modulus of complex number (original) (raw)

If we write z in polar form as z=r⁢ei⁢ϕ with r≥0,ϕ∈[0, 2⁢π), then |z|=r. It follows that the modulus is a positive real number or zero. Alternatively, if a is the real part of z, and b the imaginary part, then

| |z| | = | a2+b2, | (1) | | ---- | -- | ------ | --- |

which is simply the Euclidean norm of the point (a,b)∈ℝ2. It follows that the modulus satisfies the triangle inequality

also

| |ℜ⁡z|≤|z|,|ℑ⁡z|≤|z|,|z|≤|ℜ⁡z|+|ℑ⁡z|. | | ------------------------------------- |

Modulus is :

| |z1⁢z2|=|z1|⋅|z2|,|z1z2|=|z1||z2| | | ------------------------------------ |

Since ℝ⊂ℂ, the definition of modulus includes the real numbers. Explicitly, if we write x∈ℝ in polar form, x=r⁢ei⁢ϕ, r>0, ϕ∈[0,2⁢π), then ϕ=0 or ϕ=π, so ei⁢ϕ=±1. Thus,