Quadratic Formula
A quadratic equation is a second order polynomial:
y = ax2 + bx + c.
If a = 1, b = 0, and c = −2, then
y = x2 − 2.
The quadratic formula
√
−b ± b2 − 4ac
x=
2a
gives solutions to the following equation:
ax2 + bx + c = 0.
The solutions to x2 − 2 = 0 are:
√
√
√
± 8 ±2 2
x=
=
= ± 2.
2
2
Figure 1: Left: y = x2 − 2. Right: y = x2 + 2.
The Square Root of -1
If a = 1, b = 0, and c = 2, then y = x2 + 2. The
quadratic formula gives solutions to x2 + 2 = 0:
√
√
√
± −8 ±2 −2
x=
=
= ± −2.
2
2
We can’t take the square root of a negative number! So we√use an imaginary number, i, to represent the −1:
√
√
x = ± −2 = ±i 2.
The Square Root of -1 (contd).
Imaginary numbers can be added just like real
numbers:
7i + 5i = 12i.
Can they be multiplied?
√ 2
2
7i · 5i = 35i = 35 −1 = −35
which is real, not imaginary.
Complex Numbers
A complex number, c, has a real part, x, and an
imaginary part, y:
c = x + yi.
A complex number can be decomposed into its
real and imaginary parts:
Re(c) = x
Im(c) = y.
Two complex numbers, c1 and c2, are equal, if
and only if
Re(c1) = Re(c2)
and
Im(c1) = Im(c2).
In other words,
(x + iy) = (u + iv) iff x = u and y = v.
Adding Complex Numbers
Two complex numbers, c1 and c2, can be added:
Re(c1 + c2) = Re(c1) + Re(c2)
Im(c1 + c2) = Im(c1) + Im(c2).
The real parts are added to form the real part
of the sum and the imaginary parts are added
to form the imaginary part of the sum. In other
words, if c1 = x + yi and c2 = u + vi, then
(x + yi) + (u + vi) = (x + u) + (y + v)i.
Euler’s Equation
The Taylor series for sin x, cos x and ex are:
x0 x2 x4 x6 x8
cos x = − + − + + ...
0! 2! 4! 6! 8!
x1 x3 x5 x7 x9
sin x = − + − + + ...
1! 3! 5! 7! 9!
1
2
3
4
0
x
x
x
x
x
ex = + + + + + ...
0! 1! 2! 3! 4!
Euler’s Equation (contd.)
What is the Taylor series for eix?
ix
e =
=
=
=
=
(ix)0 (ix)1 (ix)2 (ix)3 (ix)4
+
+
+
+
+ ...
0!
1!
2!
3!
4!
i0x0 i1x1 i2x2 i3x3 i4x4
+
+
+
+
+ ...
0!
1!
2!
3!
4!
x0 x1 x2 x3 x4 x5
+ i − − i + + i − ...
0! 1! 2! 3! 4! 5!
0
2
4
1
3
5
x
x
x
x
x
x
− + − ... + i
− + − ...
0! 2! 4!
1! 3! 5!
cos x + i sin x.
Polar to Rectangular
We have just derived Euler’s equation:
eiθ = cos θ + i sin θ.
If we multiply both sides of the equation by a
we get:
aeiθ = a cos θ + ia sin θ = x + iy
where a is amplitude and θ is phase. The left
side of the equation is a complex number in polar form. The right side is a complex number in
rectangular form:
Re(aeiθ) = a cos θ = x
Im(aeiθ) = a sin θ = y.
The above equations show how we can convert
from polar to rectangular. How do we go from
rectangular to polar?
Rectangular to Polar
To solve for amplitude, a, given x + iy, we use
the fact that x = a cos θ and y = a sin θ to write
the following equation:
a2 cos2 θ + a2 sin2 θ = x2 + y2
which can be rearranged to yield
2
2
2
a cos θ + sin θ = x2 + y2
and since cos2 θ + sin2 θ = 1, it follows that
p
a = x 2 + y2 .
To solve for phase, θ, we use the fact that x =
a cos θ and y = a sin θ to write the following equation:
a sin θ y
=
tan θ =
a cos θ x
which can be directly solved for θ:
y
θ = tan−1
.
x
Imaginary
Real
Figure 2: The complex plane.
Multiplication in Rectangular Form
Using the distributive property
of multiplica√
tion, and the fact that i = −1, we can derive
the rules for multiplying two complex numbers
in rectangular form:
(x + iy) · (u + iv) = xu + ixv + iyu − yv
= (xu − yv) + i(xv + yu).
To summarize, given two complex numbers, c1
and c2:
Re(c1 · c2) = Re(c1)Re(c2) − Im(c1)Im(c2)
Im(c1 · c2) = Re(c1)Im(c2) + Im(c1)Re(c2)
...or rewriting this in matrix notation:
Re(c1 · c2)
Re(c2) −Im(c2)
Re(c1)
=
.
