Class 11 complex number basic knowledge

Basic knowlegde of complex number

Shikhakaushal.blogspot.com

Important Concepts and Formulas - Complex NumbersBasicsA complex number is any number which can be written as $a+ib$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$

$a$ is the real part of the complex number and $b$ is the imaginary part of the complex number.

Example for a complex number: 9 + i2

$i^2=-1$

$i^3=-i$

$i^4=1$

if $z=a+ib$ is a complex number, a is called the real part of z and b is called the imaginary part of z.

It can be represented as Re(z) = a and Im(z) = b

Conjugate of the complex number $z=x+iy$ can be defined as $\overline{z}=x-iy$

Example: $\overline{4+i2}=4-i2$ and $\overline{4-i2}=4+2i$

if the complex number $a+ib=0$, then $a=b=0$

if the complex number $a+ib=x+iy$, then $a=x$ and $b=y$

if $x+iy$ is a complex numer, then the non-negative real number $\sqrt{x^2+y^2}$ is the modulus (or absolute value or magnitude) of the complex number $x+iy$. It can be denoted as

$\left|x+iy\right|=\sqrt{x^2+y^2}$     (Note that modulus is a non-negative real number)

Rectangular(Cartesian) Form of Complex Numbers

A complex number when written in the form $a+ib$, it is in the Rectangular(Cartesian) form

$e^{i\theta }=\cos \theta +i\sin \theta$     (Euler's Formula)

Cube Roots of Unity = $(1{)}^{1/3}$

$=1,\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}$

$=1,w,w^2$ where $w=\frac{-1+i\sqrt{3}}{2}$

Properties of Cube Roots of Unity

(1) Cube Roots of Unity are in G.P.

(2) Each complex cube root of unity is the square of the other complex cube root of unity.

Example: $w=\frac{-1+\sqrt{3}i}{2},w^2=\frac{-1-\sqrt{3}i}{2}$

(3) $1+w+w^2=0$

(4) Product of all cube roots of unity = 1
i.e., $w^3=1$

(5) $\frac{1}{w}=w^2$ and $\frac{1}{w^2}=w$

Fourth Roots of Unity , $(1{)}^{1/4}$ are +1, -1, +i, -i

Polar and Exponential Forms of Complex Numbers

Polar and Exponential Forms are very useful in dealing with the multiplication, division, power etc. of complex numbers.

careerbless.com

navigation...

Important Concepts and Formulas - Complex NumbersBasicsA complex number is any number which can be written as $a+ib$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$

$a$ is the real part of the complex number and $b$ is the imaginary part of the complex number.

Example for a complex number: 9 + i2

$i^2=-1$

$i^3=-i$

$i^4=1$

if $z=a+ib$ is a complex number, a is called the real part of z and b is called the imaginary part of z.

It can be represented as Re(z) = a and Im(z) = b

Conjugate of the complex number $z=x+iy$ can be defined as $\overline{z}=x-iy$

Example: $\overline{4+i2}=4-i2$ and $\overline{4-i2}=4+2i$

if the complex number $a+ib=0$, then $a=b=0$

if the complex number $a+ib=x+iy$, then $a=x$ and $b=y$

if $x+iy$ is a complex numer, then the non-negative real number $\sqrt{x^2+y^2}$ is the modulus (or absolute value or magnitude) of the complex number $x+iy$. It can be denoted as

$\left|x+iy\right|=\sqrt{x^2+y^2}$     (Note that modulus is a non-negative real number)

Rectangular(Cartesian) Form of Complex Numbers

A complex number when written in the form $a+ib$, it is in the Rectangular(Cartesian) form

$e^{i\theta }=\cos \theta +i\sin \theta$     (Euler's Formula)

Cube Roots of Unity = $(1{)}^{1/3}$

$=1,\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}$

$=1,w,w^2$ where $w=\frac{-1+i\sqrt{3}}{2}$

Properties of Cube Roots of Unity

(1) Cube Roots of Unity are in G.P.

(2) Each complex cube root of unity is the square of the other complex cube root of unity.

Example: $w=\frac{-1+\sqrt{3}i}{2},w^2=\frac{-1-\sqrt{3}i}{2}$

(3) $1+w+w^2=0$

(4) Product of all cube roots of unity = 1
i.e., $w^3=1$

(5) $\frac{1}{w}=w^2$ and $\frac{1}{w^2}=w$

Fourth Roots of Unity , $(1{)}^{1/4}$ are +1, -1, +i, -i

Polar and Exponential Forms of Complex NumbersPolar and Exponential Forms are very useful in dealing with the multiplication, division, power etc. of complex numbers.
Polar Form of a Complex Number


A complex number $z=x+iy$ can be expressed in polar form as

$z=r\angle \theta =r\text{cis}\theta =r(\cos \theta +i\sin \theta )$(Please not that θ can be in degrees or radians)

where $r=\left|z\right|=\sqrt{x^2+y^2}$ (note that r ≥ 0 and and r = modulus or absolute value or magnitude of the complex number)

$\theta =\text{arg}z={\tan }^{-1}\left(\frac{y}{x}\right)$(θ denotes the angle measured counterclockwise from the positive real axis.) 
θ is called the argument of z. it should be noted that $2\pi n+\theta$ is also an argument of z where $n=\cdots -3,-2,-1,0,1,2,3,\cdots$. Note that while there can be many values for the argument, we will normally select the smallest positive value.
Please note that we need to make sure that θ is in the correct quadrant. i.e., θ should be in the same quadrant where the complex number is located in the complex plane.

careerbless.com

navigation...

