Complex Numbers Flashcards
Define a rational number.
p⁄q where p and q are integers and q != 0
Define an integer.
A whole number.
Define a real number.
A rational or irrational numbers.
If z = a + ib
Re(z) = ?
a
If z = a + ib
Im(z) = ?
b
What two conditions must be fulfilled for two complex numbers to be equal?
Their real parts are equal.
Their imaginary parts are equal.
If you have (a + ib)(a - ib), what is it equal to and what type of number is it?
a^2 + b^2
A real number
Define a natural number.
A positive, whole number.
If z = a + ib,
what does z equal?
a - ib
The conjugate of a sum of complex numbers…
is equal to the sum of the conjuagtes of the complex numbers.
___________ __ __ __
So z1 + z2 + … + zn = z1 + z2 + … + zn
The conjugate of a product of complex numbers…
is equal to the product of the conjuagtes of the complex numbers.
___________ __ __ __
So z1 x z2 x … x zn = z1 x z2 x … x zn
What equation gives you the square roots of a complex number?
___________ __________
_____ / _____ / _____
√ a + ib = √ √a2 + b2 + a ± √ √a2 + b2 - a
2 2
Where ± is the same sign as that of ib.
What equation do you simultaneously solve to find the square root of a complex number?
Let (a + ib)2 = z
∴ a2 - b2 + 2abi = z
(1) a2 - b2 = Re(z)
(2) 2ab = Im(z)
How do you factorise the sum of two squares?
x2 + y2 = ?
Use i2 to turn it into a difference of two squares, then factorise.
x2 + y2 = x2 + i2y2
= (x + iy)(x - iy)
What is √ i equal to?
√ i = ± ( 1⁄√<span> 2 </span> - 1⁄√<span> 2 </span> i )
What are the two primary forms of coordinates on the complex number plane?
Rectangular Form - (x, iy)
Mod-Arg Form - r(cosΘ + isinΘ)
What is the complex number product rule?
z1z2 = r1r2[cis(Θ1 + Θ2)]
What is the complex number quotient rule?
z1 ⁄z2 = r1 ⁄r2 [cis(Θ1 - Θ2)]
What is an Argand Diagram?
The complex number plane.
What is z on an Argand diagram?
A reflection of z in the x-axis.
What geometric effect does multiplying complex number z by real number c have?
Multiplies the distance from the origin (modulus) by c.
What geometric effect does multiplying complex number z by i have?
Rotates (the arguement of) z around the origin by 90º anticlockwise.
What geometric effect does multiplying complex number z by cisπ⁄2 have?
Rotates (the arguement of) z around the origin by 90º anticlockwise.
What geometric effect does multiplying complex number z by cisΘ have?
Rotates (the arguement of) z around the origin by Θ anticlockwise.
What geometric effect does multiplying complex number z by complex number w have?
Multiplies their modulii and adds their arguements.
What geometric effect does dividing complex number z by complex number w have?
Divides their modulii and subtracts their arguements.
z x z = ?
[z]2
[z] = ?
[z]
z + z = ?
2Re(z)
z - z = ?
2Im(z)
z + w = ?
z + w
z x w = ?
z x w
z⁄w = ?
__
(z⁄w)
[z][w] = ?
[zw]
[z]⁄[w] = ?
[z⁄w]
arg(z) + arg(w) = ?
arg(zw)
arg(z) - arg(w) = ?
arg(z⁄w)
If z = rcisΘ,
What is the value of z2?
r2cis2Θ
If z = rcisΘ,
What is the value of z-1?
1⁄rcis-Θ
If z = rcis(Θ),
What is the value of zn?
rncis(nΘ)
When are two vectors equal?
When they have the same length and direction.
What is a prime vector?
A vector that originates from the origin.
Why must [z1 + z2] ≤ [z1] + [z2]?
As vectors, they form the three sides of a triangle.
The longest side of a triangle is always smaller than the other two sides added together.
What is the complex locus equation for x = 3?
What shape is it?
Re(z) = 3
Vertical Line
What is the complex locus equation for y = -2?
What shape is it?
Im(z) = -2
Horizontal Line
What is the complex locus equation for:
x2 + y2 = 16
What shape is it?
[z] = 4
Circle, radius 4, centre (0,0)
What is the complex locus equation for:
(x - 2)2 + (y - 3)2 = 4
What shape is it?
[z - (2 + 3i)] = 2
Circle, radius 2, centre (2,3)
What is the locus of argz = π⁄6?
A line from the origin 30º above the x-axis.
Does not include the origin.
What is the equation of a line from the origin 60º above the x-axis, that doesn’t include the origin?
What is the locus of argz = π⁄3?
What is the locus of arg{z - (1 + 3i)} = π/2?
A line from (1, 3), 45º above the x-axis.
Does not include (1, 3).
What is the locus of arg{z + 2 + i)} = -2π/3?
A line from (-2, -1), 120º below the x-axis.
Does not include (-2, -1).
What is the locus of [z - z1] = [z - z2]?
A straight line equidistant from z1 and z2.
What shape does this equation form if Θ = 0?
arg(z - z1 ⁄z - z2) = Θ
A straight line passing through z1 and z2, with a gap between z1 and z2, and not including z1 and z2.
What shape does this equation form if 0 < Θ < π⁄2?
arg(z - z1 ⁄z - z2) = Θ
The major arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.
What shape does this equation form if Θ = π⁄2?
arg(z - z1 ⁄z - z2) = Θ
A semicircle, going anticlockwise from z1 to z2, not including z1 and z2.
What shape does this equation form if π⁄2 < Θ < π?
arg(z - z1 ⁄z - z2) = Θ
The minor arc of a circle, going anticlockwise from z1 to z2, not including z1 and z2.
What shape does this equation form if Θ = π?
arg(z - z1 ⁄z - z2) = Θ
A staright line from z1 to z2, not including z1 and z2.
Which direction should you always sketch when drawing a locus with the form:
arg(z - z1 ⁄z - z2) = Θ
Anticlockwise from z1 to z2.
State De Moivre’s theorem.
(cisΘ)n = cis(nΘ)
sin(-Θ) = ?
-sin(Θ)
cos(-Θ) = ?
cos(Θ)
cos3Θ = ?
1⁄4cos3Θ + 3⁄4cosΘ
sin3Θ = ?
3⁄4sinΘ - 1⁄4sin3Θ
Describe a method for finding z if:
zn = 1
zn = 1
zn = cis0
z = (cis0)1⁄n
z = (cis(0, ±2π, ±4π…))1⁄n number of terms equal to (n - 1)⁄2
z = cis(0), cis(±2π⁄n), cis(±4π⁄n)…
Simplify
Describe a method for z if:
zn = -1
zn = -1
zn = cisπ
z = (cisπ)1⁄n
z = (cis(π, ±3π, ±5π…))1⁄n number of terms equal to (n - 1)⁄2
z = cis(0), cis(±3π⁄n), cis(±5π⁄n)…
Simplify
