Download Impossible Numbers

MAT 320 Spring 2008
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You may remember from geometry that you
can perform many constructions only using a
straightedge and a compass

These include drawing circles, constructing
right angles, bisecting angles, etc.

But there are other problems that the ancient
Greeks wanted to try to solve with this
method

The Greeks wanted to know if any of the
following were possible
 Trisecting the angle: Given an angle, divide it into
three congruent angles
 Doubling the cube: Given a cube, construct
another cube with exactly twice the volume
 Squaring the circle: Given a circle, create a square
with the same area

It turns out that all of these constructions are
impossible

In order to understand why, we need to think
about how constructions really work

We start with two points, (0, 0) and (1, 0)

We say that we can “construct” a point (x, y) if
we can find that point as an intersection of lines
or circles that we can construct

The things we can construct are
 Lines: We can use our straightedge to construct a
line between any two points
 Circles: Given two points, we can construct a circle
with the center at one point and which passes
through the other
 Perpendiculars: Given a line and a point, we can
construct a perpendicular line that passes through
the point

We say that a number is “constructible” if it is
the x or y-coordinate of a constructible point

For example, all of the integers are
constructible

The number is also constructible, since the
point
is the intersection of the first two
circles on the previous slide

In fact, the set of constructible numbers is
closed under addition, subtraction,
multiplication, division, and square roots

The set of constructible numbers forms a field
that contains the rational numbers

This field contains only those numbers that
can be obtained from (possibly repeatedly)
extending Q with the roots of quadratic
polynomials

For example, Gauss showed that

Since this number is constructed out of
rational numbers and square roots, this
number must be constructible

We can use this fact to construct a regular
17-sided polygon

Let’s think about trisection of an angle,
specifically a 60-degree angle

60-degree angles are constructible: cos(60) and
sin(60) are both constructible numbers

What about 20-degree angles?

Using trig identities, it’s possible to show that
cos(20) is a root of the polynomial x3 – 3x – 1

Since the polynomial for which cos(20) is a
root has degree 3, that means that cos(20)
will involve cube roots, which aren’t allowed

So cos(20) is not a constructible number, and
60-degree angles are just one example of
angles we cannot trisect with straightedge
and compass

Given a 1 x 1 x 1 cube, we would need to
construct a x x cube to have exactly
double the volume

But is not a number we can construct, so
we wouldn’t be able to create a segment
exactly
units long to create our cube

Given a circle of radius 1 (and area π), we
would need to construct a square whose sides
have length the square root of π

Even though square roots are allowed, π is
not a rational number

It turns out π is a transcendental number,
which means it’s not the root of any
polynomial with rational coefficients

Another famous impossibility that is related
to these ideas is credited to Niels Abel (18021829)

He proved that there is no way to solve a
generic fifth-degree polynomial using radicals
(even allowing 5th roots!)

Of course, some quintics are solvable using
radicals

An example is
(twice), -1, i, and –i

But what Abel proved is that there is no
analogue to the “quadratic formula” for
quintics
, whose roots are 1

Abel’s proof is beyond what we have learned
in this course, but here are some related ideas

Have you ever noticed that roots of
polynomials tend to come in groups?

For example, if you know that
is the
root of a quadratic, you can be sure that
is also a root

It turns out that this is no accident

The roots of higher degree polynomials are
related in more complicated ways, but they
are still related

Once the degree reaches 5, the relationships
become so complicated that there is
sometimes no way to “unentangle” the roots
from one another

Keep in mind that we can still solve quintic
equations using numerical methods

The issue is that some quintic equations have
roots that we cannot express with our normal
radical notation

One example is x5 – x + 1

This does not mean that the roots don’t exist as
complex numbers