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Definition of Modulus of a Complex Number:
Let z = x + iy where x and y are real and i = β-1. Then the non negative square root of (x2+ y 2) is called the modulus or absolute value of z (or x + iy).
Modulus of a complex number z = x + iy, denoted by mod(z) or |z| or |x + iy|, is defined as |z|[or mod z or |x + iy|] = + βx2+y2 ,where a = Re(z), b = Im(z)
i.e., + βRe(z)2+Im(z)2
Sometimes, |z| is called absolute value of z. Clearly, |z| β₯ 0 for all zΟ΅ C.
For example:
(i) If z = 6 + 8i then |z| = β62+82 = β100 = 10.
(ii) If z = -6 + 8i then |z| = β(β6)2+82 = β100 = 10.
(iii) If z = 6 - 8i then |z| = β62+(β8)2 =
β100 = 10.
(iv) If z = β2 - 3i then |z| = β(β2)2+(β3)2 = β11.
(v) If z = -β2 - 3i then |z| = β(ββ2)2+(β3)2 = β11.
(vi) If z = -5 + 4i then |z| = β(β5)2+42 = β41
(vii) If z = 3 - β7i then |z| = β32+(ββ7)2 =β9+7 = β16 = 4.
Note: (i) If z = x + iy and x = y = 0 then |z| = 0.
(ii) For any complex number z we have, |z| = |Λz| = |-z|.
Properties of modulus of a complex number:
If z, z1 and z2 are complex numbers, then
(i) |-z| = |z|
Proof:
Let z = x + iy, then βz = -x β iy.
Therefore, |-z| = β(βx)2+(βy)2 = βx2+y2 = |z|
(ii) |z| = 0 if and only if z = 0
Proof:
Let z = x + iy, then |z| = βx2+y2.
Now |z| = 0 if and only if βx2+y2 = 0
β if only if x2 + y2 = 0 i.e., a2 = 0and b2 = 0
β if only if x = 0 and y = 0 i.e., z = 0 + i0
β if only if z = 0.
(iii) |z1z2| = |z1||z2|
Proof:
Let z1 = j + ik and z2 = l + im, then
z1z2 =(jl - km) + i(jm + kl)
Therefore, |z1z2| = β(jlβkm)2+(jm+kl)2
= βj2l2+k2m2β2jklm+j2m2+k2l2+2jklm
= β(j2+k2)(l2+m2
= βj2+k2 βl2+m2, [Since, j2 + k2 β₯0, l2 + m2 β₯0]
= |z1||z2|.
(iv) |z1z2| = |z1||z2|, provided z2 β 0.
Proof:
According to the problem, z2 β 0 β |z2| β 0
Let z1z2 = z3
β z1 = z2z3
β |z1| = |z2z3|
β|z1| = |z2||z3|, [Since we know that |z1z2| = |z1||z2|]
β |z1z2 = |z3|
β |z1||z2| = |z1z2|, [Since, z3 = z1z2]
11 and 12 Grade Math
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