2.4 Complex Numbers
I. A complex number is written in standard form: a + bi
where a ε set of real numbers
and
bi is a pure imaginary number
Therefore a + bi is an imaginary number
√(-1) = i
(√(-1))2 = i2 = -1
(√(-1))3 = i3 = -i
(√(-1))4 = (√(-1))2(√(-1))2 = (-1)(-1) = 1
(√(-1))0 = 1
Therefore you can see the pattern that:
i0 = 1
i 1= √(-1) = i
i2 = -1
i3 = -i
i4 = 1
so i5 = i
i6 = -1
i7 = - i
i8 = 1
so what would i63 = ?
i60i3
i60 = 1 and so therefore
i60i3 = (1) i3 = - i
B. Additive Identity is zero
what do you add to a number to get that number? of course, zero so
a + bi + ________ = 0
______ = -a - bi
C. Adding and Subtracting complex Numbers:
Example 1: (3 + 4i) + (7 + 2i)
= (3 + 7) + (4i + 2i) by grouping like terms
= 10 = 6i
Example 2: (3 + 4i) - (7 + 2i)
= (3 - 7) + (4i - 2i)
= (-4) + (2i)
= -4 + 2i
D. Multiplying Complex Numbers
Example 1: (3 + 4i)(7 + 2i)
= (3)(7) + (3)(2i) + (4i)(7) + (4i)(2i)
= 21 + 6i + 28i + 8i2
= 21 + 34i + 8(-1)
= 21 + 34i - 8
= 13 + 34i
E. Dividing Complex Numbers - multiply by the complex conjugate
Given "a + bi", the complex conjugate would be "a - bi"
Example 1: complex #1
but recall that "a + bi" has to be in standard form so...
29/53 + 22i/53 would be the answer!
F. Applications:
1. Fractal Geometry
Mandelbrot Set: to draw this, consider: Sequence of numbers
c, c2 + c, (c2 + c)2 + c, [(c2 + c)2+ c ]2 + c, ...
For some values, it is Bounded, which means that all elements in the sequence are less than some fixed number N. Therefore, complex number "c" is in the Mandelbrot Set.
For other values, it is Unbounded, which means that the elements in the sequence become infinitely large. Therefore, complex number "c" is not in the Mandelbrot Set.
I. A complex number is written in standard form: a + bi
where a ε set of real numbers
and
bi is a pure imaginary number
Therefore a + bi is an imaginary number
√(-1) = i
(√(-1))2 = i2 = -1
(√(-1))3 = i3 = -i
(√(-1))4 = (√(-1))2(√(-1))2 = (-1)(-1) = 1
(√(-1))0 = 1
Therefore you can see the pattern that:
i0 = 1
i 1= √(-1) = i
i2 = -1
i3 = -i
i4 = 1
so i5 = i
i6 = -1
i7 = - i
i8 = 1
so what would i63 = ?
i60i3
i60 = 1 and so therefore
i60i3 = (1) i3 = - i
B. Additive Identity is zero
what do you add to a number to get that number? of course, zero so
a + bi + ________ = 0
______ = -a - bi
C. Adding and Subtracting complex Numbers:
Example 1: (3 + 4i) + (7 + 2i)
= (3 + 7) + (4i + 2i) by grouping like terms
= 10 = 6i
Example 2: (3 + 4i) - (7 + 2i)
= (3 - 7) + (4i - 2i)
= (-4) + (2i)
= -4 + 2i
D. Multiplying Complex Numbers
Example 1: (3 + 4i)(7 + 2i)
= (3)(7) + (3)(2i) + (4i)(7) + (4i)(2i)
= 21 + 6i + 28i + 8i2
= 21 + 34i + 8(-1)
= 21 + 34i - 8
= 13 + 34i
E. Dividing Complex Numbers - multiply by the complex conjugate
Given "a + bi", the complex conjugate would be "a - bi"
Example 1: complex #1
but recall that "a + bi" has to be in standard form so...
29/53 + 22i/53 would be the answer!
F. Applications:
1. Fractal Geometry
Mandelbrot Set: to draw this, consider: Sequence of numbers
c, c2 + c, (c2 + c)2 + c, [(c2 + c)2+ c ]2 + c, ...
For some values, it is Bounded, which means that all elements in the sequence are less than some fixed number N. Therefore, complex number "c" is in the Mandelbrot Set.
For other values, it is Unbounded, which means that the elements in the sequence become infinitely large. Therefore, complex number "c" is not in the Mandelbrot Set.
Check out this website:
http://mathworld.wolfram.com/MandelbrotSet.html
Example 1: Let's let c = -1
So the sequence would be:
-1, (-1)2 + (-1) = 0, (0)2 + (-1) = -1, (-1)2 + (-1) = 0
or -1, 0, -1, 0
so this is bounded!
Example 2: Let's let c = - i
-i, (-i)2 + (-i) = -1 + -i, (-1 -i)2 + (-i) = -3i, (-3i)2 + (-i) = -9 - i
or -i, -1 - i, -3i, - 9 - i
so this is not bounded.
2. Impedance - the opposition to current in an electrical circuit.
Equation of 2 pathways:
1/z = 1/z1 + 1/z2
where
z1 is the impedance of pathway 1
z2 is the impedance of pathway 2
