IBMaths: AI HLDPRevision Notes1. Number & Algebra1.6 Further Complex Numbers1.6.1 Geometry of Complex Numbers

Geometry of Complex Numbers (DP IB Maths: AI HL)

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Geometry of Complex Addition & Subtraction

What does addition look like on an Argand diagram?

  • In Cartesian form two complex numbers are added by adding the real and imaginary parts
  • When plotted on an Argand diagram the complex number z1  + z2 is the longer diagonal of the parallelogram with vertices at the origin, z1 , z2 and z1  + z2

1-9-1-ib-aa-hl-geometrical-addition-of-cns-diagram-1

 

What does subtraction look like on an Argand diagram?

  • In Cartesian form the difference of two complex numbers is found by subtracting the real and imaginary parts
  • When plotted on an Argand diagram the complex number z1  - z2 is the shorter diagonal of the parallelogram with vertices at the origin, z1 , - z2 and z1  - z2

 1-9-1-ib-aa-hl-geometrical-subtraction-of-cns-diagram-2

What are the geometrical representations of complex addition and subtraction?

  • Let w be a given complex number with real part a and imaginary part b
    • w equals a plus b straight i
  • Let z be any complex number represented on an Argand diagram
  • Adding w to z results in z being:
    • Translated by vector open parentheses a b close parentheses
  • Subtracting w from z results in z being:
    • Translated by vector open parentheses negative a minus b close parentheses

Examiner Tip

  • Take extra care when representing a subtraction of a complex number geometrically
    • Remember that your answer will be a translation of the shorter diagonal of the parallelogram made up by the two complex numbers

Worked example

Consider the complex numbers z1 = 2 + 3i and z2 = 3 - 2i.  

On an Argand diagram represent the complex numbers z1, z2, z1  + z2 and z1  - z2 .

1-9-1-ib-aa-hl-geometry-cn-we-solution-1-addition

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Geometry of Complex Multiplication & Division

What do multiplication and division look like on an Argand diagram?

  • The geometrical effect of multiplying a complex number by a real number, a, will be an enlargement of the vector by scale factor a
    • For positive values of a the direction of the vector will not change but the distance of the point from the origin will increase by scale factor a
    • For negative values of a the direction of the vector will change and the distance of the point from the origin will increase by scale factor a
  • The geometrical effect of dividing a complex number by a real number, a, will be an enlargement of the vector by scale factor 1/a
    • For positive values of a the direction of the vector will not change but the distance of the point from the origin will increase by scale factor 1/a
    • For negative values of a the direction of the vector will change and the distance of the point from the origin will increase by scale factor 1/a
  • The geometrical effect of multiplying a complex number by i will be a rotation of the vector 90° counter-clockwise
    • i(x + yi) = -y + xi
  • The geometrical effect of multiplying a complex number by an imaginary number, ai, will be a rotation 90° counter-clockwise and an enlargement by scale factor a
    • ai(x + yi) = -ay + axi
  • The geometrical effect of multiplying or dividing a complex number by a complex number will be an enlargement and a rotation
    • The direction of the vector will change
      • The angle of rotation is the argument
    •  The distance of the point from the origin will change
      • The scale factor is the modulus

What does complex conjugation look like on an Argand diagram?

  • The geometrical effect of plotting a complex conjugate on an Argand diagram is a reflection in the real axis
    • The real part of the complex number will stay the same and the imaginary part will change sign

Examiner Tip

  • Make sure you remember the transformations that different operations have on complex numbers, this could help you check your calculations in an exam

Worked example

Consider the complex number z = 2 - i.

On an Argand diagram represent the complex numbers z, 3z, iz, z* and zz*.

1-9-1-ib-aa-hl-geometry-cn-we-solution-2-multiplication

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    Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.