Prove the multiplicative inverse property of the complex numbers?

Answer 1

Multiplicative inverse of #a+ib# is #a/(a^2+b^2)-ib/(a^2+b^2)#

Multiplicative inverse of a number system is also called reciprocal.

Multiplicative identity in a number system is a number, which when multiplied by number, say #x#, returns the number itseIf. This number is #1# and while dealing with complex numbers, it is #1+i0#.
Multiplicative inverse of a number #x#, is a number #x'#, which when multiplied by #x# leads to multiplicative identity i.e. in complex numbers #1+i0#.
Let the complex number be #a+ib# and its multiplicative inverse be #c+di#, then
#(a+ib)(c+id)=1+i0#
or #(ac-bd)+i(ad+bc)=1+i0#
i.e. #ac-bd=1# and #ad+bc=0#
This when solved for #c# and #d# would give
#c=a/(a^2+b^2)# and #d=-b/(a^2+b^2)#
and hence multiplicative inverse of #a+ib# is #a/(a^2+b^2)-ib/(a^2+b^2)#
Recall that for a complex number #z=a+ib#, its complex conjugate is #barz=a-ib# and modulus of #z# is #|z|^2=(a^2+b^2)#
Hence we can write multiplicative inverse as #barz/|z|^2# too.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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