IB DP Maths: AA HL复习笔记1.9.1 Geometry of Complex Numbers
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?
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- Let w be a given complex number with real part a and imaginary part b
- 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 .
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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