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IB DP Maths: AA HL复习笔记1.9.1 Geometry of Complex Numbers

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月11日 下午4:03

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

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

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

Exam 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

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

Exam 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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