Outer Limits of Reason (57 page)

Furthermore, if we assume that
a
and
b
are the smallest such numbers, there are no smaller squares that make this true.

When the two smaller squares are placed into the larger one, there must be some overlap and two missing parts, as in
figure 9.3
.

There are two problems in 
figure 9.3
: there is an overlap that shows we counted an area twice and there are two smaller regions of the
a
box that are not covered by either of the two
b
boxes. In order for the areas of the larger square to equal the sum of the two smaller ones, the two missing parts must collectively have the same area as the overlap—that is, the two lesser (missing) squares must fit into the larger (overlap) square. These are necessarily smaller than our original squares. But wait! We assumed that the squares we started with (of size
a
and
b
) were the smallest ones such that the two lesser ones fit into the larger one, but now we've found smaller ones. This is a contradiction. There must be something wrong with our assumption that such numbers exist. Conclusion: the square root of 2 is not a rational number. It is an irrational number.

 

Since classical times, many other math problems have also defied human reason. These problems continued to be pursued through medieval times and were finally shown to be unsolvable in modern times.

The ancient Greeks approached mathematics with a geometric slant. Their methods continue to be employed today in high school geometry classes. For the classical Greeks, mathematics was all about constructing geometric objects using a straightedge and a compass. If a shape could be constructed in this manner, the Greeks were able to deal with it, and if not, the shape was considered unreasonable. The constructions of shapes were easy to describe: given two points, one can use a straightedge to draw a line connecting the points. Given a point that is the center and another point that indicates the radius, one can use the compass to make a circle.
⁵
Iterating these two operations can lead to many different shapes. We, of course, are interested in what geometric objects
cannot
be constructed.

Three of the most famous construction problems from classical times are depicted in
figure 9.4
.

The first problem (a) is called
squaring the circle
. Given a circle with a certain area, use a straightedge and a compass to construct a square that has the same area as the circle. In order for someone to perform such a construction, they would need to construct a line segment comparable to the number π. Another famous problem (b) is to take a given angle and split it into three equal parts. This problem is called
trisecting an angle
. The third problem (c) is called
doubling the cube
, and entails taking a cube of a certain size and forming another cube twice its volume. To construct such a cube, one needs to construct a line segment proportional to the cube root of 2. We will soon see that all three of these constructions are impossible using a straightedge and a compass.

The Greeks declared that any number for which a line segment of that size can be constructed using a straightedge and a compass is called a
constructible number
(also called a
Euclidean number
). All whole positive numbers can be constructed. It can be shown that we can multiply and divide using those instruments, hence all rational numbers are constructible. There are, however irrational numbers that are also constructible. For example, if you make a square whose sides are length 1, then the diagonal is the square root of 2, which can be constructed.

Modern mathematicians have defined a larger class of numbers called
algebraic numbers
. These are numbers that are solutions to polynomial equations of the form

a
_n
x
ⁿ
+
a
_n
–1
x
^n
–1
+ ··· +
a
₂
x
²
+
a
₁
x
+
a
₀
= 0

where all the coefficients are whole numbers. Since every rational number
a/b
is the solution to the equation

bx – a =
0,

every rational number is algebraic. It is known that every constructible number is algebraic. However, there are more algebraic numbers. For example, the cube root of 2 is not constructible, but it is algebraic because it is the solution to

x
³
–
2 = 0.

However, not every real number is an algebraic number. Real numbers that are not algebraic are called
transcendental numbers
. Such numbers are not solutions to an algebraic equation. In a sense these numbers cannot be described by the usual algebraic operations. If a number is transcendental, then it is not algebraic and definitely not constructible. It turns out that it is very hard to prove that a number is a transcendental number. It was not until 1844 that mathematicians proved that any transcendental number exists. Then, in 1882, Ferdinand von Lindemann (1852–1939) proved that π is transcendental. This shows that π cannot be constructed, and hence it is impossible to square the circle using a straightedge and a compass. It has also been proved that it is impossible to trisect an angle and double a cube.

Very few numbers are known to be transcendental. One might be tempted to say that since there are so few known examples, few exist. But a short counting argument shows this premise is totally wrong. Consider the hierarchy of types of numbers in
figure 9.5
.

Every algebraic number has some integer polynomial equation for which it is a solution. Using a clever counting technique (like the zigzag theorem or the necklace theorem that we saw in
section 4.2
), one can show that the set of integer polynomial equations is countably infinite. This shows that there is only a countably infinite number of algebraic numbers. Therefore, since there is an uncountably infinite number of real numbers, the numbers that are transcendental real numbers are uncountably infinite. Another way of looking at this is that the numbers we can describe with the usual operations—algebraic numbers—are countably infinite, but there are vastly more numbers that cannot be described by the usual operations. From this we can conclude that the quantity of shapes and numbers that cannot be constructed is vastly larger than the quantity of shapes and numbers that can be constructed.

9.2  Galois Theory

Paris. The night of May 29, 1832. A young man was frantically writing a long letter. He had to write quickly because there was much to say and he knew he would be killed the next day. The letter contained a summation of his mathematical research, and he wanted to put it all on paper before it was too late. He finished the letter with a plea to his friend to “ask [world-famous mathematicians] Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.”
⁶
The following day he fought a duel of honor for the love of a woman, and as he predicted, he was mortally wounded. Brought to the local hospital, he survived one more day. Supposedly, his last words were to his brother: “Don't cry, Alfred! I need all my courage to die at twenty.”
⁷
The young man's name was Évariste Galois and his work will always be a major part of modern mathematics.

What was in this letter? Hermann Weyl (1885–1955), one of the greatest mathematicians of the twentieth century, wrote that “this letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”
⁸
Perhaps Weyl was being a little bombastic in his assessment. Nevertheless, Galois's work contains ideas that are essential for modern mathematics and physics. What can a twenty-year-old possibly have to say that is so important?

Born in 1811, during post–French Revolutionary fervor, Galois had a short and tragic life. His father was the mayor of a small town outside Paris who committed suicide after a bitter political dispute. Évariste was a passionate and complicated youth who was not an easy person to deal with. At a young age he became obsessed with mathematics to the exclusion of his other studies. He failed to get into the École Polytechnique, the most prestigious school in France. He eventually was admitted to a second-tier school, but his brilliance was mostly misunderstood by his teachers. Galois submitted two articles for publication and both were supposedly lost by the editors. At some point he became involved with radical political groups, which led to his expulsion from school. It is not known who the other player was in the fatal duel. The identity of the woman the duel was fought over is also unknown. One can only speculate about the other works this young genius might have accomplished if his life had not been tragically cut short.

Galois's work dealt with solving polynomial equations. Before we can understand his contributions, we have to study some history. Consider the following simple equation:

ax
+
b
= 0.

Such an equation is called a “linear” equation, and most ninth-grade students know how to solve for x:

x
= –
b
/
a
.

More complicated, “quadratic” equations take the following form:

ax
²
+
bx
+
c
= 0.

The solutions to such equations were known in ancient times and are still taught to high school students using the “quadratic formula.” There are actually two solutions:

and

.