Download 1 Imaginary Numbers 2 Quiz 24A 3 Complex Numbers

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MA 1165 - Lecture 24
4/8/09
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Imaginary Numbers
One basic fact is that there is no real number that is the square root of −1. As a result, the quadratic
equation
x2 + 1 = 0
(1)
has no real number solutions. This is a shortcoming of the real numbers, since the very similar looking
equation
x2 − 1 = 0
(2)
has two solutions, x = 1, −1.
To address this shortcoming, mathematicians define the imaginary number i to be a square root of −1. Or
in other words, i is a number such that
i2 = −1.
(3)
The choice of the word imaginary is unfortunate, since all numbers are simply abstract concepts with useful
patterns. That’s what math is. When we do computations, the main thing to remember is that when we
square i, we get −1. For example,
i3 = i2 · i = (−1)i = −i,
(4)
and
(−2i)2 = (−2)2 · i2 = 4 · (−1) = −4.
2
(5)
Quiz 24A
Simplify the following expressions. Choose your answers from
(a) 1
(b) −1
QA1.
i4 .
QA2.
4i2 .
QA3.
−4i2 .
QA4.
(−i)2 .
3
(c) 4
(d) −4
(e) none of these
Complex Numbers
Any real number times i, like −3i, will be called an imaginary number. A real number plus an imaginary
number, like 2 − 3i, will be called a complex number. Since −3i = 0 − 3i for example, every imaginary
number is also a complex number. Any real number, like 2 = 2 + 0i, is also complex.
Again, we simply treat i as if it were any other number, and we can always replace i2 with −1. For example,
(2 + 3i)2 = (2 + 3i)(2 + 3i) = 4 + 6i + 6i + 9i2 = 4 + 12i − 9 = −5 + 12i,
(6)
(1 − 2i)(1 + 2i) = 1 + 2i − 2i − 4i2 = 1 + 0 + 4 = 5.
(7)
and
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4 QUIZ 24B
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Quiz 24B
Simplify the following expressions. Choose your answers from
(a) 21 − 14i
(b) −8i
QB1.
(2 − 3i)(1 + i).
QB2.
(2 − 2i)2 .
QB3.
(7)(3 − 2i).
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(c) 8 − 8i
(d) 5 − i
(e) none of these
Graphing Complex Numbers
It is useful to have a graphical representation for complex numbers. We’ll use the familiar coordinate axes,
but here the horizontal axis will represent the real numbers and be called the real axis, and the vertical axis
will represent the imaginary numbers and be called the imaginary axis. These are shown in Figure 1. We
will graph a complex number in the same way we would plot a point. For example, the complex number
2 + i will be in the position above the 2 on the real axis and across from the i on the imaginary axis. Some
other points are also shown in Figure 1.
imaginary axis
3i
2i
1 + 3i
2i
i
2+i
−2
−3
−2
real axis
1
−1
2
3
−i
−2 − 2i
−2i
−3i
Figure 1: The plots of several complex numbers.
In this context, it also useful to be able to describe a complex number in terms of how far it is from the
origin, and the angle it makes with the positive real axis. If we have a point described this way (see Figure
2), then we already know how to write it as a real number plus an imaginary number.
If the point were on the unit circle, the real and imaginary parts of the number would be the same as the xand y-coordinates in the xy-plane. Therefore, in the complex version, we have cos(θ) + i sin(θ). If we’re not
on the unit circle, we’re on a circle of radius r. To get that, we just multiply by r. The complex number
lying a distance r from the origin, and making an angle θ with the real axis is
r(cos(θ) + i sin(θ)).
(8)
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6 QUIZ 24C
imaginary axis
r(cos(θ) + i sin(θ))
r
θ
real axis
Figure 2: Plotting a complex number in terms of an angle and distance from the origin.
For example, the complex number that is a distance r = 3 from the origin and makes an angle of
real axis is
√
√
3(cos( π3 ) + i sin( π3 )) = 3( 12 + i 2 3 ) = 32 + 3 23 i .
6
π
3
with the
(9)
Quiz 24C
Find the complex number that is r from the origin and makes an angle θ from the positive real axis. Choose
your answers from
(a)
3
2
−
√
3 3
2
i
(b) 2i
QC1.
r = 3 and θ =
π
4.
QC2.
r = 2 and θ =
π
2.
QC3.
r = −3 and θ =
2π
.
3
(c) −2
(d)
√
3 2
2
+
√
3 2
2
i
(e) none of these
(For negative r’s, go the opposite direction on the ray.)
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7 HOMEWORK 24
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Homework 24
Simplify the following expressions.
For problems 1-4, choose your answers from
(a) −3
1.
i6 .
2.
2i3 .
3.
−3i4 .
4.
(−i)5 .
(b) −2i
(c) −i
(d) −1
(e) none of these
For problems 5-7, choose your answers from
(a) 10 − 6i
(b) −8 − 6i
5.
(5 − 2i)(7 + i).
6.
(1 − 3i)2 .
7.
(7i)(2 − 5i).
(c) 37 − 9i
(d) 35 + 14i
(e) none of these
Find the complex number that is r from the origin and makes an angle θ from the positive real axis. For
problems 8-10, choose your answers from
√
(a) 2
(b) −2
(c) −5
(d) 3 + i
(e) none of these
π
6.
8.
r = 2 and θ =
9.
r = 5 and θ = π.
10.
r = −2 and θ = 3π. (For negative r’s, go the opposite direction on the ray.)