Complex Number

Question

If |2^z| = 1 for a non - zero complex number z then which one of the following is necessarily true.

• A.  Re(z) = 0
• B. |z| = 1
• C. Re(z) = 1
• D. No such z exists

Answer 1

We can write 2=e^(ln(2)) and we can write z=a+ib.

So 2^z=

(e^(ln(2)))^z=

e^(zln(2))=

e^(ln(2)(a+ib))=

e^(aln(2))×e^(ibln(2)). Let r=e^(aln(2)).

But e^(ibln(2)) can be expressed as cosθ+isinθ where θ=bln(2).

So we have re^iθ to replace the original 2^z, where r is the modulus |2^z|.

So r=1=e^(aln(2)) making a=0. Since a is the real part of z, Re(z)=0, answer A.