Im(c1 · c2)
Im(c2) Re(c2)
Im(c1)
But this is pretty hard to remember.
Multiplication in Polar Form
Given two complex numbers in polar form:
c1 = a1eiθ1
c2 = a2eiθ2 .
We start with the rules for multiplying two complex numbers in rectangular form:
Re(c1 · c2)
Re(c2) −Im(c2)
Re(c1)
=
.
Im(c1 · c2)
Im(c2) Re(c2)
Im(c1)
Now substitute a2 cos θ2 for Re(c2), a2 sin θ2 for
Im(c2), a1 cos θ1 for Re(c1), and a1 sin θ1 for Im(c1)
to get:
Re(c1 · c2)
a2 cos θ2 −a2 sin θ2
a1 cos θ1
=
.
Im(c1 · c2)
a2 sin θ2 a2 cos θ2
a1 sin θ1
Multiplying this out yields:
Re(c1 · c2) = a1a2 cos θ1 cos θ2 − a1a2 sin θ1 sin θ2
Im(c1 · c2) = a1a2 cos θ1 sin θ2 + a1a2 sin θ1 cos θ2.
Multiplication in Polar Form (contd.)
Using the identities:
cos(θ1 + θ2) = cos θ1 cos θ2 − sin θ1 sin θ2
sin(θ1 + θ2) = cos θ1 sin θ2 + sin θ1 cos θ2
we see that
Re(c1 · c2) = a1a2 cos(θ1 + θ2)
Im(c1 · c2) = a1a2 sin(θ1 + θ2).
It follows that
a1eiθ1 · a2eiθ2 = a1a2ei(θ1+θ2).
To summarize, we multiply the amplitudes and
sum the phases.
Observation
It is easy to add two complex numbers in rectangular form but hard in polar. Conversely it is
easy to multiply two complex numbers in polar
form but hard in rectangular.
Imaginary
Real
Figure 3: Multiplication of complex numbers.
Square root of a complex number
• A complex number always has two square
roots.
• We find the first square root by taking the
square root of the amplitude, a, and dividing
the phase, θ, by 2.
• The second square root has the same amplitude, but is at θ/2 + π.
iθ
• Consequently,
the
square
roots
of
ae
are
√ iθ/2
√ i(θ/2+π)
ae
and ae
.
N-th roots of unity
• Question How many square roots does 1 have?
• Answer Two. They are both real.
• Question How many cube roots does 1 have?
• Answer Three. Only one of which is real.
• Question How many fourth roots does 1 have?
• Answer Four. Two are real. Two are imaginary.
Do you see a pattern?
Imaginary
Real
Figure 4: The square roots of aei θ are
√
aeθ/2 and
√
ae(θ/2+π) .
Complex conjugate
c = x + iy
c∗ = x − iy
...or, in polar form:
c = aeiθ
c∗ = ae−iθ
The complex conjugate has the same amplitude,
but the phase is multiplied by minus one.
Imaginary
Imaginary
e^(1i 2pi / 3)
2 pi / 2
2 pi / 3
Real
e^(1i 2pi / 2)
Real
e^(0i 2pi / 2)
e^(0i 2pi / 3)
e^(2i 2pi / 3)
Imaginary
Imaginary
e^(1i 2pi / 5)
e^(1i 2pi / 4)
e^(2i 2pi / 5)
2 pi / 4
2 pi / 5
Real
e^(2i 2pi / 4)
Real
e^(0i 2pi / 5)
e^(0i 2pi / 4)
e^(3i 2pi / 5)
e^(3i 2pi / 4)
e^(4i 2pi / 5)
Imaginary
e^(2i 2pi / 6)
Imaginary
e^(2i 2pi / 7)
e^(1i 2pi / 6)
e^(1i 2pi / 7)
e^(3i 2pi / 7)
2 pi / 6
2 pi / 7
Real
e^(3i 2pi / 6)
Real
e^(0i 2pi / 6)
e^(0i 2pi / 7)
e^(4i 2pi / 7)
e^(4i 2pi / 6)
e^(6i 2pi / 7)
e^(5i 2pi / 6)
e^(5i 2pi / 7)
Figure 5: The N-th roots of unity.
Complex conjugate (contd.)
The sum of a conjugate pair:
c + c∗ = (x + iy) + (x − iy)
= 2x
= 2Re(c)
The product of a conjugate pair:
cc∗ = (x + iy)(x − iy)
= x2 + iy − iy − i2y2
= x 2 + y2
= a2
...or, in polar form:
cc∗ = aeiθae−iθ
= a2ei(θ−θ)
= a2
The amplitude of a complex number is the square
root of the product of the complex number and
its conjugate:
√
|c| = cc∗.
Imaginary
Real
Figure 6: Complex conjugate.