argand diagrams

  • complex numbers donโ€™t fit on the real line, but they can be represented using a second axis called the argand diagram
  • ![[notes/images/Pasted image 20230912095941.png|imaginary axis perpendicular to the โ€œrealโ€ axis|300]]
  • here is the imaginary axis perpendicular to the โ€œrealโ€ axis
  • so the number 3+2i is on the plane somewhere!
  • every point in the plane is a complex number - and (conversely) any complex number is a point in the plane!
  • called the complex plane but traditionally called an argand diagram
  • all real numbers are a subset of the complex plane! as a real number e.g. 3 is a complex number, which is 3 + 0i, so any real number can be thought of as special kinds of complex numbers!
  • leads to some interesting mathematics (amazing and spectacular) which is going to get taught later on by dr pearce !!

vectors

  • complex numbers can be alternatively be viewed as 2d vectors, each point has a position vector which determines its positions
  • can be visualised the way you add and subtract vectors.

rotation about the origin

  • when the complex number is multiplied by , the result is -2 - 3i. this is achieved through a rotation of 180deg about the origin.
  • when complex number 2+3i is multiplied by i, we obtain
    • i(2+3i) = 2i+3i
    • = 2i-3
    • = -3 + 2i
  • results in a rotation of 90deg anticlockwise
  • explored further in year 12 course

reflection in the horizontal axis

  • the conjugate of a complex number z = a + bi is z = a - bi (?)