Complex numbers

Question 1

i =

Answer 1

(-1)^1/2

Question 2

i^2=

Answer 2

-1

Question 3

Cartesian form of imaginary numbers.

Answer 3

z=a+ib
Where a is real number and b is the imaginary part of z

Question 4

For equation with no real roots how to work out?

Answer 4

Work to point where all numbers are fully substituted in.
Under root, make it -1 * n. Take root n outside the root. Solve remembering that root -1 =i

Question 5

Complex conjugate of z is

Answer 5

ž=a-ib

Question 6

How to add complex numbers

Answer 6

Add real parts and imaginary parts separately.

Question 7

How to multiply complex numbers.

Answer 7

Do like expanding two brackets, bearing in mind i^2=-1

Question 8

Argand diagram

Answer 8

X axis is real, y axis is imaginary

Question 9

How to work out the modulus of z or |z| or r

Answer 9

Pythagoras, |z| = root of a^2+b^2

Question 10

How to work out argument, Ø

Answer 10

Arg=tan^-1(b/a)

Question 11

How to write polar coordinates

Answer 11

z=r(cosØ+isinØ)

Question 12

Loci

Answer 12

Relates to a set of points satisfying a particular condition, often forming a line,curve or circle.

Question 13

How to solve loci questions

Answer 13

Magnitude of z+n = q
Expand z to x+ iy
Group together real and imaginary parts.
Go reverse through Pythagoras to unsolve magnitude by having it = root of real^2 +imaginary ^2
Square both sides to get rid of a root to give final answer usually = to the equation of a circle.

Question 14

De Moivres theorem states:

Answer 14

Z^n=[r(cosØ+isinØ)]^n = r^n(cosnØ+isinØ*n)

Question 15

In an argand diagram what are the values of an angle between?

Answer 15

pi,-pi

Question 16

If an angle is not between pi and -pi what should we do to it

Answer 16

+2pi

Question 17

Complex roots process

Answer 17

If given complex root, complex conjugate will be another root.
Make both into factors.
Multiply factors together, put real and imaginary parts together. Use difference of two squares to obtain answer.
Put this answer through algebraic long division with polynomial. If remainder is 0 then top line is a factor.
Make equal to a root.

Question 18

How to change root to a factor

Answer 18

Make it equal 0

Question 19

How to show that a complex root is a root.

Answer 19

We raise root to all the powers that are found in question, e.g. z^3 and z^2 and find what they are equal to.
Substitute into our equation and solve, proving that it’s = to 0 hence it is a root

Question 20

How to help solve fraction with annoying bottom line

Answer 20

Multiply both top and bottom by what makes the bottom difference of 2 squares.
Or a value that makes it easier to solve.

Question 21

What must our argand diagrm for our polar form and when we find argument do?

Answer 21

They must match, if not we + or subtract pi to make them do so.

Question 22

Simplifying in polar form when negative Ø

Answer 22

If Ø is negative then we remove the - from cosØ as it doesn’t matter, and make it - isin+Ø

Question 23

Roots of unity

Answer 23

1 can be written as cos2pi + isin2pi.
We do de moivres theorem on it. Then we add 2pi/n to angles, keeping it in range.
If it’s 4th root we have 4 values.