IB DP Maths: AI HL復習筆記1.6.1 Geometry of Complex Numbers
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
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?
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?.
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
- The direction of the vector will change
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*.
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