Geometry of Complex Numbers (DP IB Analysis & Approaches (AA)): 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 and

    • first travel along the vector

    • then travel along the vector

  • is the resultant vector

• This works in either order

  • or

• Geometrically, adding to is the same as

  • a translation of by the vector  

What does subtraction look like on an Argand diagram?

• Subtraction can be seen using subtraction of vectors

  • To find

    • first travel along the vector

    • then travel along the vector (the reverse of )

• You cannot swap the order

  • is not the same as

• Geometrically, subtracting from is the same as

  • a translation of by the vector  

Geometry of complex multiplication & division

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

• If the complex number is multiplied by the complex number then

  • will be enlarged by a scale factor of

  • will be rotated by an angle of

• If the complex number is divided by the complex number then

  • will be enlarged by a scale factor of

  • will be rotated by an angle of

What special cases do I need to know?

• Some special cases are

  • multiplying / dividing by a real number,

    • enlarges by a scale factor of /

    • where could be negative

  • multiplying / dividing by an imaginary number,

    • rotates the point 90° counter-clockwise / clockwise

    • enlarges by a scale factor of /

    • where could be negative

What does complex conjugation look like on an Argand diagram?

• For the complex number , the complex conjugate

  • is a reflection of in the real axis