I need to find modulus and argument of complex numbers

I have some complex numbers I need to find modulus and argument for.

• (1+√3i)(1-√3i)
  • z = a+bi = 1-√3i+√3i+3 = 4
  • lzl  = √(a^2 + b^2) = √4^2 = √16 = 4
  • arg(z) = tan-1(b/a) = tan-1(0/4) = 0 degrees
• conjugate -i
  • conjugate -i = i
  • lzl  = √(a^2 + b^2) = √1^2 = √1 = 1
  • arg(z) =  tan-1(b/a) = tan-1(1/0) = 90 degrees
• 1-i/1-i
  • z = a+bi = 1-i/1-i = (1-i)*(1+i)/(1-i)*(1+i) = 1+i-i-i^2/1+i-i-i^2 = 1+i-i+1/1+i-i+1 = 2/2 = 1
  • lzl  = √(a^2 + b^2) = √1^2  = √1 = 1
  • arg(z) =  tan-1(b/a) = tan-1(1/0) = 90 degrees

Is the calculations I did correct? - I think that the last one is wrong, but I just cant figure out why ..