Complex Numbers

1. A number of the form z= x + iy where x, y$\in$R is called a complex number. Here x is called real part and y is called imaginary part, i= $\sqrt{-1}$

ie Re (z)= x, Im (z)= y.

2. A complex number z= x+iy, x, y$\in$R is called purely real if y=0, i.e. Im (z)= 0 and is purely imaginary if x= 0 i.e. Re (z) = 0


3. Equal Complex Numbers

Let $z_1$= $x_1$+ ${\mathrm{iy}}_1$and $z_2$= $x_2$+ ${\mathrm{iy}}_2$
Then $z_1$= $z_2$, if $x_1$+ $x_2$;$y_1$+ $y_2$
4. z = x + iy = 0 $\iff$ x = 0, y = 0
5. If z = x + iy, then $\stackrel{-}{z}$= x-iy is called complex conjugate of z.

6. Modulus and Argument of a Complex Number

Modulus of Z= x+iy, is denoted as |z| and |z|= $\sqrt{x^2+y^2}$

(2) |$\overline{Z}$| = $\sqrt{x^2+y^2}$

(3) Z $\overline{Z}$ = $x^2$ + $y^2$ = ${\left|Z\right|}^2$

(4) |$Z_1+Z_2$| $\le$ |$Z_1$| + |$Z_2$|

(5) |$Z_1-Z_2$| $\ge$ |$Z_1$| - |$Z_2$|

(6) |$Z_1\mathrm{.}{Z}_2$| = |$Z_1$| . |$Z_2$|

(7) $\left|\frac{Z_1}{Z_2}\right|$= $\frac{\left|Z_1\right|}{\left|Z_2\right|}$

(8) Arg ($Z_1\mathrm{.}{Z}_2$) = Arg ($Z_1$) + Arg ($Z_2$)

(9) Arg ($Z_1$/ $Z_2$) = Arg ($Z_1$) - Arg ($Z_2$)

(10) From (6) (7) (8) and (9) we see that in a product, the modolus = product of

Modullii and amplitude=sum of the amplitudes and in a quotient, modulus=ration of

the modullii and amplitude=Difference of the amplitudes.
8. Square roots of a complex number (V & E)
9. Argand diagram
10. Laplace - Demovre's formula ${(\cos \theta +i\sin \theta )}^n$= Cos n$\theta$ + i sin n$\theta$
Note:

1) ${(\mathrm{Cos}\theta +i\sin \theta )}^{-n}$ = Cos n$\theta$ - i sin n$\theta$

2) ${(\mathrm{Cos}\theta -i\sin \theta )}^n$ = Cos n$\theta$ - i sin n$\theta$

3) ${(\mathrm{Cos}\theta -i\sin \theta )}^{-n}$= Cos n$\theta$ + i sin n$\theta$
11. The cube roots of unity are:

w= -$\frac{1}{2}$+$\frac{i\sqrt{3}}{2}$ and $w^2$= -$\frac{1}{2}$-$\frac{i\sqrt{3}}{2}$
12. If w is a cube root of unity then:

a) $w^3$ = 1

b) 1 + w+ $w^2$= 0

The argument or amplitude of z ($\ne$0) = ${\tan }^{-1}$ $\frac{y}{x}$ ie. solution of the system of

equations cos$\theta$= $\frac{x}{\sqrt{x^2+y^2}}$, sin$\theta$= $\frac{y}{\sqrt{x^2+y^2}}\mathrm{.}$

If $\theta$ is a valve of the argument, then 2n$\pi$ +$\theta$ where n$\in$l are also valves of the

arguments of z. $\therefore$ arguments of z are in A.P.

7. Principal valve of the argument of z- The argument $\theta$, which satisfy -$\pi$ $<$ $\theta$ $\le$ $\pi$. This is denoted bu Arg z. The principal valve of arg z is alculated by the formula arg(z) = ${\tan }^{-1}$ $\frac{y}{x}$, z$\ne$0 and using the quadrant in which z lies.

8. Calculation of Argz

(i) Argz = ${\tan }^{-1}$ $\left|\frac{y}{x}\right|$, if x$>$0, y$>$0.

(ii) argz= $\pi$- ${\tan }^{-1}$$\left|\frac{y}{x}\right|$, if x$<$0, y$>$0.

