Mathematics
Grade 12
15 min
Subtraction in the complex plane
Subtraction in the complex plane
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Introduction & Learning Objectives
Learning Objectives
Subtract complex numbers algebraically by subtracting their corresponding real and imaginary parts.
Represent complex numbers as position vectors in the Argand diagram.
Interpret the subtraction of two complex numbers, z1 - z2, as the vector from the point z2 to the point z1.
Demonstrate complex subtraction geometrically by adding the additive inverse (z1 + (-z2)).
Calculate the modulus of the difference between two complex numbers, |z1 - z2|.
Relate the modulus of the difference, |z1 - z2|, to the distance between the points representing z1 and z2 in the complex plane.
How can subtracting one imaginary point from another give you a real, measurable distance? 🗺️ Let's find out how subtraction becomes a geometric tool in the complex plane.
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Key Concepts & Vocabulary
TermDefinitionExample
Complex Plane (Argand Diagram)A two-dimensional plane used to represent complex numbers. The horizontal axis is the real axis (Re), and the vertical axis is the imaginary axis (Im).The complex number z = 3 + 4i is represented by the point (3, 4) on the complex plane.
Vector RepresentationA complex number z = a + bi can be represented as a position vector originating from (0,0) and terminating at the point (a, b) in the complex plane.z = -2 + i is a vector from the origin to the point (-2, 1).
Additive InverseFor any complex number z = a + bi, its additive inverse is -z = -a - bi. Geometrically, this is a vector of the same length but pointing in the opposite direction.The additive inverse of z = 5 - 3i is -z = -5 + 3i.
Geometric Interpretation of SubtractionThe diffe...
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Core Formulas
Algebraic Subtraction Rule
Let z₁ = a + bi and z₂ = c + di. Then: z₁ - z₂ = (a - c) + (b - d)i
To subtract two complex numbers, subtract the real parts and subtract the imaginary parts separately.
Subtraction as Addition of the Inverse
z₁ - z₂ = z₁ + (-z₂)
Subtracting a complex number is equivalent to adding its additive inverse. This is the key to visualizing subtraction using the head-to-tail or parallelogram method for vector addition.
Distance Formula in the Complex Plane
The distance d between z₁ = a + bi and z₂ = c + di is: d = |z₁ - z₂| = \sqrt{(a-c)^2 + (b-d)^2}
This formula directly connects the modulus of the difference to the Euclidean distance between the points representing the complex numbers on the Argand diagram.
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Challenging
The set of points z in the complex plane that satisfy the equation |z - z₁| = |z - z₂| for two distinct fixed points z₁ and z₂ is:
A.circle passing through z₁ and z₂
B.The line segment connecting z₁ and z₂
C.The perpendicular bisector of the line segment connecting z₁ and z₂
D.point midway between z₁ and z₂
Challenging
Given z₁ = 6 + 8i and z₂ = 3 + 4i, which of the following inequalities relating |z₁ - z₂|, |z₁|, and |z₂| must be true according to the triangle inequality?
A.|z₁ - z₂| ≥ |z₁| + |z₂|
B.|z₁ - z₂| ≤ ||z₁| - |z₂||
C.|z₁ - z₂| = |z₁| - |z₂|
D.|z₁ - z₂| ≥ ||z₁| - |z₂||
Challenging
Let z₁ = 2 and z₂ = 2i. Find the complex number z₃ in the first quadrant such that the points representing z₁, z₂, and z₃ form an isosceles right-angled triangle with the right angle at the origin (0,0).
A.z₃ = -2 + 2i
B.z₃ = 2 + 2i
C.z₃ = 2 - 2i
D.This is not possible.
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