers (a, b) that satisfies the following condition and the following laws of operations
⬜(a, b) = (c, d) if and only if a = c, b = d (condition of equality)
⬜(a, b) + (c, d) = (a+ c, b+ d) (definition of addition)
⬜(a, b).(c, d) = (ac-bd, ad+bc) (definition of multiplication)
The first component a of the complex number (a, b) is its real part while the second component b is its imaginary part.
🌑 Any complex number of the form (a, 0), where ‛a‘ is a real number can be identified with real number a. Such a complex number is said purely real.
🌑Any complex number of the form (0, b) where b is a real number is said to be purely imaginary and denoted by ib or bi.
🌑Any complex number is denoted by z= a + ib, a is known as real part and b is known as imaginary part of z.
● In notation, Re(z) = a, Im(z) = b and z = Re(z) + iIm(z)
🔘Note :
ロ Im(z) is b not ib
ロ 0 = 0 + i0 is the only complex number which is purely real as well as purely imaginary
ロ Any real number is also a complex number
ロ Set of complex number is denoted by C.
ロ i= square root of -1. It is not a real number. It is called the fundamental imaginary unit.
☛ Properties of Complex Number :
🌑 If a + ib = 0, a and b being real, then a=0 and b=0.
ロ We have a + ib = 0 ⇒ a= -ib ⇒ a² = i²b², whence a² + b²=0. Therefore a = 0 and b = 0, since a and b are real.
🌑 If a + ib = c + id, then a = c and b = d
ロ We have a + ib = c + id ⇒(a-c) + i(b-d) = 0. Hence by the above property a – c = 0 and b – d =0, whence a=c and b=d
🔘 Note:
The complex number doesn’t posses the law of trichotomy (i.e., there is no ordering among complex numbers). a + ib > c + id or a + ib < c + id are completely meaningless unless b = d = 0.
🌑 The algebraic sum, difference, product or ratio of two complex numbers is a complex number.
ロ If Z₁ = a + ib and Z₂ = c + id, then,
ロ Z₁ + Z₂
= (a + c) + i(b + d), which is a complex number.
ロ Z₁ - Z₂
= (a – c) + (b – d), which is a complex number.
ロ Z₁ .Z₂
= (ac – bd) + i(ad + cb), which is a complex number
ロ`Z_1/Z_2`
`= (a + ib)/(c + id) `
` =((a + ib)(c – id))/((c + id)(c – id) )`
` = (ac + bd)/(c² + d²) + i( (cb – ad)/(c² + d²))
`
, which is a complex number. [ if c + id not equals to zero]
🌑 Conjugate of a Complex Number :
Conjugate z : If z=x+ iy, then it's conjugate is
z =x+ iy = x- iy
`●z\overline{z}=x^2+y^2`
`●Re(\overline{z})=(z+\overline{z})/2=x`
`●\overline{z_1\pm z_2}=\overline{z_1}\pm \overline{z_2}`
`●\overline{z_1+ z_2+ z_3+...+z_n}`=`\overline{z_1} + \overline{z_2} + \overline{z_3} +...+ \overline{z_n}`
`●(\overline{z_1z_2})=(\overline{z_1})(\overline{z_2})`
`●(\overline{z_1z_2z_3...z_n})`=`\overline{z_1} \ \overline{z_2} \ \overline{z_3}...\overline{z_n}`
`●\overline{(\overline{z})}=z`
`●z_1\overline{z_2}+\overline{z_1}+z_2`= `2Re(\overline{z_1}z_2)=2Re(z_1\overline{z_2})`
`●(\overline{z^n})=(\overline{z})^n`
`If \ P(z)=a_0+a_1z+a_2z^2+...+a_nz^n, ``where \ a_0,a_1,a_2,a_3,...,a_n and z ` are complex numbers, then`\overline{P(z)}=``\overline{a_0}+\overline{a_1} \ \overline{z}+\overline{a_2} \ (\overline{z})^2+\overline{a_3} \ (\overline{z_3})^3+...+\overline{a_n} \ (\overline{z})^n``=\overline{P}(\overline{z}),`` \ where \ \overline{P}(z)= ``\overline{a_0}+\overline{a_1}z+\overline{a_2}z^2+...+\overline{a_n}z^n`
`● If \ Re(z)=(P(z))/(Q(z))``, \ where \ P(z) and \ Q(z) \ are \ polynomial in \ z, \ Q(z)\ne 0,``\ then \ \overline{R(z)}=[\overline{P}(\overline{z})]/[\overline{Q}(\overline{z})]`
`● If \ z=[[x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]], ``\ then \ (\overline{z})=[[\overline{x_1},\overline{y_1},\overline{z_1}],[\overline{x_2},\overline{y_2},\overline{z_2}],[\overline{x_3},\overline{y_2},\overline{z_3}]]`
🌑 If w= ƒ(z) where ƒ(z) is a polynomial in z then w= ƒ(z), where ƒ is the polynomial whose coefficients are conjugate of coefficient of ƒ.