(iii) Argz = $\pi$+ ${\tan }^{-1}$$\left|\frac{y}{x}\right|$, if x$<$0, y$<$0.

(iv) Argz = 2$\pi$- ${\tan }^{-1}$$\left|\frac{y}{x}\right|$, if x$>$0, y$<$0. or

= $-{\tan }^{-1}$$\left|\frac{y}{x}\right|$, if x$>$0, y$<$0

(v) arg z = $\pi$/2 if x = 0, y $>$0

(vi) arg z= 3$\pi$/2 if x=0, y $<$0

9. Properties of conjugate and modulii

(i) |z| = |$\stackrel{-}{z}$|

(ii) ${\left|z\right|}^2$= z$\overline{z}$

(iii) $\overline{(z_1\pm z_2)}$= $\overline{z_1}$ $\pm$ $\overline{z_2}$

(iv) $\overline{z_1{z}_2}$= $\overline{z_1}$ $\mathrm{.}$ $\overline{z_2}$

(v) $\overline{\left(\frac{z_1}{z_2}\right)}$= $\frac{\overline{z_1}}{\overline{z_2}}$

(vi) | z | = 0 if z = 0

(vii) $|z_1+z_2|$ $\le$ $\left|z_1\right|$ + $\left|z_2\right|$; $|z_1-z_2|$$\le$$\left|z_1\right|$ + $\left|z_2\right|$

(viii) $|z_1-z_2|$$\ge$$||z_1|$ - $\left|z_2\right||$

(ix) Re(z) = $\frac{z+\overline{z}}{2}$ ; Im (z) $\frac{z+\overline{z}}{2i}$

z+$\overline{z}$ is purely real

z-$\overline{z}$ is purely imaginary

(x) ${|z_1+z_2|}^2$ + ${|z_1-z_2|}^2$= 2${\left|z_1\right|}^2$+ 2${\left|z_2\right|}^2$

10. Properties of arg(z)

(i) arg($z_1$ $z_2$) = arg$z_1$ + arg$z_2$

(ii) arg $\frac{z_1}{z_2}$= arg$z_1$- arg$z_2$

(iii) argz is not defined when z = 0

(iv) argz + arg$\overline{z}$= 0 or 2$\pi$

(v) arg$\overline{z}$= -argz = arg$\frac{1}{z}$

(vi) arg iz = arg i + arg z= $\pi$/2 + arg z

11. (i) If z lies in the first quadrant, then $\overline{z}$
lies in the fourth quadrant and vice versa.

(ii) If z lies in the second quadrant, then $\overline{z}$
lies in the 3rd quadrant and vice versa.

12. De- Moivre's Theorem

(i) ${(\cos \theta +\mathrm{isin}\theta )}^n$= cos n$\theta$ + i sin n$\theta$, where n is any integer.

(ii) con n$\theta$ + i sin n$\theta$ is one of the valves of ${(\cos \theta +i\sin \theta )}^n$ where n is any fraction

Particular cases:

${(\mathrm{Cos}\theta +i\mathrm{Sin}\theta )}^{-n}$= Cos n $\theta$ - Sin n$\theta$

${(\mathrm{Cos}\theta +i\mathrm{Sin}\theta )}^n$= Cos n $\theta$ - Sin n$\theta$

${(\mathrm{Cos}\theta +i\mathrm{Sin}\theta )}^n$= Cos n $\theta$ - Sin n$\theta$

Note. $e^{i\theta }$= cos$\theta$ + i sin$\theta$;

$e^{-i\theta }$ = cos$\theta$ - i sin$\theta$

13. Cube roots of Unity.

Cube roots of unity are 1, $\omega$, $w^2$ where $\omega$ = $\frac{-1+\sqrt{3}i}{2}$, ${\omega }^2$ = $\frac{-1-\sqrt{3}i}{2}$

14. The sum of the cube roots of unity i.e, 1 + $\omega$
+ ${\omega }^2$= 0 and their product = 1.$\omega$.${\omega }^2$=1 i.e., ${\omega }^3$=1.

15. Each complex root of unity is the square of the other.
16. The three cube roots of unity are the vertices of an equilateral triangle inscribed in the circle |z|= 1.

$n^{\mathrm{th}}$ roots of unity.

