Complex numbers Flashcards
i =
(-1)^1/2
i^2=
-1
Cartesian form of imaginary numbers.
z=a+ib
Where a is real number and b is the imaginary part of z
For equation with no real roots how to work out?
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
Complex conjugate of z is
ž=a-ib
How to add complex numbers
Add real parts and imaginary parts separately.
How to multiply complex numbers.
Do like expanding two brackets, bearing in mind i^2=-1
Argand diagram
X axis is real, y axis is imaginary
How to work out the modulus of z or |z| or r
Pythagoras, |z| = root of a^2+b^2
How to work out argument, Ø
Arg=tan^-1(b/a)
How to write polar coordinates
z=r(cosØ+isinØ)
Loci
Relates to a set of points satisfying a particular condition, often forming a line,curve or circle.
How to solve loci questions
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.
De Moivres theorem states:
Z^n=[r(cosØ+isinØ)]^n = r^n(cosnØ+isinØ*n)
In an argand diagram what are the values of an angle between?
pi,-pi
If an angle is not between pi and -pi what should we do to it
+2pi
Complex roots process
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.
How to change root to a factor
Make it equal 0
How to show that a complex root is a root.
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
How to help solve fraction with annoying bottom line
Multiply both top and bottom by what makes the bottom difference of 2 squares.
Or a value that makes it easier to solve.
What must our argand diagrm for our polar form and when we find argument do?
They must match, if not we + or subtract pi to make them do so.
Simplifying in polar form when negative Ø
If Ø is negative then we remove the - from cosØ as it doesn’t matter, and make it - isin+Ø
Roots of unity
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.