🔘Note:
In a complex number replacing i by -i we can get the conjugate of that complex number. The conjugate of that complex number is the image of its about real axis on Argand Plane(Complex plane)
Hence the modulus of the conjugate and the complex number is same
arg(conjugate of z) = - arg (z), except negative real number z.
☛Integral power of i :
🌑 i⁰= 1 , i¹ = i, i² = -1, i³= -i, i⁴= 1
🌑 i4m = 1 , i4m+1 = i , i4m+2= -i , i4m+3 = -i ( for all m belongs to natural number set)
🌑 The sum and the product of two conjugate complex numbers are both real.
ロ Let the two conjugate complex numbers be (a + ib) and (a – ib). Then (a + ib) + (a – ib)= 2a, which is real, and (a + ib)(a – ib) = a² +b², which is real.
☛Normal Form :
⬜The complex number (a, b) can be expressed as (a, b) = (a, 0) + (0, b).
⬜Again, (0, b) = (0, 1).(b, 0) = (b, 0).(0, 1).
⬜Therefore (a, b) = (a,0) + (0, 1).(b, 0)
= (a, 0) + (b, 0).(0, 1)
☛Geometrical Representation :
⬜ Just as real numbers are represented as points on a line, complex numbers(which are also ordered pairs of real numbers) can be represented as points on a plane. With respect to a given rectangular co-ordinate system in a plane, the complex number a + ib can be represented by the point with co-ordinates (a, b).
⬜ The first co-ordinate axis is called the real axis and the second co-ordinate axis is called the imaginary axis.
⬜ The plane in which all the complex numbers are presented is said to be the Complex Plane, or the Gaussian Plane, after the name of Gauss, a renowned German Mathematician.
⬜ The origin represents the zero complex number ( 0 + i0). The points on the real axis represent all real numbers of the form (a, 0) and the points on the imaginary axis, other than the origin, represent all imaginary numbers of the form (0, b).
The complex number Z = x + iy is represented by a point P, whose co-ordinates are (x, y).
R=|z|= (x² + y²)½ = square root of (Re(z)² + Im(z))² is known as modulus of z.
The angle ㄥPOX is measured in such a way that its value is between π and -π, then it is called argument or amplitude( principal argument or principle amplitude) of z denoted by arg(z) or amp(z).
🔘Note:
ロ If Im(z) > 0, then θ is measured in anticlock wise sense.
ロ If Im(z) < 0, then θ measured in clockwise sense.
ロ If Im (z) = 0 and Re(z) > 0, then arg(z) = 0
ロ If Im(z) = 0 and Re (z) < 0, then arg(z) = π
ロ If Re(z) = 0 and Im (z) >0, then arg (z) = π/2
ロ If Re (z) = 0 and Im (z) < 0, then arg (z) = -π/2
ロ If z = 0 then amplitude is not defined.
☛ Method to find Argument:
◻ When z lies in first quadrant 
arg(z) = tan-1(y/x)
x>0, y>0
◻ When z lies in second quadrant
arg(z) = π - tan-1(y/|x|)
= π +tan-1(y/x)
x<0, y>0
◻ When z lies in third quadrant
arg(z) =-[ π - tan-1(|y|/|x|)]
=- π + tan-1(y/x)
x<0, y<0
◻ When z lies in fourth quadrant
arg(z) = - tan-1(|y|/x)
=tan-1(y/x)
x>0, y<0
Remember: A mistake that students often do is by taking argument z =x+ iy as tan-1(y/x) but this is true only for x>0.
☛Polar (modulus – amplitude) form of a complex number:
If z= x + iy, and |z| = r, arg z = θ then x = rcosθ
And y = r sinθ. So r(cosθ + isinθ), which is known as the polar form of z.
☛ Properties of Modulus of Complex Number:
If z, z1, z2 are complex numbers, then
◻ |z1z2|=|z1||z2|
◻ |z1/z2|= |z1|/|z2|, given z2 ≠ 0
◻ |z|=0 ⇔ z=0
◻ |z| = |-z| = |z| = |-z|
◻ -|z| ≤ Re(z) ≤ |z|, -|z| ≤ Im(z) ≤ |z|
◻ |z1 + z2|2 + |z1 - z2|2 = 2(|z1|2 + |z2|2)
◻ If z₂≠ 0, then |z₁+z₂|²=|z₁|² + |z₂|² if and only if z₁/z₂ is purely imaginary.
◻ |z₁ + z₂| ≤ |z₁| + |z₂|
◻ |z₁ + z₂| = |z₁| + |z₂| if and only z₁z₂ >0
◻ |z₁-z₂| ≤ |z₁| + |z₂|
◻ |z₁-z₂| = |z₁| + |z₂| if and only z₁z₂ >0
◻ |z₁-z₂| ≥ ||z₁| - |z₂||
☛Properties of argument of complex numbers :
🌑 If Y, Z are two complex numbers then,
ロ arg (YZ) = arg(Y) + arg(Z) + 2kπ, where k belongs to {0, 1, -1} so that arg(YZ) will be lies in the interval (-π, π]
ロ arg(Y/Z) = arg( YZ)= arg(Y) – arg(Z) + 2kπ, where k belongs to {0, 1, -1} so that the arg will be lies in the interval (-π, π].
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