$n^{\mathrm{th}}$ roots of unity are 1, $\alpha$, ${\alpha }^2$......... ${\alpha }^{n-1}$ where $\alpha$= Cos $\frac{2\pi }{n}$+ i sin $\frac{2\pi }{n}$= $e^{\frac{i2\pi }{n}}$

17. The four fourth roots of unity are 1, -1, i, -i
18. Geometrical Interpretation of Multiplication of a Complex Number by i

Let z= x + iy be a complex number. Put x = r cos$\theta$, y= r sin $\theta$.

$\therefore$ z= r(cos$\theta$ + i sin $\theta$) Then r= |z|, $\theta$= arg z.

r= cos $\pi$/2 + i sin $\pi$/2= i

Again iz= $(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})$ r (cos $(\theta$) + i sin $\theta$)= r $[\cos (\frac{\pi }{2}+\theta )+\mathrm{isin}(\frac{\pi }{2}+\theta )]$

19. Complex form of various loci

(1) |z- $z_1$| = r represents a circle with centre $z_1$ and radius r

(2) |z- $z_1$|$<$r represents the interior of the circle with centre $z_1$and radius r

(3) $\left|\frac{z-z_1}{z-z_2}\right|$= k represents a circle if k $\ne$ 1

When k=1, it represents a straight line, and is the right bisector of the line joining of $z_1$ and $z_2$.

(4) |z- $z_1$| + |z- $z_2$|= 2a where a$\in$ $R^{+}$ represents an ellipse having foci $z_1$, $z_2$.

Whose major axis is 2a.

(5) |z- $z_1$| - |z- $z_2$|= 2b where b$\in$$R^{+}$ represents a hyperbola having foci $z_1$and $z_2$ .

Whose major axis is 2a and transverse axis is 2b.

(6)$\alpha \overline{z}$ + $\overline{\alpha }$z + r= 0 (r$\in$R) represents a straight line.

(7) z$\overline{z}$ + $\alpha \overline{z}$+ $\overline{\alpha }$z+ $\beta$= 0, represents a circle, $\beta$ $\in$ R

20. Let $z_1$, $z_2$, $z_3$ be the affixes of points A, B, C respectively in the Argand diagram. Then the angle between AB and AC is given by arg $\left(\frac{z_3-z_1}{z_2-z_1}\right)$

21. If $z_1$, $z_2$, $z_3$ are collinear, Then arg $\left(\frac{z_3-z_1}{z_2-z_1}\right)$= 0 or $\pi$

ie. $\frac{z_3-z_1}{z_2-z_1}$is purely real

22. If $z_1$, $z_2$, $z_3$, $z_4$ be affixes of four points A, B, C, D respectively, in the Argand diagram, then DC is inclined to BA at an angle arg $\left(\frac{z_3-z_4}{z_1-z_2}\right)$

If DC $\perp$ BA then arg $\left(\frac{z_3-z_4}{z_1-z_2}\right)$= $\pm$ $\frac{\pi }{2}$

Notes:

(1) The order relation is not defined on the set C of all complex numbers.

(2) The area of the triangle whose vertices are z, iz and z + iz is 1/2 ${\left|z\right|}^2$

(3) If $z_1$, $z_2$, $z_3$ be the vertices of a triangle then the area of the triangle is $\mathrm{\Sigma }$

$\frac{(z_2-z_3){\left|z_1\right|}^2}{4i{z}_1}$ (b) Area of the triangle with vertices $z_1$, wz and z + wz is $\frac{\sqrt{3}}{4}$ ${\left|z\right|}^2$.

(4) If $z_1$, $z_2$, $z_3$ be the vertices of an equilateral triangle and $z_0$ be the circumcentre,

then $z_1^2$ + $z_2^2$ + $z_3^2$= 3$z_0^2$

(5) If $z_1$, $z_2$, $z_3$...........$z_n$ be the vertices of a regular polygon of n sides and $z_0$ be its

centroid, then $z_1^2$ + $z_2^2$ +........+ $z_n^2$= n$z_0^2$.

(6) If $z_1$, $z_2$, $z_3$ be the vertices of a triangle, then the triangle is equilateral if ${(z_1-z_2)}^2$+ ${(z_2-z_3)}^2$ + ${(z_3-z_1)}^2$= 0 or $z_1^2$ + $z_2^2$+$z_3^2$ =$z_1$$z_2$+ $z_2$ $z_3$+$z_3$, $z_1$.