The polar form of a complex number a+ib is a+ib = r(Cos t + i Sin t), where r is its modulus and t is its argument. As we operate on the real numbers, we can perform operations on complex numbers also. The operations on the complex numbers are as follows.
Equal numbers in polar form:
If two complex numbers are same then their modulus are same and their arguments differ by 2kπ.
If r(Cos t + i Sint) = R (Cos T + i Sin T)
Then r = R and t = 2kπ + T.
Conjugate of a complex number in polar form:
As we know, the conjugate of the complex number a+ib is a-ib. So, the conjugate of a complex number in r(Cos t + i Sint) is r(Cos (-t) + i Sin(-t)).
Inverse of a complex number in polar form:
We have,
So, for inverting a complex number in polar form, we just need to invert its modulus and change the sign of its argument.
Product in polar form:
We have,
r(Cos t + i Sint) * R (Cos T + i Sin T)
= rR (Cost Cos T- Sin t Sin T) + i rR (Sint CosT + Cos t Sin T)
= rR [Cos(t+T) + i Sin (t+T)]
So, for multiplying two complex numbers in polar form, we just need to multiply the moduli and add the arguments.
Quotient in polar form:
We have,
So, for dividing the complex numbers in polar form, we just need to divide their moduli and subtract their arguments in order.
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