Download Applications of imaginary numbers

Up until now, you've been told that you can't take the square root of a negative number. That's
because you had no numbers which were positive after you'd squared them (so you couldn't "go
backwards" by taking the square root). Every number was positive after you squared it. So you
couldn't very well square-root a negative and expect to come up with anything sensible.
Now, however, you can take the square root of a negative number, but it involves using a new
number to do it. This new number was invented (discovered?) around the time of the
Reformation. At that time, nobody believed that any "real world" use would be found for this new
number, other than easing the computations involved in solving certain equations, so the new
number was viewed as being a pretend number invented for convenience sake.
(But then, when you think about it, aren't all numbers inventions? It's not like numbers grow on
trees! They live in our heads. We made them all up! Why not invent a new one, as long as it
works okay with what we already have?)
Anyway, this new number was called "i", standing for "imaginary", because "everybody knew"
that i wasn't "real". (That's why you couldn't take the square root of a negative number before:
you only had "real" numbers; that is, numbers without the "i" in them.) The imaginary is defined to
be:
Then:
Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
Now, you may think you can do this:
But this doesn't make any sense! You already have two numbers that square to 1; namely –1 and
+1. And i already squares to –1. So it's not reasonable that i would also square to 1. This points
out an important detail: When dealing with imaginaries, you gain something (the ability to deal
with negatives inside square roots), but you also lose something (some of the flexibility and
convenient rules you used to have when dealing with square roots). In particular, YOU MUST
ALWAYS DO THE i-PART FIRST!

Simplify sqrt(–9).
(Warning: The step that goes through the third "equals" sign is "
is outside the radical.)

Simplify sqrt(–25).
", not "
". The i

Simplify sqrt(–18).

Simplify –sqrt(–6).
In your computations, you will deal with i just as you would with x, except for the fact that x2 is just
x2, but i2 is –1:

Simplify 2i
+ 3i.
2i + 3i = (2 + 3)i = 5i

Simplify 16i
– 5i.
16i – 5i = (16 – 5)i = 11i

Multiply and simplify (3i)(4i).
(3i)(4i) = (3·4)(i·i) = (12)(i2) = (12)(–1) = –12

Multiply and simplify (i)(2i)(–3i).
(i)(2i)(–3i) = (2 · –3)(i · i · i) = (–6)(i2 · i)
=(–6)(–1 · i) = (–6)(–i) = 6i
Note this last problem. Within it, you can see that
This pattern of powers, signs, 1's, and i's is a cycle:
, because i2 = –1. Continuing, we get:
In other words, to calculate any high power of i, you can convert it to a lower power by taking the
closest multiple of 4 that's no bigger than the exponent and subtracting this multiple from the
exponent. For example, a common trick question on tests is something along the lines of
"Simplify i99", the idea being that you'll try to multiply i ninety-nine times and you'll run out of time,
and the teachers will get a good giggle at your expense in the faculty lounge. Here's how the
shortcut works:
i99 = i96+3 = i(4×24)+3 = i3 = –i
That is, i99 = i3, because you can just lop off the i96. (Ninety-six is a multiple of four, so i96 is just 1,
which you can ignore.) In other words, you can divide the exponent by 4 (using long division),
discard the answer, and use only the remainder. This will give you the part of the exponent that
you care above. Here are a few more examples:

