Complex Numbers

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)

How do you describe the geometric effect of multiplying a complex number by z by w e.g w = -3i

You find the modulus and argument of that q transformation. e.g -3i has modulus 3 and argument -π/2 so the geometric effect of multiplying a number by this is enlarging the vector of z by scale factor 3 and rotating it π/2 clockwise

i^2=

-1

How do you multiply in modulus argument form?

Multiply the moduli and add the arguments

What can you deduce about the subtraction of two conjugate pairs?

it is completely imaginary z - z* = 2Im(z)

What is the modulus argument form of a complex number?

r(cosϴ+isinϴ) (Sometimes written as [r,ϴ] or r cis ϴ ) where 0< ϴ < ± π

In modulus argument form what values can r take?

r>0

x^2 = -9

±3i

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 does |z-a| = r represent in loci?

A circle of points |z| where 'a' is the centre and 'r' is the radius.

How would you represent this |Z-6+4i| = 3

A circle where 6-4i (swap signs) is the centre and has a radius of 3

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 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)

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 divide in modulus argument form?

Divide the moduli and subtract the arguments

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

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

If your argument is not between 0 and ± π what do you do?

If it is less than -π then keep adding 2 π until you are within the range if it is more than π keep subtracting 2 π until you are within the range

What does Im(z) mean?

Imaginary part of the complex number (contains i)

What does it mean if two complex numbers are equal?

Imaginary parts are equal and real parts are equal

What does Re(z) mean?

Real part of the complex number

What does cos-θ equal?

Remember the cos graph is symmetical at the y axis so negative values of cos will equal positive values of cos so cos-θ = cos θ

What does arg(z-a) = ϴ represent in loci?

Start at point 'a' and move anticlockwise through the angle ϴ. The points z lie on this angle line

How do you subtract complex numbers? (6-9i) - (1+6i) = ???

Subtract imaginary parts and subtract complex parts (6-1) + (-9-6)i = 5-15i

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

What does -π/4 < arg(z-3+4i) < π/4 look like?

The area of point in-between -π/4 and π/4 from the point 3-4i

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-a| = |z-b| represent in loci?

The perpendicular bisector of a and b

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 does sin-θ equal?

The sin graph has rational symmetry not mirror symmetry at the y axis so for modulus of sin θ = sin-θ but the signs are different so sin-θ = -sin θ

What can you deduce about the sum and product of two conjugate pairs?

They will be real z + z* = 2 x real part of z zz* = |z|^2

Where will z and z* lie on an argand diagram?

They will be reflections of each other in the x axis

How do you represent complex numbers geometrically on an Argand diagram?

x axis is real and y axis is imaginary

What is the complex conjugate of z = x+yi?

z* = x- yi

What does |z|^2 equal?

zz*

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 )