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CIE A-Level Maths Study Notes

2.9.4 Argand Diagram Representation

In this section, focus into the Argand diagram, a crucial tool for visualizing and understanding complex numbers. This diagram not only represents complex numbers graphically but also provides a clear insight into their operations, making it an essential concept for students.

Argand Diagram Basics

1. Argand Diagram: A 2D plane to represent complex numbers, where the horizontal axis (Real Axis) shows the real part and the vertical axis (Imaginary Axis) shows the imaginary part of a complex number.

2. Real Axis: The 'Re' axis. In a complex number z=a+biz = a + bi, 'a' is plotted here.

3. Imaginary Axis: The 'Im' axis. In z=a+biz = a + bi, 'bi' goes here.

4. Point Representation: Each complex number z=a+biz = a + bi is a unique point a,ba, b on the diagram, useful for visualizing complex numbers as vectors.

5. Modulus and Argument:

  • Modulus z|z|: Distance from the point to the origin.
  • Argument arg(z)\arg(z): Angle from the positive real axis to the line connecting the point and the origin.
Argand Diagram

Image courtesy of Online Math Learning

Operations on the Argand Diagram

1. Addition: Add complex numbers by adding their real and imaginary parts separately. Geometrically, it's like vector addition: the sum z1+z2z_1 + z_2 is the diagonal of the parallelogram formed by z1z_1 and z2z_2.

2. Subtraction: Subtract by taking the difference of real and imaginary parts. Geometrically, it's the vector from the head of the subtracted number to the other number's head.

3. Multiplication: Multiply by rotating and scaling. Multiply the angle of z1z_1 by the argument of z2z_2 and the modulus of z1z_1 by the modulus of z2z_2. Results in rotating and stretching z1sz_1's vector.

4. Division: Inverse of multiplication. Divide the angle of z1z_1 by the argument of z2z_2 and the modulus of z1z_1 by the modulus of z2z_2. Leads to rotating and shrinking z1sz_1's vector.

5. Conjugation: Conjugate of z=a+biz = a + bi is z=abiz* = a - bi. Reflect the point a,ba, b across the real axis to get a,ba, -b.

Example Questions

Question 1: Representation of 3+4i3 + 4i on an Argand Diagram


  • Understand: 3+4i3 + 4i has real part 3, imaginary part 4i.
  • Plot: Point 3,43, 4 on diagram, 3 on real axis, 4 on imaginary axis.
  • Modulus: Distance from origin. Calculate as 32+42=5√3² + 4² = 5.
  • Argument: Angle with positive real axis. Calculate as tan1(43)53.13\tan^{-1}\left(\frac{4}{3}\right) ≈ 53.13^\circ.
Argand Diagram

Question 2: Adding 1+2i1 + 2i and 3+4i3 + 4i on the Argand Diagram


  • Understand: Add real parts (1 and 3) and imaginary parts (2i and 4i) separately.
  • Plot: Points (1, 2) and (3, 4) on diagram.
  • Parallelogram: Draw using (1, 2) and (3, 4) as vertices.
  • Sum: Diagonal from origin to opposite vertex of parallelogram. Algebraically, 4+6i4 + 6i.
  • Interpretation: Sum is like vector addition of (1, 2) and (3, 4).
Argand Diagram
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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