Simplify i17.
i17 = i16 + 1 = i4 · 4 + 1 = i1 = i

Simplify i120.
i120 = i4 · 30 = i4· 30 + 0 = i0 = 1

Simplify i64,002.
i64,002 = i64,000 + 2 = i4 · 16,000 + 2 = i2 = –1
Now you've seen how imaginaries work; it's time to move on to complex numbers. "Complex"
numbers have two parts, a "real" part (being any "real" number that you're used to dealing with)
and an "imaginary" part (being any number with an "i" in it). The "standard" format for complex
numbers is "a + bi"; that is, real-part first and i-part last.
Imaginary Numbers
What is
? A symbol thrust upon earthlings by an alien culture? Well, not quite...
(unless you view mathematics as representative of an alien culture... but we won't go
there).
The "usual" numbers "normal" people work with are called real numbers. They include
whole numbers, fractions, decimals, negative numbers, square roots of positive numbers,
and numbers like pi (3.14159265...). They include basically anything that a calculator
will turn into a decimal, or anything that has a position on the number line.
But
does not have a position on the number line. There is no number we can square
to get negative one. Squaring a positive number always produces a positive number, and
squaring a negative number also always produces a positive number. So how do we
square a number to get a negative one?
Answer: We can't use a real number for
. We must use a different kind of number.
The sixteenth century mathematicians who first seriously worked with
weren't sure
what its significance was, so they called it an imaginary number, a very unfortunate
term. In fact, there is nothing imaginary about
. (But it did take two centuries
before mathematicians generally understood its significance.)
Today, we often use the symbol i to represent
. Mathematicians usually use one of
two statements as the definition of i, either
or
. (Some electrical
engineers will use j instead of i, because they use i to represent inductive force, alias
current. But the idea is the same.)
We don't need any other definitions for square roots of other numbers, having i is
enough. For instance, we can write
, since the square root of 9 is 3, and the
square root of a negative number will produce the number i.
Classifying the New Types of Numbers
The collection of numbers created by taking square roots of the negative numbers are
sometimes called the pure imaginary numbers. In symbols, we might write a generic
pure imaginary number as , the product of a nonzero real number b with the imaginary
number i. None of the pure imaginary numbers have a position on the number line.
We can also talk about adding a real number with a pure imaginary number. The result is
called a complex number. In symbols, complex numbers are numbers of the form
, where a is a real number, and is a pure imaginary number.
If these numbers aren't on the number line, can we draw pictures of complex or
imaginary numbers? Actually, yes. But a one-dimensional number line will not
sufficient. The pure imaginary part of the complex number needs to be represented on a
second number line. In other words, we need a two-dimensional picture to represent
complex numbers. Here is a picture of the number
, represented by a point.
So the real numbers are points on the horizontal axis. Imaginary numbers are those
numbers which are not real, and are located somewhere other than the horizontal axis.
Pure imaginary numbers are points on the vertical axis (other than the origin). Complex
numbers include all possible points in the picture. (Real numbers are complex numbers.
So are imaginary numbers. Just like a whole number is also a fraction.)
Imaginary Numbers, numbers formed by multiplying a real number times i,
where i is the square root of minus 1. The square root of any negative number can
be expressed using i. The square root of -16, for example, is equal to i times the
square root of 16 (i):
=  = i
Certain equations, such as x2 = -1, have no real number solutions. There simply is
no real number that can be substituted for x in order to fulfill this equation because
the square of any real number, positive or negative, is always positive. During the
16th century mathematicians invented the concept of a whole new set of numbers,
the imaginary numbers, to deal with such equations.
Any sum of a real number and an imaginary number, such as 3.2 + 2i, is a complex
