Square root of a negative real
number is called the imaginary number.
Complex Number:
All real and non- real numbers are
called complex number
OR
A number of the form a + ib is
called a complex number. Here a and b are the real numbers and i is called the
iota.
For example Z = a + ib,
where a is called real part and b is called imaginary part.
Value of iota is √-1
i = √-1 , i2 = -1, i3 = -
i, i4 = 1
Geometrical Representation of the values of iota.
Two complex numbers are equal if their real and imaginary parts are equal
If Y and Z are two complex numbers such that
Y = a + ib and Z = c + id,
Then Y = Z ⇒ a = c and b = d
Addition of two complex numbers
Two complex numbers are added by
simply adding their real and imaginary parts
If
Y = a + ib and Z = c + id then
Y
+ Z = (a + c) + i(b + d)
Properties of addition of two complex numbers
i) Addition of two complex numbers
holds the closure property. This means that addition of two complex numbers is
also a complex number.
ii) Addition of complex numbers is commutative. ⇒ Y + Z = Z + Y
iii) Addition of complex numbers holds associative law ⇒ (X + Y) + Z = X + (Y + Z)
iv)Existence of additive identity:
0 is the additive identity for the
addition of complex number. ∵
Z + 0 = Z
v) Existence of additive inverse :
If Z is a complex number then - Z is
called the additive inverse of Z. ∵ Z + ( - Z) = 0
Subtraction of two complex numbers :
Two complex numbers are subtracted
by subtracting their corresponding real and imaginary parts.
If
Y = a + ib, Z = c + id then
Y - Z = (a + ib) - (c + id)
= (a - c) + i(b - d)
Multiplication of two
complex numbers
If
Y = a + ib, Z = c + id then
YZ
= (a + ib)(c + id)
= ac + iad + ibc + i2bd
= ac + i(ad + bc) + (-bd)
= (ac - bd) + i(ad + bc)
Properties of
Multiplication of two complex numbers
i) Closure property
:
Product of two complex numbers is
also a complex number.
ii) Commutative law :
Multiplication of complex
numbers is commutative. ⇒
YZ = ZY
iii) Associative Law:
Multiplication of complex
numbers is associative. ⇒
(XY)Z = X(YZ)
iv) Existence of multiplicative identity:
1 is called the multiplicative
identity for the product of complex number.
∵ Z x 1 = 1 x Z = Z
v) Multiplicative inverse :
If Z is a complex number
then Z-1 is called the inverse of Z. ∵ Z x Z-1 = 1
vi) Distributive law :
Product of complex numbers
holds the distributive property. ⇒
X(Y + Z) = XY + XZ
Conjugate of a complex number \[If\: Z = a+ib \; is\; a\; complex\; number\; then \; conjugate\; of\; Z\; is \; denoted\; by\; \overline{Z}\; and \; is\; given\; by\]\[\overline{Z}=\overline{a+ib}=a-ib\]
Note:- Conjugate of the conjugate of complex number is the complex number itself. \[If\: z=a+ib,\: then\: \overline{z}=\overline{a+ib}=a-ib\]\[\overline{\overline{z}}=\overline{a-ib}=a+ib=z\: \Rightarrow \overline{\overline{z}}=z\]
Modulus of complex number
If Z = a + ib is a complex number then modulus
of Z is denoted by |Z| and is given by \[\left | Z \right |=
\sqrt{a^{2}+b^{2}}\]
Argand Plane :
The plane having a
complex number assigned to each of its point is called a complex plan or Argand
plane.
Every point of the form (x,y) can be represented in the cartesian coordinate
plane.
When a complex number is represented
in the plan then the plane is called a complex plane or Argand plane.
In Argand plane x- axis is called
real axis and y-axis is called imaginary axis.
If Z = x + iy then \[\left | Z \right |= \sqrt{x^{2}+y^{2}}\]
|Z| is the distance of the point (x,y) from the origin (0,0)
Polar Representation of a complex
number z = x + iy
Let point P represent the complex
number Z = x + iy. Let directed line OP = |Z| = r and θ is the angle which OP
makes with the positive direction of x - axis. \[In\; \Delta OAP,\; \; \;
\frac{OA}{OP}=cos\theta\]\[\frac{x}{r}=cos\theta \Rightarrow x=rcos\theta\]
\[In\; \Delta OAP,\; \; \;
\frac{AP}{OP}=sin\theta\]\[\frac{y}{r}=sin\theta \Rightarrow y=rsin\theta\]
Putting x = rcosθ and y = rsinθ
in Z = x + iy we get
Z = rcosθ + irsinθ
or Z = r(cosθ + isinθ)
This is called polar form of a
complex number.
