4 - complex numbers

Question 1

i

Answer 1

√-1

Question 1

complex number

Answer 1

has real and imaginary parts
x + yi

Question 2

imaginary number

Answer 2

multiple of i
bi

Question 3

i²

Answer 3

-1

Question 4

realising imaginary numbers

Answer 4

same as rationalising surds - use DOTS
eg 5+4i/2-3i * 2+3i/2+3i

Question 5

complex conjugate

Answer 5

if z = a + bi
z* = a-bi

Question 6

zz*

Answer 6

a² + b² (always real)

Question 7

argand diagrams

Answer 7

can represent real num on num line
for complex add another axis perp to the real num line
Re = x
Im = y

Question 8

conjugate on argand diagram

Answer 8

reflection in the real axis
x + yi has coordinates (x,y)

Question 9

radians

Answer 9

1rad = the angle subtended by an arc of length r
1rad = 360/2π = 180/πº
a full turn = 2π rad = 360º
rad> degrees = *180/π
degrees > rad = *π/180

Question 10

modulus - |z|

Answer 10

the distance from the origin to z
if z = x + yi
|z| = √x²+ y²

Question 11

argument - arg(z)

Answer 11

the anti-clockwise angle from the real axis in rad
usually in the range -π < θ <= π (principle argument)
arg (z) = tan-1(y/x)

Question 12

mod-arg form

Answer 12

x = r cos θ
y = r sin θ
z = x + i y
z = r(cos θ + i sin θ) or r cis θ
when r = |z| and θ = arg(z)

Question 13

-sin θ = sin - θ

Question 14

cos θ = cos - θ

Question 15

multiplying imaginary numbers - effect on mod and arg

Answer 15

multiply mod together
add arguments
|z1z2| = |z1| |z2|
arg (z1z2) = arg(z1) + arg(z2)

Question 16

dividing imaginary numbers - effect on mod and arg

Answer 16

divide mods
subtract arguments
|z1/z2| = |z1| / |z2|
arg (z1/z2) = arg(z1) - arg(z2)

Question 17

loci

Answer 17

set of points that satisfy a rule (eg equidistant from centre = circle)

Question 18

loci |z-z1| = r

Answer 18

is a circle centre (x1,y1) with radius r where z1 = x1+ i y1
if |z-z1| = r
z = √(x-x1) ² + (y-y1)²

Question 19

cartesian eq

Answer 19

in terms of x and y

Question 20

loci |z-z1 | = |z-z2|

Answer 20

perpendicular bisector of the line segment that joins z1 to z2
find eq by doing
(x-x1) ² + (y-y1)² = (x-x2) ² + (y-y2)²

Question 21

loci arg (z-z1) = θ

Answer 21

half line
at angle of θ rad (anticlockwise from real)
at point (x1, y1)

Question 22

min/ max arg(z) and |z|

Answer 22

draw locus and find the max/min angle anticlockwise from real and distance from the origin to a point on the loci