number. More generally, complex numbers are all of the form a + bi, where a and b
are real numbers but bi (the product of a real number and i) is an imaginary
number. Complex numbers include all real numbers because a + bi = a when b is
equal to zero. Complex numbers also include all imaginary numbers because a + bi
= bi (an imaginary number) when a is equal to zero.
In mathematics, an imaginary number (or purely imaginary number) is a complex
number whose squared value is a real number less than or equal to zero.[1] The imaginary
unit, denoted by i or j, is an example of an imaginary number. If y is a real number, then
i·y is an imaginary number, because:
Imaginary numbers were defined in 1572 by Rafael Bombelli. At the time, such numbers
were thought not to exist, much as zero and the negative numbers were regarded by some
as fictitious or useless. Many other mathematicians were slow to believe in imaginary
numbers at first, including Descartes who wrote about them in his La Géométrie, where
the term was meant to be derogatory.[2]
Although Descartes originally used the term imaginary number to mean what is currently
meant by the term complex number, the term imaginary number today usually means a
complex number with a real part equal to 0, that is, a number of the form i·y. Zero (0) is
the only number that is both real and imaginary.
Geometric interpretation
Geometrically, imaginary numbers are found on the vertical axis of the complex number
plane, allowing them to be presented orthogonal to the real axis. One way of viewing
imaginary numbers is to consider a standard number line, positively increasing in
magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this xaxis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary
numbers then "increase" in magnitude upwards, and "negative" imaginary numbers
"decrease" in magnitude downwards. This vertical axis is often called the "imaginary
axis" and is denoted
, , or simply Im.
In this representation, multiplication by −1 corresponds to a rotation of 180 degrees about
the origin. Multiplication by i corresponds to a 90-degree rotation in the "positive"
direction (i.e. counter-clockwise), and the equation i2 = − 1 is interpreted as saying that if
we apply 2 90-degree rotations about the origin, the net result is a single 180-degree
rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also
satisfies this interpretation. This reflects the fact that −i also solves the equation x2 = − 1
— see imaginary unit.
[edit] Applications of imaginary numbers
For most human tasks, real numbers (or even rational numbers) offer an adequate
description of data. Fractions such as ⅔ and ⅛ are meaningless to a person counting
stones, but essential to a person comparing the sizes of different collections of stones.
Negative numbers such as −3 and −5 are meaningless when measuring the mass of an
object, but essential when keeping track of monetary debits and credits[2]. Similarly,
imaginary numbers have essential concrete applications in a variety of sciences and
related areas such as signal processing, control theory, electromagnetism, quantum
mechanics, cartography, vibration analysis and many others.
In electrical engineering, for example, the voltage produced by a battery is characterized
by one real number (called amplitude), such as +12 volts or −12 volts. But the "AC"
voltage in a home requires two parameters. One is an amplitude, such as 120 volts, and
the other is an angle (called phase). The voltage is said to have two dimensions. A 2dimensional quantity can be represented mathematically as either a vector or as a
complex number (known in the engineering context as phasor). In the vector
representation, the rectangular coordinates are typically referred to simply as X and Y.
But in the complex number representation, the same components are referred to as real
and imaginary. When the complex number is purely imaginary, such as a real part of 0
and an imaginary part of 120, it means the voltage has an amplitude of 120 volts and a
phase of 90°, which is physically very real.
Some programming languages have built-in support for imaginary numbers. For example,
in the Python interpreter, one may use them by appending a lowercase or uppercase J to
the number[3]:
IN THE LESSON ON RADICALS, we saw how to solve any equation in
