Complex Numbers/argand diagrams Flashcards
i
√-1
Written as b i
Or a + b i as a complex number where a is real
√-36
√36 x √-1
6i
√-7
√7 x √-1
√7 i
i
i^2
i^3
i^4
i
-1
-i
1
(2+3i)(3-2i)
Expand as normal collecting like terms 6 - 4i + 9i -6i^2 6 + 5i - 6(-1) 6 + 5i + 6 12 + 5i
Complex conjugate
The same as the original complex number but with the second sign reversed
What does the complex conjugate do
Multiplying or dividing it by the complex number gives you a real number
Symbol for the conjugate of z
z*
How to use the complex conjugate
Typically it is used the same way as rationalising a denominator
What do you know if one root is a complex number
Another root must be the complex conjugate
What to remember if a root is complex
The bracket is (x-(a + bi))
x - a - bi
Roots if a cubic touches the x axis 3/2/1 time(s)
3 times : 3 real roots
2 times: 3 real roots - 2 of them the same
1 time : 1 real root - 2 complex (conjugates)
Roots if a quartic touches the x axis
4/3/2/1/0 time(s)
4 times: 4 real roots
3 times: 4 real roots - 2 identical
2 times: 2 real roots - 2 complex (conjugates)
1 time: 2 identical real - 2 complex (conjugates)
0 times: 4 complex (2 conjugate pairs)
Argand Diagrams
Used for plotting complex numbers
x-axis is real, y is imaginary
Plot like coordinates and draw a line to the origin
Argand diagram solutions trick
The real (x)-axis is a line of symmetry for solutions to a polynomial
|z|
Use pythagoras on the real part and the coefficient of i to find the modulus of z
arg(z)
Argument of z - anti-clockwise angle in radians from the real axis, give as negative value if more than pi
arg(z) when x and y values are positive
tan-1(y/x)
Quadrant 1
USE POSITIVE VALUES OF Y AND X
arg(z) when x is positive and y is negative
-(tan-1(y/x))
Quadrant 4
USE POSITIVE VALUES OF Y AND X
arg(z) when x is negative and y is positive
π - tan-1(y/x)
Quadrant 2
USE POSITIVE VALUES OF Y AND X
arg(z) when x and y are both negative
-π + tan-1(y/x))
Quadrant 3
USE POSITIVE VALUES OF Y AND X
Modulus argument form
If r = |z| and o = arg(x)
z = r(cos(o) + i sin(o))
MUST BE PLUS SIN AND COS
How to find the point from the modulus argument form
x of z = r cos theta
y of z = r sin theta
Modulus of multiplied complex numbers
The combined modulus is each individual modulus multiplied together
Argument of multiplied complex numbers
The combined argument is the sum of the individual arguments
Modulus of divided complex numbers
The combined modulus is the division of the individual moduluses
Argument of divided complex numbers
The combined argument is the same as subtracting the individual arguments
How to go from modulus argument form to complex number
Solve by putting the modulus argument form in your calculator
What to know for negatives arguments
Cos(-θ) = Cos(θ) Sin(-θ) = -Sin(θ)
IF YOU CHANGE ONE ARG THE OTHER MUST CHANGE WITH IT
How to get an argument into the principal argument form
The argument is the smallest difference (positive or negative) between the worked value and an even coefficient of pi
Principal argument form
-π < arg(x) < π
How to solve for the complex solutions when not given any
Complete the square
How to adjust arg depending on what direction you have it measured from
If anti-clockwise from up: add half pi
If anti-clockwise from east: subtract pi
If anti-clockwise from down: subtract half pi
How to go from two multiplied mod arg to x+ yi
Multiply mods
Add args
Carry out that
For divide: divide mods and subtract args
How to solve from one complex root
Another is the complex conjugate
Do (x- a - bi)(x - a + bi)
Divide the equation by the answer (by inspection) and solve normally
