Complex Numbers/argand diagrams Flashcards
i
√-1
Written as b i
Or a + b i as a complex number where a is real
√-36
√36 x √-1
6i
√-7
√7 x √-1
√7 i
i
i^2
i^3
i^4
i
-1
-i
1
(2+3i)(3-2i)
Expand as normal collecting like terms
6 - 4i + 9i -6i^2
6 + 5i - 6(-1)
6 + 5i + 6
12 + 5i
If z = x + yi, what is the complex conjugate of z?
z* = x - yi
What does the complex conjugate do
Reflects the complex number in the real axis. Multiplying or dividing it by the complex number gives you a real number.
Symbol for the conjugate of z
z*
How to use the complex conjugate
Typically it is used the same way as rationalising a denominator
What do you know if one root is a complex number and the coefficients of the polynomial are real?
Another root must be the complex conjugate
What to remember if a root is complex
The bracket is (x-(a + bi))
x - a - bi
Roots if a cubic touches the x axis 3/2/1 time(s)
3 times : 3 real roots
2 times: 3 real roots - 2 of them the same
1 time : 1 real root - 2 complex (conjugates)
Roots if a quartic touches the x axis
4/3/2/1/0 time(s)
4 times: 4 real roots
3 times: 4 real roots - 2 identical
2 times: 2 real roots - 2 complex (conjugates)
1 time: 2 identical real - 2 complex (conjugates)
0 times: 4 complex (2 conjugate pairs)
Argand Diagrams
Used for plotting complex numbers
x-axis is real, y is imaginary
Plot like coordinates and draw a line to the origin for plotting complex numbers
Argand diagram solutions trick
The real (x)-axis is a line of symmetry for solutions to a polynomial