this form:
x² = a.
The solution is
x=±
.
If we apply that rule to this equation -x² = −1
-- then
x=±
.
But
is not a real number. There is no positive or negative
number whose square will be negative. Nevertheless, it turns out to
be extremely useful in mathematics and science to say that the
equation
x² + 1 = 0
has a solution.
is called a complex number or an imaginary number. It is the
imaginary unit. Its symbol is i.
i =
.
The complex number i is purely
algebraic. That is, we call it a "number"
because it will obey all the rules we
normally associate with a number. We
may add it, subtract it, multiply it, and
so on.
The basic algebraic property of i is the following:
i² = −1
Example 1. 3i· 4i = 12i² = 12(−1) = −12.
Example 2. −5i· 6i = −30i² = 30.
We can see, then, that the factor i² changes the sign of a
product.
Problem 1. Evaluate the following.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
a) i² = −1
b) i· 2i = 2i² = 2(−1) = −2
c) (3i)² = 3²i² = −9
d) −5i· 4i = −20i² = 20
Negative radicand
If a radicand is negative -, where a > 0,
-- then we can simplify it as follows:
=
=
Examples 3.
= i
.
= i
= i
= i
= 2i
= 2i
Problem 2. Express each of the following in terms of i.
a)
= i
b)
= 3i
c)
= 7i
d)
= i
g)
= 3i
e)
= i
h)
= 5i
f)
= 2i
i)
= 7i
Powers of i
Let us begin with i0, which is 1. (Any number with exponent 0 is
1.) Each power of i can be obtained from the previous power by
multiplying it by i. We have:
i =1
0
i =i
1
i = −1
2
i = −1· i = −i
3
i = −i· i = −i² = −(−1) = 1
4
And we are back at 1 -- the cycle of powers will repeat! Any
power of i will be either
1, i, −1, or −i
-- according to the remainder upon dividing the exponent n by 4.
Examples 4 .
i
9
i
18
=
i, because on dividing 9 by 4, the remainder is 1. i = i .
9
1
= −1, because on dividing 18 by 4, the remainder is 2. i =
i.
18
2
= −i, because on dividing 35 by 4, the remainder is 3. i =
i.
35
i
35
i
40
3
=
1, because on dividing 40 by 4, the remainder is 0. i = i .
40
Note: Even powers of i will be either 1 or −1, according as
the exponent is a multiple of 4 or 2 more than a multiple of 4.
While odd powers will be either i or −i.
Problem 3. Evaluate each power of i.
a) i = −i
b) i = 1
d) i = i = i
e) i = i = 1
f) i = i = i
g) i = i = −i
h) i = i = −1
i) i = i = 1
3
9
1
27
3
c) i = i = −1
4
12
30
6
0
2
2
17
100
1
0
Algebra with complex numbers
Complex numbers follow the same rules as real numbers. For
example, to multiply
(2 + 3i)(2 − 3i)
the student should recognize the form (a + b)(a − b) -- which will
produce the difference of two squares. Therefore,
(2 + 3i)(2 − 3i) = 4 − 9i²
= 4 − 9(−1)
=4+9
= 13.
Again, the factor i² changes the sign of the term.
Problem 4. Multiply.
0
a) (1 + i
)(1 − i
) = 1 − 2i² = 1 + 2 = 3
b) (3 − i
)² =
9 − 6i
+ 2i²,
=
9 − 6i
−2
=
7 − 6i
c) (2 + 3i)(4 − 5i) =
upon squaring the binomial,
8 − 10i + 12i − 15i²
=
8 + 2i + 15
=
23 + 2i
Problem 5. (x + 1 + 3i)(x + 1 − 3i)
a) What form will that produce? The difference of two squares.
b) Multiply out.
(x + 1 + 3i)(x + 1 − 3i) =
c) (x − 2 − i
)(x − 2 + i
)=
(x + 1)² − 9i²
=
x² + 2x + 1 + 9
=
x² + 2x + 10
(x − 2)² − 2i²
=
x² − 4x + 4 + 2
=
x² − 4x + 6
The real and imaginary components
Here is the standard form of a complex number:
a + bi,
where both a and b are real. For example,
3 + 2i.
a -- that is, 3 in the example -- is called the real component (or
the real part). b (2 in the example) is called the imaginary
component (or the imaginary part). Again, the components are real.
Problem 6. Name the real component a and the imaginary
component b.
a) 3 − 5i a = 3, b = −5.
b) 1 + i
a = 1, b =
c) i a = 0, b = 1.
d) −6 a = −6, b = 0.
.
Complex conjugates
The complex conjugate of a + bi is a − bi. The main point about
a conjugate pair is that when they are multiplied -(a + bi)(a − bi)
-- a positive real number is produced. For, that form is the
difference of two squares:
(a + bi)(a − bi) = a² − b²i² = a² + b²
The product of a conjugate pair
is equal to the sum of the squares of the
components.
Problem 7. Calculate the positive real number that results from
multiplying each number with its complex conjugate.
a) 2 + 3i. (2 + 3i)(2 − 3i) = 2² + 3² = 4 + 9 = 13
b) 3 − i . (3 − i )(3 + i ) = = 3² + ( )²= 9 + 2 = 11
c) u + iv. (u + iv)(u − iv) = u² + v²
d) 1 + i. (1 + i)(1 − i) = 1² + 1² = 2
e) −i.
(−i)(i) = −i² = 1