Geometry of Complex Addition, Subtraction & Conjugation (Cambridge International A-Level Mathematics)

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 $z_{1}=a+bi$and $z_{2}=c+di$, then $z_{1} + z_{2} = ( a + b \mathrm{i} ) + ( c + d \mathrm{i} )$
  • This can be written as

$( \frac{a}{b} ) + ( \frac{c}{d} ) = ( \frac{a + c}{b + d} )$

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

• A complex number $x+yi$ can be represented by the position vector$( \frac{x}{y} )$

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 $z_{1}=a+bi$ and $z_{2}=c+di$, then $z_{1} - z_{2} = ( a + b \mathrm{i} ) - ( c + d \mathrm{i} )$
  • This can be written as

$( \frac{a}{b} ) - ( \frac{c}{d} ) = ( \frac{a - c}{b - d} )$

• An alternative is to write $( a + b \mathrm{i} ) - ( c + d \mathrm{i} )$ as $( a - c ) + ( b - d ) \mathrm{i}$, 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

  • $w=a+bi$

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

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

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

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

Worked Example

Exam Hint Read questions carefully; is it asking to plot the complex number as a point or as a vector?

Be extra careful when representing subtraction geometrically, remember that the solution will be a translation of the shorter diagonal of the parallelogram made up by the two vectors.