argand diagrams ยง
- complex numbers donโt 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 โrealโ axis|300]]
- here is the imaginary axis perpendicular to the โrealโ 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 ยง
- 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 ยง
- when the complex number 2+3i is multiplied by โ1, 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 ยง
- the conjugate of a complex number z = a + bi is z = a - bi (?)