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Argand diagrams - Chapter 2 - Further maths - Coggle Diagram
Argand diagrams - Chapter 2 - Further maths
Purpose and basics of Argand diagrams
what argand diagrams represent
plotted on a graph with axes relabelled
x-axis is the real part, Re
y-axis is the imaginary part, Im
a complex number z = x + iy can be represented by a point P(x,y) or by the vector OP
argand diagrams used to represent complex numbers and to see the effects of different moduli and arguments
they provide a new way to complete complex number sums and to solve equations and inequalities graphically using loci
vector view and operations on the diagram
using the vector OP for a complex number allows addition and subtraction to be shown geometrically
use vectors and drawing tip-to-tail to visualise addition and subtraction
Modulus and argument
modulus-argument form of a complex number
for |z| = r and arg z = 0, the number can be written in modulus-argument form
z = r(cos 0 + i sin 0)
operations in modulus-argument form
for complex numbers z1 and z2
|z1 / z2| = |z1| / |z2| and arg(z1 / z2) = arg z1 - arg z2
for any real n, arg(z^n) = n arg z
|z1 z2| = |z1| |z2| and arg(z1 z2) = arg z1 + arg z2
definitions and how to compute them
the modulus of z = x + iy, written |z|, is the distance from the origin to the point and equals the square root of (x^2 + y^2)
found using pythagoras on the argand diagram
the argument of z, written arg z = 0, is the angle (in radians) from the positive real axis to OP
for nonzero x, tan 0 = y/x, and a quick sketch ensures the correct quadrant is chosen
Loci and regions on the Argand diagram
standard loci descriptions
complex numbers can represent loci of points on an argand diagram
|z - z1| = |z - z2| is the perpendicular bisector of the segment joining z1 to z2
arg(z - z1) = 0 is a half-line from but not including z1, at angle 0 to a line through z1 parallel to the real axis
|z - z1| = r is a circle with centre at z1 = x1 + iy1 and radius r
regions defined by argument inequalities
inequalities involving arg specify angular sectors with vertex at the reference point
solid boundaries indicate included angles; dashed boundaries indicate excluded boundaries
when the statement excludes arg = 0, points where the argument is zero are not included