If Z = a + ib, then to convert it
into polar form we put a = rcosθ and b = rsinθ
tanθ = b/a and
θ = tan-1(b/a)
Sign convention while converting a complex number from Cartesian form to polar form.
DISCUSSION AND CONCLUSIONS
If (a, b) lie in first quadrant then Argument = θ
If (a, b) lie in second quadrant then Argument = π - θ
If (a, b) lie in third quadrant then Argument = -π + θ
If (a, b) lie in fourth quadrant then Argument = - θ
If complex number Z is purely +ve real number then Argument is 0
If complex number Z is purely -ve real number then Argument is π
If complex number Z is purely +ve imaginary number then Argument is π/2
If complex number Z is purely -ve imaginary number then Argument is - π/2
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Method to convert a complex number into a polar form
\[z=-\sqrt{3}+i\]\[r=|z|=\sqrt{\left ( \sqrt{3} \right )^{2}+\left ( 1 \right
)^{2}}=2\]Let polar form of complex number is \[z=rcos\theta
+irsin\theta\]Putting r = 2 we get\[z=2cos\theta +i2sin\theta\]Compairing this
with the first equation we get \[2cos\theta =-\sqrt{3}\; \; and\; \;
2sin\theta =1\]\[\Rightarrow cos\theta =-\frac{\sqrt{3}}{2}\; \; and\; \;
sin\theta =\frac{1}{2}\]\[tan\theta =\frac{sin\theta }{cos\theta
}=\frac{1/2}{-\sqrt{3}/2}=\frac{-1}{\sqrt{3}}\]\[tan\theta
=\frac{-1}{\sqrt{3}}=-tan\left ( \frac{\pi }{6} \right )\]\[since\; point \; \;
(-\sqrt{3},1)\; \; lie \; in \; the\; second\; quadrant\]\[\Rightarrow Argument
= \pi -\theta =\pi -\frac{\pi }{6}=\frac{5\pi }{6}\] Hence required polar form
of a complex number is \[z=2cos(\frac{5\pi }{6})+i\: 2sin(\frac{5\pi
}{6})\]
General quadratic
equation is ax2 + bx + c = 0
Discriminant
(D) = b2 - 4ac
Nature Of The Roots Of The Quadratic Equation
\[If D> 0 \; then \; roots\; are \; real\; and\; unequal\;
or\; distinct\; or\; different)\]
\[If D= 0 \; then \; roots\; are \; real\; and\; equal\]
\[If D< 0 \; then \; roots\; are \; not\; real\]
\[If D\geq 0 \; then \; roots\; are \; \; real\]
Quadratic Formula For Solving The Quadratic Equations
\[x=\frac{-b\pm \sqrt{D}}{2a}\; \; \; or\; \; \; x=\frac{-b\pm
\sqrt{b^{2}-4ac}}{2a}\]
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Method of finding the square root of a complex number
This can be
explained by taking an example
\[Find\; the\;
square \; root\; of\; the\; complex\; number\; \; z=\sqrt{3+i4}\]\[Let\; \;
a+ib=\sqrt{3+i4}\]Squaring on both side we
get,\[(a)^{2}+(ib)^{2}+2iab=3+i4\]\[(a)^{2}-b^{2}+2abi=3+i4\]Compairing real
and imaginary parts we ger, \[(a)^{2}-b^{2}=3 .......(1),\: \: and\:
2ab=4\]\[Now\: using\: the\: formula\: \:
(a^{2}+b^{2})^{2}=(a^{2}-b^{2})^{2}+(2ab)^{2}\]\[(a^{2}+b^{2})^{2}=(3)^{2}+(4)^{2}=9+16=25\]\[\Rightarrow
(a^{2}+b^{2})=5 ......(2)\]Adding eqn(1) and eqn(2) we
get\[2a^{2}=8\Rightarrow a^{2}=4\Rightarrow a=\pm 2\]From eqn(1) - eqn(2) we
get\[2b^{2}=2\Rightarrow b^{2}=1\Rightarrow b=\pm 1\]\[\sqrt{3+i4}=a+ib=\pm
2\pm 1i=\pm (2+i)\]Here between 2 and i we apply +ve sign because 2ab is
positive. If the value of 2ab is -ve then between a and b we apply -ve sign
in the square root of the complex number.
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Question:\[If\: x=-5+2\sqrt{-4}, \: then\: find \: the\: value\: of\\
x^{4}+9\: x^{3}+35\: x^{2}-x+4\]
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