Using Complex Numbers Flashcards
i^2=
-1
x^2 = -9
±3i
What does Re(z) mean?
Real part of the complex number
What does Im(z) mean?
Imaginary part of the complex number (contains i)
What is a complex number, what form does it take?
Contains real and imaginary parts usually in the form x+yi where x and y are real (denoted using letter z).
How do you add complex numbers?
(3+4i) + (2-8i) = ???
Add imaginary parts and add complex parts
(3+2) + (4-8)i = 5-4i
How do you subtract complex numbers?
(6-9i) - (1+6i) = ???
Subtract imaginary parts and subtract complex parts
(6-1) + (-9-6)i = 5-15i
How do you multiply complex numbers?
(7+2i)(3-4i) = ???
Use foil to multiply brackets in normal way but remember i^2 is -1
21 - 28i + 6i - 8i^2
21 - 22i + 8
29-22i
What does it mean if two complex numbers are equal?
Imaginary parts are equal and real parts are equal
How would u find a and b and find the two complex numbers?
z1 = (3-a) + (2b-4)i z2 = (7b-4) + (3a-2)i
By equating Re(z) and Im(z)
Re(z) 3-a = 7b-4
Im(z) 2b-1 = 3a-2
Solve these simultaneously to find a and b. Sub a and b back into originals to find the two complex numbers.
(Answer : a = 0 and b = 1 so z1= 3-2i and z2= 3-2i)
How do you find the square root of a complex number?
e.g find the square roots of 21-20i
1) let (a+bi)^2 = 21-20i
2) expand LHS
3) equate imaginary and real parts
4) solve simultaneously by substitution method
5) REMEMBER A AND B ARE REAL
6) put values for a and b back in (a+bi)
(Answer 5-2i and -5+2i)
What is the complex conjugate of z = x+yi?
z* = x- yi
If the coefficients of a quadratic equation are real and the roots are complex what can you deduce about the roots?
The roots will be complex conjugates pairs so z and z*
What can you deduce about the sum and product of two conjugate pairs?
They will be real
How do you divide complex numbers?
e.g (2+7i) / (3-5i)
zz* = real
so multiply top and bottom by complex conjugate of denominator so here multiply by 3+5i
(answer -29/34 + 31/34i )
How do you represent complex numbers geometrically on an Argand diagram?
x axis is real and y axis is imaginary
What is the modulus of z= x+yi, |z|?
The distance from the origin- given as √(x^2+y^2)
Think of it as finding the distance between two coordinates where you use pythagorus theorem.
What does |z|^2 equal?
zz*
What is the argument?
The angle the complex number makes on the Argand diagram with the x axis - angle ranges from -π to π measured anticlockwise in radians
How would you find the argument of (-5-5i)?
1) arctan(y/x) ignoring any negative signs
2) work out what quadrant you will be in
3) this is quadrant 3 so do π-ans and make it negative as you are going in the clockwise direction
(answer -3π/4)
What is the modulus argument form of a complex number?
r(cosϴ+isinϴ)
Sometimes written as [r,ϴ] or r cis ϴ
How would you derive the modulus argument form?
Draw a triangle: hypotenuse = r ϴ = angle adjacent = x opposite = y generate equations for sinϴ and cosϴ and rearrange to get x and y - stick these values into x+yi
In modulus argument form what values can r take?
r>0
How do you multiply in modulus argument form?
Multiply the moduli and add the arguments
