# Solution of the Equations

How do we know linear equations have solutions? Well, we
can solve them by

algebra. For example:

17x + 4 = 1

17x = 1 − 4 = −3

We can do the same for many quadratic equations, using the quadratic formula:

If ax^{2} + bx + c = 0, then

As long as the expression under the radical sign is not
negative, we’ll get one or

two real-number solutions. (We’ll think more about what happens if it is
negative

later.) It turns out there are similar, but longer, formulas for degree three,
and

even degree four polynomials. But they seem to get harder and harder as the

degree gets larger, and last week we saw that there are some polynomials of

degree five and higher that are impossible to solve by an explicit formula. But
we

can try another approach: Consider the polynomial equation 3x^{5} − 15x + 5 = 0

(One of the “impossible” equations featured last week.).

Figure 1. Graph of p(x) = 3x^{5} − 15x + 5

Its graph certainly appears to cross the x-axis in several
places. And indeed

it does: We can calculate that p(0) = 5, p(1) = −7, and then apply the

Intermediate Value Theorem to conclude that its graph must cross the x-axis

somewhere in between:

Theorem. Let f be a continuous,
real-valued function defined on aclosed interval [a, b]. If y is any number between f(a) and f(b), then there is at least one c in [a, b] with f(c) = y. |

If you take the time to understand what this theorem is
saying, it might seem

obvious: to move continuously from f(a) to f(b), you need to pass through

each point in between. It is obvious. But the fact that it is both obvious and

true is deep, because it shows how closely the real number system models our

intuitive notions of continuity and connectedness. The statement would be false,

for instance, if we restricted attention only to rational numbers or integers.

What about other polynomials? If p is any polynomial of
odd degree, we can

generalize the geometrically “obvious” thinking we saw above:

• If
then for large x-values, p(x)

behaves much like its highest degree term:
(Of course, the

meaning of “ ≈ ” is relative: the difference between p(x) and a_{n}x^{n} is
only

small compared to the size of p(x) itself.) Let’s assume a_{n} is positive from

now on, just for convenience.

• If a_{n} is positive, then, a_{n}x^{n} is positive when x is
positive, and – because

n is odd – negative when x is negative.

• Thus for large positive x-values, p(x) is positive, and
for large negative

x-values, p(x) is negative.

• So, by the Intermediate Value Theorem, a root exists
somewhere between

these large negative and large positive x-values.

What about even degree polynomials? To take a simple case,
let p(x) = x^{2} −

2x + 2:

Figure 2. Graph of p(x) = x^{2} − 2x + 2

As you can see, the parabola “goes upward” on both sides,
and does not cross

the x-axis anywhere. But the quadratic formula does give us the two “solutions”

and
Do they have a geometric interpretation terms

of intersecting graphs?

In fact, they do. The numbers
are not real (in the sense of having

addresses where they live on the real number line). We can think of them as new

kinds of numbers, and if we visualize the reals as extending horizontally, then
we

can pick one of the two (arbitrarily), call it i, call the other −i, and place
them

vertically above and below the real number line along the y-axis.

Once i becomes a number, it has the same rights and duties
to be added, subtracted,

multiplied, and divided as any of its “real” brethren do. Just as multiplication

by a positive number a sends a real number horizontally a times as far

from 0 as it used to be, the same operation with a real number b sends i
vertically

b times further away. In this way, we label each point (0, b) along the y-axis
with

the new number bi. Thus every “address” (a, b) in the xy plane becomes the

home of the so-called “complex number” a+bi. Moreover, we can add complex

numbers simply by adding their horizontal and vertical coordinates separately:

(a + bi) + (c + di) is defined as (a + c) + (b + d)i.

And we can multiply them, too, simply by treating i^{2},
whenever we see it, as −1.

We thus extend ordinary arithmetic with real numbers to the complex realm.

Just as the set of real numbers is often denoted R, we use
C for the complex

numbers.

Now the fact that the two complex numbers
and

appeared as roots of the polynomial equation x^{2}−2x+2 = 0 means, algebraically,

that when we apply the extended arithmetic of the complex numbers to plug them

into the polynomial p, the result is 0. What about geometrically?

We graph a function f on the real line R using two copies
of R which are

perpendicular to each other, intersect in a point, and together form a plane,

R2. Numbers on the first (horizontal) copy represent inputs, and numbers on

the second are outputs. Each point (x, y) which lies on the curve of a graph

corresponds to the function value f(x) = y. And the intersection of two curves

(like a graph and the x-axis) corresponds to a common solution to a pair of

equations.

Naturally, then, we would graph a function on the complex
plane C using two

copies of C which are perpendicular, intersect in a point, and together form. .
.**a
what?**

. . . Aye, there’s the rub. If we put two perpendicular
planes together, we should

get a four-dimensional space, but humans have a tough time visualizing such a

thing. That’s where mathematicians come in.

Instead of a “curved line” in R^{2}, the graph of a
complex-valued function will

consist of a “curved surface” in C^{2}. The fact that the quadratic formula gives

the two roots x = 1±i for the equation p(x) = 0 means that the curved surface

graph of p intersects the first copy of C in C^{2} (the 2-dimensional counterpart
to

the x-axis in the real case) in those two points.

Notice that, in this case, at least, jumping from the
reals to the complex numbers

has insured that our even-degree polynomial has roots. Can we generalize that

argument using the intermediate value theorem to show that every polynomial,

of odd or even degree, has a root among the complex numbers? Let’s try. Our

first step is a nice bit of geometry: We can rewrite the “addresses” of complex

numbers using polar coordinates:

We label points in the plane by indicating the direction
(in terms of an angle

measured from the x-axis) and distance from the origin, rather than by their
horizontal

and vertical displacements from 0. It isn’t too hard to calculate formulas

for the translation in each direction:

The corresponding re-labeling of complex numbers has some nice properties:

• The number whose polar coordinates are (r, θ ) is actually
equal to re^{iθ} .

(Yes, that’s the familiar e ≈ 2.71828 from calculus. In the name of equal

rights for complex numbers, we’re extending our familiar exponentiation

function to allow for imaginary exponents.)

• Using the rules for multiplying exponentials, we have

• Thus to get the polar-coordinate “address” of the
product

we multiply the distances from 0 and ADD the angles.

In particular, we can geometrically understand the
function p(x) = x^{n} by noting

that

It stretches distances from 0, and multiplies angles by n.
For instance, p(x) = x^{2}

will carry each point on the unit circle to a corresponding point with twice the

angle, and hence wrap the circle around itself twice. And it would wrap the

radius 2 circle twice around the radius 4 circle, the radius 3 circle twice
around

the radius 9 circle, and so forth. . . . So in general,

The function p(x) = x^{n} wraps the whole complex
plane n times arounditself, and stretches it radially via |

Now here is the basic principle which generalizes the Intermediate Value Theorem:

Theorem. Let f be a continuous,
complex-valued function defined ona closed disk D. If f maps the boundary of the disk n > 0 times around the boundary of another disk D0, and y is any point in the interior of D0, then there is at least one c in D with f(c) = y |

This may not seem quite as obvious as the Intermediate
Value Theorem itself. But

it reflects how closely the complex number system models our intuitive notions

of continuity and two-dimensional connectedness: If you were to stretch the

boundary of a rubber disk around another disk, with no rips or punctures, it

would have to cover every point in the second disk.

Now we are in a position to understand the first version
(the regular-strength

version, we might say) of the