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11.. NQuumadberartSicysEteqmusations eeLLeeaarrnn..PPuunnjjaabb

=z1 z3 (=2 + i) (1 + 3i) (2 - i) (1 - 3i)
z2 3 - 2i 3 - 2i

= (=2 - 3) + (-6 -1)i -1 - 7i
3 - 2i 3 - 2i

= (-1 - 7i)(3 + 2i)
(3 - 2i) (3 + 2i)

= (-3 + 14) + (-2 - 21)i= 11 - 23 i
32 + 22 13 13

Example 3: Show that, ∀ z1, z2 ∈C, z1 z2 =z1 z2

Solution: Let z1 =a + bi, z2 =c + di

z1zz12z2==(a(a++bbi)i()c(c++ddi)i)==(a(acc--bbdd)()a(add++bbcc)i)i

==(a(acc--bbdd) )--(a(add++bbcc)i)i (1)
(2)
z1.zz1.2z=z21.=(za2(a+=+b(aib)i+=) b=(ic)(+c=+d(idc)+i) di)
==(a(a-=-b(aib)i--)(b(cia)-d(cd+i-)bdci))i

= =(ac(-acbd-)b+d )(-+a(d-a-dbc-)bi c)i

z1.z2 =(a + bi) (c + di)
=(a - bi) (c - di)

= (ac - bd ) + (-ad - bc)i

Thus from (1) and (2) we have, z1 z2 = z1 z2

Polar form of a Complex number: Consider adjoining diagram

representing the complex number z = x + iy. From the diagram, we

see that x = rcosq and y = rsinq where r = |z| and q is called argument

of z.

Hence x + iy = rcosq + rsinq ....(i)

where=r x2 + y2 aannddtanq-1=xy tan -1 y
x

Equation (i) is called the polar form of the complex number z.

*In any triangle the sum of the lengths of any two sides is greater than the length of the third
side and difference of the lengths of any two sides is less than the length of the third side.

version: 1.1

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