How do you calculate the complex number?

Question

|(π + i)^100 / (π - i)^100|

Answer 1

Hello De D.,

Those | | indicate the modulus of the number. First, |ab| = |a|*|b|, so in particular, |a¹⁰⁰| = |a|¹⁰⁰. Now, |z| = |z^*| where z^* is the complex conjugate of z. (π+1) = (π-1)^* , so |π+1| = |π-1|. This means that |(π+1)/(π-1)| = |(π+1)|/|(π-1)| = 1. So |(π+1)¹⁰⁰/(π-1)¹⁰⁰| = |(π+1)/(π-1)|¹⁰⁰ = (|π+1|/|π-1|)¹⁰⁰ = 1¹⁰⁰ = 1.

Regards,

John