Geometry of Complex Addition, Subtraction & Conjugation (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Geometry of complex addition, subtraction & conjugation

How do I sketch complex addition on an Argand diagram?

• The addition of complex numbers can be shown by the addition of corresponding column vectors 

  • If $z_1=a+b\mathrm{i}$and $z_2=c+d\mathrm{i}$, then  $z_1+z_2=\left(a+b\mathrm{i}\right)+\left(c+d\mathrm{i}\right)$

  • This can be written as

$\left(\genfrac{}{}{0pt}{}{a}{b}\right)+\left(\genfrac{}{}{0pt}{}{c}{d}\right)=\left(\genfrac{}{}{0pt}{}{a+c}{b+d}\right)$

• An alternative is to write $\left(a+b\mathrm{i}\right)+(c+d\mathrm{i})$ as $\left(a+c\right)+(b+d)\mathrm{i}$, adding the respective real and imaginary parts separately

• A complex number $x+y\mathrm{i}$ can be represented by the position vector $\left(\genfrac{}{}{0pt}{}{x}{y}\right)$

How do I sketch complex subtraction on an Argand diagram?

• As with addition we can use knowledge of vectors to represent subtraction of complex numbers

  • If $z_1=a+b\mathrm{i}$ and $z_2=c+d\mathrm{i}$, then $z_1-z_2=\left(a+b\mathrm{i}\right)-\left(c+d\mathrm{i}\right)$

  • This can be written as

$\left(\genfrac{}{}{0pt}{}{a}{b}\right)-\left(\genfrac{}{}{0pt}{}{c}{d}\right)=\left(\genfrac{}{}{0pt}{}{a-c}{b-d}\right)$

• An alternative is to write $\left(a+b\mathrm{i}\right)-(c+d\mathrm{i})$ as $\left(a-c\right)+(b-d)\mathrm{i}$, subtracting the respective real and imaginary parts separately

Which geometric transformations represent complex addition and subtraction?

• Let w be a given complex number with real part a and imaginary part b

  • $w=a+b\mathrm{i}$

• Let z be any complex number represented on an Argand diagram

• Adding w to z results in z being translated by vector $\left(\begin{array}{c}a\\ b\end{array}\right)$

• Subtracting w from z results in z being translated by vector $\left(\begin{array}{c}-a\\ -b\end{array}\right)$

What geometric transformation represents complex conjugation?

• If we plot complex conjugate pairs on an Argand diagram, we notice the points are reflections of each other in the real axis

• Let z be any complex number represented on an Argand diagram

• Complex conjugating z results in z being reflected in the real axis