2 - Argand Diagrams Flashcards
What are the real and imaginary axis?
x and y.
How is a complex number represented on the diagram?
point P(x,y) where x and y are Cartesian coordinates.
How else can complex numbers be represented on argand diagrams?
As vectors (x)
(y)
What is the modulus of a complex number?
The distance from the origin to the number on the diagram. √x^2 + y^2.
What is the argument of a complex number?
The angle between the positive real axis and the line joining the number to the origin on the diagram, measured in the anti-clockwise diagram - tan0 = y/x.
What is the principle argument?
Argument given in the rang -pi< 0 <= pi.
What is arg z in the first quadrant, if a is the acute angle?
a.
What is arg z in the second quadrant, if a is the acute angle?
pi - a.
What is arg z in the third quadrant, if a is the acute angle?
-(pi-a).
What is arg z in the fourth quadrant, if a is the acute angle?
-a.
What is the complex-argument form of z?
z = r(cos0 + isin0).
What is mod z1z2 equal to?
mod z1 mod z2.
What is arg(z1z2) equal to?
argz1 + argz2.
What is modz1/z2 equal to?
modz1 / modz2.
What is arg(z1/z2) equal to?
argz1 - argz2.
What is the distance between two points on an argand diagram?
mod z1 - z2.
What is the locus of a point z, such that mod z -z1 = r or mod z(x + iy) = r?
A circle with centre (x,y) and radius r.
What is the Cartesian equation of a circle?
(x - x1)^2 + (y - y1)^2 = r^2.
What is the locus of points z where mod z - z1 = mod z - z2?
A perpendicular bisector of a line segment joining z1 and z2.
What is the locus of points z such that arg(z - z1) = theta?
A half-line from but not including the fixed point z, making an angle theta from the fixed point parallel to the x-axis.
What is the Cartesian equation of a half-line?
y - y1 = tan theta(x - x1).
What else can you do with an argand diagram?
Represent inequalities.
