How do I prove |z|=0 if and only if z=0
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depends how you define |z|.
If you have |x + iy| = √(x^2 + y^2) = 0, then you know that:
√(x^2 + y^2) = 0 --->
x^2 + y^2 = 0
now if x and y are real, then x^2 ≥ 0 and y^2 ≥ 0. The only way you can have zero is if both x and y are zero.
If you have |x + iy| = √(x^2 + y^2) = 0, then you know that:
√(x^2 + y^2) = 0 --->
x^2 + y^2 = 0
now if x and y are real, then x^2 ≥ 0 and y^2 ≥ 0. The only way you can have zero is if both x and y are zero.