Important Concepts and Formulas - Complex NumbersBasicsA complex number is any number which can be written as $a+ib$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$

$a$ is the real part of the complex number and $b$ is the imaginary part of the complex number.

Example for a complex number: 9 + i2

$i^2=-1$

$i^3=-i$

$i^4=1$

if $z=a+ib$ is a complex number, a is called the real part of z and b is called the imaginary part of z.

It can be represented as Re(z) = a and Im(z) = b

Conjugate of the complex number $z=x+iy$ can be defined as $\overline{z}=x-iy$

Example: $\overline{4+i2}=4-i2$ and $\overline{4-i2}=4+2i$

if the complex number $a+ib=0$, then $a=b=0$

if the complex number $a+ib=x+iy$, then $a=x$ and $b=y$

if $x+iy$ is a complex numer, then the non-negative real number $\sqrt{x^2+y^2}$ is the modulus (or absolute value or magnitude) of the complex number $x+iy$. It can be denoted as

$\left|x+iy\right|=\sqrt{x^2+y^2}$     (Note that modulus is a non-negative real number)

Rectangular(Cartesian) Form of Complex Numbers

A complex number when written in the form $a+ib$, it is in the Rectangular(Cartesian) form

$e^{i\theta }=\cos \theta +i\sin \theta$     (Euler's Formula)

Cube Roots of Unity = $(1{)}^{1/3}$

$=1,\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}$

$=1,w,w^2$ where $w=\frac{-1+i\sqrt{3}}{2}$

Properties of Cube Roots of Unity

(1) Cube Roots of Unity are in G.P.

(2) Each complex cube root of unity is the square of the other complex cube root of unity.

Example: $w=\frac{-1+\sqrt{3}i}{2},w^2=\frac{-1-\sqrt{3}i}{2}$

(3) $1+w+w^2=0$

(4) Product of all cube roots of unity = 1
i.e., $w^3=1$

(5) $\frac{1}{w}=w^2$ and $\frac{1}{w^2}=w$

Fourth Roots of Unity , $(1{)}^{1/4}$ are +1, -1, +i, -i

Polar and Exponential Forms of Complex NumbersPolar and Exponential Forms are very useful in dealing with the multiplication, division, power etc. of complex numbers.
Polar Form of a Complex Number


A complex number $z=x+iy$ can be expressed in polar form as

$z=r\angle \theta =r\text{cis}\theta =r(\cos \theta +i\sin \theta )$(Please not that θ can be in degrees or radians)

where $r=\left|z\right|=\sqrt{x^2+y^2}$ (note that r ≥ 0 and and r = modulus or absolute value or magnitude of the complex number)

$\theta =\text{arg}z={\tan }^{-1}\left(\frac{y}{x}\right)$(θ denotes the angle measured counterclockwise from the positive real axis.) 
θ is called the argument of z. it should be noted that $2\pi n+\theta$ is also an argument of z where $n=\cdots -3,-2,-1,0,1,2,3,\cdots$. Note that while there can be many values for the argument, we will normally select the smallest positive value.
Please note that we need to make sure that θ is in the correct quadrant. i.e., θ should be in the same quadrant where the complex number is located in the complex plane. This will be clear from the next topic where we will go through various examples to convert complex numbers between polar form and rectangular form. It is strongly recommended to go through those examples to get the concept clear.
$x=r\cos \theta$
$y=r\sin \theta$

If $-\pi <\theta \le \pi ,\theta$ is called as principal argument of z(In this statement, θ is expressed in radian)

Exponential form of a Complex Number

We have already seen that in polar form , a complex number can be expressed as $z=r(\cos \theta +i\sin \theta )$. By Euler's Formula, we have $e^{i\theta }=\cos \theta +i\sin \theta$

Hence, we can express a complex number in Exponential form as $z=r{e}^{i\theta }$ (Note that θ is in radians)

While there may be many values of θ satisfying this, we will normally select the smallest positive value.

Note that radians and degrees are two units for measuring angles.

$360°=2\pi$ radian

Convert Complex Numbers from Rectangular Form to Polar Form and Exponential FormExample 1: Convert $z=1+i\sqrt{3}$ to polar form$r=\left|z\right|=\sqrt{x^2+y^2}=\sqrt{1^2+(\sqrt{3}{)}^2}=\sqrt{1+3}=\sqrt{4}=2$
$\text{arg}z=\theta ={\tan }^{-1}\left(\frac{y}{x}\right)={\tan }^{-1}\left(\frac{\sqrt{3}}{1}\right)={\tan }^{-1}\left(\sqrt{3}\right)=\frac{\pi }{3}$
Here the complex number is in first quadrant in the complex plane. The angle we got, $\frac{\pi }{3}$ is also in the first quadrant. Hence we select this value.
Hence, the polar form is $z=2\angle \left(\frac{\pi }{3}\right)=2[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)]$
Similarly we can write the complex number in exponential form as $z=r{e}^{i\theta }=2e^{\left(\frac{i\pi }{3}\right)}$
(Please note that all possible values of the argument, arg z are $2\pi n+\frac{\pi }{3}\text{where}n=0,\pm 1,\pm 2,\cdots$

$<br>$