Core Pure 1 - Chapter 2: Argand diagrams Flashcards
What is an argand diagram
New coordinate system that can plot complex numbers
What is the structure of an argand axis
Real along the bottom imaginary along the top
What is the geometric property for the modulus of z
The length of the line z
How do you find the modulus of a complex number
Apply pythagoras
What is the geometric property of arg(z)
Anticlockwise rotation in radians from the positive real axis
How do you find the arguement?
arctan(y/x)
How do you find the argument in all 4 quadrants
1st: z = α
2nd: z = π - α
3rd : z = -(π - α)
4th : z = -α
What is the modulus-argument form of a complex number?
r(cosθ + isinθ)
What does r and θ represent in the modulus-argument form a complex number?
r = modulus
θ = arguement
How do you convert from modulus argument form back to cartesian form?
Expand the brackets of the modulus argument form
How do you multiply complex numbers in modulus argument form
Multiply the moduli, add the arguements
How do you divide complex numbers in modulus argument form
Divide the moduli, subtract the arguments
How do you convert r(cosθ - isinθ) back into standard form
Negate sin and negate the argument for both
What is meant by a loci
Set of points that satisfy a restriction
How do you convert from modulus to cartesian form
Let z = x+iy, then group the real and imaginary terms together
What is the general form a circle on an argand diagram
┃z - z₁┃= r
Where z is a general complex number and z₁ is a fixed complex number
What geometric property does the loci form of a circle describe
The distance between two complex numbers at a fixed point
How do you find the maximum/minimum argument for a circle
The maximum/minimum argument occurs at the point of tangency from the circle to the origin, so draw a tangent in all possible places then use geometry to work out the value of arg(z)
How do you find the maximum/minimum modulus for a circle
Occurs when z passes through the centre of the circle so you will have to use the radius and Pythagoras to maximise/minimise
What is the general form of a perpendicular bisector on an argand diagram
┃z - z₁┃= ┃z - z₂┃where both can vary
How do you find the cartesian equation of a perpendicular bisector
Let z = x+iy
Group terms
Apply the definition of the modulus on both sides
Rearrange for y
How do you minimise the modulus of a perpendicular bisector
The minimum modulus occurs at the perpendicular distance from the origin to the bisector, so work out the equation of the perpendicular line and any useful coordinates
What is the general formula of half lines on an argand diagram
arg(z - z₁) = θ
What must you include when sketching a half line?
Open circle at the point of argument and always state the restriction on
How do you find a complex number from 2 different loci
Convert both loci to cartesian form and solve simultaneously
How do you find a range of values for half lines
Find the value of the argument at 1 solution then state a range for which the amount of solutions would change
How do you solve region problems
Consider each locus individually on one diagram then shade depending on intersection or union
