Core Pure 1 - Chapter 2: Argand diagrams Flashcards

1
Q

What is an argand diagram

A

New coordinate system that can plot complex numbers

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What is the structure of an argand axis

A

Real along the bottom imaginary along the top

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What is the geometric property for the modulus of z

A

The length of the line z

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

How do you find the modulus of a complex number

A

Apply pythagoras

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What is the geometric property of arg(z)

A

Anticlockwise rotation in radians from the positive real axis

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

How do you find the arguement?

A

arctan(y/x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

How do you find the argument in all 4 quadrants

A

1st: z = α
2nd: z = π - α
3rd : z = -(π - α)
4th : z = -α

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What is the modulus-argument form of a complex number?

A

r(cosθ + isinθ)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What does r and θ represent in the modulus-argument form a complex number?

A

r = modulus
θ = arguement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

How do you convert from modulus argument form back to cartesian form?

A

Expand the brackets of the modulus argument form

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

How do you multiply complex numbers in modulus argument form

A

Multiply the moduli, add the arguements

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

How do you divide complex numbers in modulus argument form

A

Divide the moduli, subtract the arguments

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

How do you convert r(cosθ - isinθ) back into standard form

A

Negate sin and negate the argument for both

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What is meant by a loci

A

Set of points that satisfy a restriction

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

How do you convert from modulus to cartesian form

A

Let z = x+iy, then group the real and imaginary terms together

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What is the general form a circle on an argand diagram

A

┃z - z₁┃= r
Where z is a general complex number and z₁ is a fixed complex number

17
Q

What geometric property does the loci form of a circle describe

A

The distance between two complex numbers at a fixed point

18
Q

How do you find the maximum/minimum argument for a circle

A

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)

19
Q

How do you find the maximum/minimum modulus for a circle

A

Occurs when z passes through the centre of the circle so you will have to use the radius and Pythagoras to maximise/minimise

20
Q

What is the general form of a perpendicular bisector on an argand diagram

A

┃z - z₁┃= ┃z - z₂┃where both can vary

21
Q

How do you find the cartesian equation of a perpendicular bisector

A

Let z = x+iy
Group terms
Apply the definition of the modulus on both sides
Rearrange for y

22
Q

How do you minimise the modulus of a perpendicular bisector

A

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

23
Q

What is the general formula of half lines on an argand diagram

A

arg(z - z₁) = θ

24
Q

What must you include when sketching a half line?

A

Open circle at the point of argument and always state the restriction on

25
How do you find a complex number from 2 different loci
Convert both loci to cartesian form and solve simultaneously
26
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
27
How do you solve region problems
Consider each locus individually on one diagram then shade depending on intersection or union
28