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

Geometry of Complex Addition, Subtraction & Conjugation

What does addition look like on an Argand diagram?

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

  • If and , then  

  • This can be written as

• An alternative is to write  as , adding the respective real and imaginary parts separately

• A complex number  can be represented by the position vector 

What does subtraction look like on an Argand diagram?

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

  • If  and , then 

  • This can be written as

• An alternative is to write  as , subtracting the respective real and imaginary parts separately

What are the geometric representations of complex addition and subtraction?

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

  • 

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

• Adding w to z results in z being translated by vector 

• Subtracting w from z results in z being translated by vector 

What is the geometric representation of 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