Modulus, Conjugate and the Argand Plane
|z| = √(x2 + y2) is the distance of z from the origin on the Argand plane and conj(z) = x - iy is its reflection in the real axis.
What is Modulus, Conjugate and the Argand Plane?
|z| = √(x2 + y2) is the distance of z from the origin on the Argand plane and conj(z) = x - iy is its reflection in the real axis.
Key formula / rule: |z|^2 = z * conj(z), which converts modulus equations into algebra.
Key points
- Compute the modulus and conjugate of a complex number.
- Derive the locus described by a given modulus equation.
Common exam trap
Writing |z1 + z2| = |z1| + |z2| as an identity instead of an inequality.
Definitions
- Term
Modulus, Conjugate and the Argand Plane
- Meaning
|z| = √(x2 + y2) is the distance of z from the origin on the Argand plane and conj(z) = x - iy is its reflection in the real axis.
- Term
Modulus, Conjugate and the Argand Plane — explanation
- Meaning
Placing z at the point (x, y) turns algebraic identities into geometry: the modulus becomes a length, the conjugate a reflection, and |z1 - z2| the distance between two points. Most JEE questions on loci exploit exactly this translation.
Learning objectives
Compute the modulus and conjugate of a complex number.
Derive the locus described by a given modulus equation.
Formulae
- Key point
|z|^2 = z * conj(z), which converts modulus equations into algebra.
- Key point
|z1 z2| = |z1||z2| and |z1 + z2| ≤ |z1| + |z2|.
- Key point
|z - a| = r is the circle of centre a and radius r.
Prerequisites
MAT-U1-CQ-T1-S1-C1
Common mistakes
Writing |z1 + z2| = |z1| + |z2| as an identity instead of an inequality.
Squaring a modulus equation without checking that both sides are non-negative.
Keywords
Modulus
Conjugate
Argand
Plane
Practice preview
If z = 3 + 4i, what is the modulus of z?…
easy
The complex number z = 1 - i is represented by a point in which quadrant of the Argand plane?…
easy
If z is a complex number such that z * conj(z) = 25, then which of the following statements is true?…
medium