Geometry of Complex Numbers (DP IB Analysis & Approaches (AA): HL): Revision Note

Geometry of complex addition & subtraction

What does addition look like on an Argand diagram?

• Addition can be seen using addition of vectors

  • To add the complex numbers $z$ and $w$

    • first travel along the vector $z$

    • then travel along the vector $w$

  • $z+w$ is the resultant vector

• This works in either order

  • $z+w$ or $w+z$

• Geometrically, adding $w=a+b\mathrm{i}$ to $z=x+y\mathrm{i}$ is the same as

  • a translation of $z$ by the vector $\left(\begin{array}{c}a\\ b\end{array}\right)$ 

What does subtraction look like on an Argand diagram?

• Subtraction can be seen using subtraction of vectors

  • To find $z-w$

    • first travel along the vector $z$

    • then travel along the vector $-w$ (the reverse of $w$)

• You cannot swap the order

  • $z-w$ is not the same as $w-z$

• Geometrically, subtracting $w=a+b\mathrm{i}$ from $z=x+y\mathrm{i}$ is the same as

  • a translation of $z$ by the vector $\left(\begin{array}{c}-a\\ -b\end{array}\right)$ 

Geometry of complex multiplication & division

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

• If the complex number $z_1$ is multiplied by the complex number $z_2$ then

  • $z_1$ will be enlarged by a scale factor of $\left|z_2\right|$

  • $z_1$ will be rotated by an angle of $\arg z_2$

• If the complex number $z_1$ is divided by the complex number $z_2$ then

  • $z_1$ will be enlarged by a scale factor of $\frac{1}{\left|z_2\right|}$

  • $z_1$ will be rotated by an angle of $-\arg z_2$

What special cases do I need to know?

• Some special cases are

  • multiplying / dividing $z$ by a real number, $k$

    • enlarges $z$ by a scale factor of $k$ / $\frac{1}{k}$

    • where $k$ could be negative

  • multiplying / dividing $z$ by an imaginary number, $k\mathrm{i}$

    • rotates the point 90° counter-clockwise / clockwise

    • enlarges $z$ by a scale factor of $k$ / $\frac{1}{k}$

    • where $k$ could be negative

What does complex conjugation look like on an Argand diagram?

• For the complex number $z$, the complex conjugate $z^{*}$

  • is a reflection of $z$ in the real axis