Modulus of a complex number — A/L Combined Mathematics

What must I remember about Modulus of a complex number?

• Definition of Modulus: For complex number z = x + iy, its modulus (absolute value) is |z| = √(x² + y²); geometrically represents Euclidean distance from origin to point (x, y) in Argand plane.
• Non-negativity: |z| ≥ 0 for all z ∈ ℂ; |z| = 0 if and only if z = 0 (x = 0 and y = 0).
• Multiplicative and Division properties: |z1 * z2| = |z1| * |z2|; |z1 / z2| = |z1| / |z2| (z2 ≠ 0); |zⁿ| = |z|^n.
• Relationship with conjugate: |z| = |z*| = |-z| = |-z*| = √(z * z*).
• Triangle Inequality in ℂ: For any complex numbers z1 and z2, ||z1| - |z2|| ≤ |z1 ± z2| ≤ |z1| + |z2|; upper bound equality holds iff z1 and z2 lie on the same ray from origin (arg(z1) = arg(z2)).

Test yourself (1 minute)

1. Distance between two complex numbers

   The Euclidean distance between points z1 and z2 in the Argand plane is |z1 - z2| = √[(x1 - x2)² + (y1 - y2)²].

2. Maximum and minimum values using Triangle Inequality

   Given |z - z0| = r, the maximum value of |z| is |z0| + r and the minimum value of |z| is ||z0| - r|.

3. Parallelogram identity in ℂ

   |z1 + z2|^2 + |z1 - z2|^2 = 2(|z1|^2 + |z2|^2); sum of squares of diagonals of a parallelogram equals sum of squares of its four sides.

Practise Modulus of a complex number questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

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Year 2 · Term 1