Find a complex number z such that the real part and imaginary part of z are both integers, and such that zz = 89

waffles
Oct 29, 2017

#1**+1 **

Find a complex number z such that the real part and imaginary part of z are both integers,

and such that zz = 89

I assume, \( z\bar{z} = 89\)

89 has one solution, because 89 is a prime number and \(89 = 1 \pmod{4}\)

\(89 = 5^2 + 8^2\)

so

\(z = 5 + 8i \text{ and } \bar{z} = 5-8i \text{ so } z\bar{z} = 5^2+8^2 = 89\\ \text{or} \\ z= 8 + 5i \text{ and } \bar{z} = 8-5i \text{ so } z\bar{z} = 8^2+5^2 = 89\)

heureka
Oct 30, 2017