1 - Complex Numbers (FP2)
For |z - 12 - 5i| = 3 Find the minimum and maximum values of |z|
You can see that shortest distance between the locus of z and the origin is 13 - 3 = 10 and the longest is 13 + 3 = 16 So minimum |z| = 10 and maximum |z| = 16
For loci, what happens when the equation is an inequality instead?
You get a region of possible values rather than a locus
What is the locus represented by |z - z~1| = r?
A circle with centre z~1 and radius r
How do you draw the locus for arg z = theta?
A straight line with an angle of theta to the positive real axis Only draw the line where x > 0 and y > 0
If z^4 = a + bi How do you find the 4 solutions for z?
Convert to mod-arg form Find fourth root of r and divide theta by 4 to get the first value for z Now find the 4 z values which are in the range of arguments specified by adding or subtracting (2pi / x) from the argument successively where x is the number of solutions (the power of z)
How would you draw |z - z~1| = 2|z - z~2| as a locus?
Distance between z and z~1 is twice the distance between z and z~2
How do you divide 2 complex numbers that are in modulus-argument form or exponential form?
Divide the modulus values Subtract the arguments
If you don't know how to draw a region, what should you do?
Draw the locus that you would get if there was an equals sign rather than an inequality Test points to see which side of the locus should be shaded as the region
How would you draw the locus |2 - 5i - z| = 3? Notice the -z
Flip all signs and leave r as 3 (because it's a modulus)
What is the locus for |z - z~1| = |z - z~2|?
If you draw a line between z~1 and z~2 The locus is the perpendicular bisector of that line
What is the equation for the circle locus?
If z~1 = a + bi Equation is: (x - a)^2 + (y - b)^2 = r^2
How do you find the Cartesian equation for the locus |z - z~1| = |z - z~2|?
Inside the modulae, group the real and imaginary parts (bringing i outside a bracket) Now apply Pythagoras on both sides Remove the square roots Rearrange
If you have cos theta - i sin theta, how do you convert it into the required format?
It is important to get a positive sign and for both theta values to still be the same as each other
How do you multiply 2 complex numbers that are in modulus-argument form or exponential form?
Multiply the modulus values Add the arguments
How do you draw the locus for Arg(z - z~1) = theta?
Same as arg z = theta but the line starts at z~1 rather than the origin
What is principal argument?
The argument which is between -pi and pi
When a transformation is applied to z, what do we say happens?
The z-plans is transformed to the w-plane (If w is the transformed function of z)
How would you find a formula for cos 3x in terms of powers of cos x?
Use De Moivre's theorem on (cos x + i sin x)^3 to get an expression with cos 3x Binomially expand (cos x + i sin x)^3 and simplify Equate the expanded expression and the De Moivre expression Take out a factor of i out of everything possible Remove all things with i around them (equating coefficients) You now have a completely real equation with cos 3x and powers of cos x
How can you write a and b in terms of the modulus and argument?
a = r cos theta b = r sin theta
What is the identity which allows you to go between modulus-argument form and exponential form?
e^(i theta) = cos theta + i sin theta
What are the formulas for r and theta for z = a + bi?
r = sqrt(a^2 + b^2) Theta = arctan(b / a)
What are r and theta? Give the alternative notations for these
r or |z| is the modulus, the distance between the complex number and the origin Theta or Arg z is the argument, the angle between the positive real axis and the line between the origin and the complex number
What is w in real-imaginary form?
w = u + iv
How do you draw the locus: Arg((z - a) / (z - b) = theta
x > 0 and y > 0
How do you find the Cartesian equation for the locus of Arg(z - z~1) = theta?
y - y~1 = tan theta (x - x~1)
How do you find the Cartesian equation for a locus formed from arg z = theta?
y = x tan theta
If z = cos x + i sin x Find z + 1/z
z + 1/z = z + z^-1 = cos x + i sin x + cos(-x) + i sin(-x) = cos x + i sin x + cos x - i sin x = 2 cos x
What are the 4 identities related to z if z = cos x + i sin x?
z + 1/z = 2 cos x z - 1/z = 2i sin x z^n + 1 / z^n = 2 cos nx z^n - 1 / z^n = 2i sin nx
What is the Cartesian form of complex numbers?
z = a + bi
What is the exponential form of complex numbers?
z = r e^(i theta)
What is modulus-argument form?
z = r(cos theta + i sin theta)
What is De Moivre's theorem in exponential form?
z^n = r^n e^(i n theta)
