Complex Numbers. Aim: Identify parts of complex numbers. Imaginary NumbersReal Numbers.
Complex Numbers? What’s So Complex?
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Transcript of Complex Numbers? What’s So Complex?
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Complex Numbers?
What’s So Complex?
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Complex numbers are vectors
represented in the complex plane as
the sum of a Real part and an Imaginary part:
z = a + biRe(z) = a; Im(z) = b
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Just like vectors!
|z| = (a2 + b2)1/2 is length or magnitude, just like vectors.
= tan-1 (b/a) is direction, just like vectors!
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Just like vectors!
For two complex numbers a + bi and c + di:
Addition/subtraction combines separate components,
just like vectors.
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Useful identities
Euler: eix = cos x + i sin x
cos x = (eix + e-ix)/2
sin x = (eix - e-ix)/2i
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Sure, he’s French, but we must give
props:DeMoivre:
(cos x + i sin x)n = cos (nx) + i sin (nx)
cos 2x + i sin 2x = ei2x
cos 2x = (1 + cos 2x)/2 sin 2x = (1 - cos 2x)/2
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What about multiplication?
Just FOIL it!
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Scalar multiples of a complex number: a
line
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Multiplication:
the hard way!
z1z2= r1 (cos1 + i sin1) r2 (cos2 + i sin2)
= r1 r2 (cos1 cos2 - sin1 sin2) +
i r1 r2 (cos1 sin2 + cos2 sin1)
= r1 r2 [cos(1 + 2) + i sin(1 + 2)]
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Multiplication:
the easy way!
1 2
1 2
1 2 1 2
( )1 2
i i
i
z z re r e
r r e
“Neither dot nor cross do you multiply complex numbers by.”
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Multiplication: by i
(cos sin )
sin cos
iiz ir e
ir i
r ir
Rotate by 90o and swap Re and Im
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i ‘s all over the Unit Circle!
Note i4 = 1 does not mean that 0 = 4
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i ‘s all over the Unit Circle!
Did you see i½?
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Square root of i?Find the square root of 7+24 i.
(Hint: it’s another complex number, which we’ll call u+vi).
2
2 2
2 2
i 7+24i
( i) 7 24i
2 i - 7 24i
- 7; 2 24
u v
u v
u uv v
u v uv
Which can be solved by ordinary means to yield 4+3i and -4 - 3i.
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Complex Conjugates
22 2
z a bi
z a bi
zz a b z
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Complex Conjugates
1
1
; tan ( / )
; tan ( / )
z a bi b a
z a bi b a
Complex conjugates reflect in the Re axis.
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Complex Reciprocals
22 2
2
1
zz a b z
z
z z
The reciprocal of a complex number lies on the same ray as its
conjugate!
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Powers of z
The graph of f(z)=zn for |z|<1 is called
an exponential
spiral.
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This shape is at the heart of the computation of fractals!
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The basic geometry
of the solar system!
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It shows up in nature!
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And the decorative arts!
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The rotation comes from our old buddy
DeMoivre:
(cos x + i sin x)n = cos (nx) + i sin (nx)
Raising a unit z to the nth power is multiplying its
angle by n.
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How about a slice of :
Roots of z
/1231
9 /1232
17 /1233
4.24
4.24
4.24
i
i
i
z e
z e
z e
Each successive nth root is another 2/n around the circle.
If z3 = 3+3i = 4.24ei
then
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Find the roots of the complex equation z2 + 2i z +
24 = 0Sounds like a job for the
quadratic formula!
22 (2 ) 4(24)
2
4 96
25 6 ,4
i iz
i
i i i i
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Was that so complex?
And never forget, ei = -1