Simplify the following:
abs(-12 i + 5) + abs(-4 i + 3)
Absolute value gives the distance to the origin.
abs(5 - 12 i) = sqrt(Re(5 - 12 i)^2 + Im(5 - 12 i)^2) = sqrt(5^2 + (-12)^2):
sqrt(5^2 + (-12)^2) + abs(3 - 4 i)
sqrt(25 + (-12)^2) + abs(3 - 4 i)
sqrt(144 + 25) + abs(3 - 4 i)
sqrt(169) + abs(3 - 4 i)
Simplify radicals.
sqrt(169) = sqrt(13^2) = 13:
13 + abs(3 - 4 i)
Absolute value gives the distance to the origin.
abs(3 - 4 i) = sqrt(Re(3 - 4 i)^2 + Im(3 - 4 i)^2) = sqrt(3^2 + (-4)^2):
sqrt(3^2 + (-4)^2) + 13
sqrt(9 + (-4)^2) + 13 = 18
