1. CHANGES IN MATHEMATICAL LAWS
A Thesis submitted for Albion College Honors
Brian G. Wu
April 1, 2014
Albion College
2. Abstract
Over the years, mathematicians have changed the rules of mathematics to compute
what previously seemed to be impossible. Mathematicians initially thought that
one could not take the square root of a negative number. To solve this problem,
mathematicians introduced the complex number system to calculate the roots of a
polynomial, including those involving square roots of a negative number. Geometers
initially first studied Euclidean geometry, where parallel lines never intersect. Looking
from a new perspective in projective geometry, mathematicians realized that parallel
lines intersect at a point at infinity. Division by zero also seemed impossible. Recently,
mathematicians invented two numbers, V and u, which result from division by zero.
Adding these two elements to a field results in a new algebraic structure called a
wheel. This thesis presents the background of the historical development of numbers,
projective geometry, and the basics of a wheel. In particular, it includes a discussion
of properties that are true in a field but fail to be true in a wheel, for example, the
division algorithm for polynomials.
I
6. Chapter 1
Introduction
What are some of the bottlenecks mathematicians face in arithmetic and geometric
systems? One of them is closure under operations. An operation of at least one
element in a set satisfies closure if the operation yields another element in the set.
However, operating on an element from a set often results in an element that does not
belong in the set. For example, taking the square root of a non-negative real number
results in a real number, but taking the square root of a negative real number does not.
To solve this problem, mathematicians expanded the number system. Additionally,
one realizes that some geometric properties, especially the notion of parallel lines
never intersecting, do not hold true in all situations. To solve this problem, geometers
looked away from the Euclidean space and invented projective geometry.
Expanding and generalizing a mathematical concept is a common theme in math-
ematics. The familiar rules of algebra forbids division by zero because if a divided by
zero equals b, then b times zero would have to equal a, which is not true. As shown
in Figure 1.1, the graph of y 1
x
is unbounded when x 0, making the function
1
7. Figure 1.1: In this graph of y 1
x
, when x 0, the function is unbounded. There is
a vertical asymptote at x 0.
undefined at x 0. To fill in this gap, mathematicians recently defined two new
elements, V 1
0
and u 0
0
, representing the two cases that arise when dividing by
zero. To understand the algebraic properties that are lost because of division by zero,
one must understand the motivations that led to figuring out a solution to division by
zero as well as the notion of familiar structures. This work discusses past examples
in the history of mathematics where this kind of a “gap” has been filled.
2
8. Chapter 2
Numbers
2.1 Natural Numbers
As a young student, one first deals with the set of natural numbers N, which one uses
for counting (e.g. how many cookies are in a box) and ordering (e.g. the third largest
city in the world). They are 1, 2, 3, 4, . . .. Some authors choose to include zero in N.
However, including zero may lead to counting errors because one never counts from
zero. One can always add natural numbers and still end up with a natural number,
which satisfies closure under addition. One can subtract a number from a larger
number and end up with a natural number. However, one cannot subtract a number
from a smaller number and produce a natural number, making this operation lack
closure under subtraction. Subtracting a number from a smaller number would yield
a negative number. For example, 5 ¡ 4 1, but 4 ¡ 5 ¡1, which is not a natural
number. The integers are an extension of the natural numbers that allow negative
numbers.
3
9. 2.2 Integers
The set N is a subset of Z, the set of integers. An integer is a number that is expressed
without fractional or decimal components. In other words, Z includes positive or
negative whole numbers. Examples of integers include 2, 0, and -2355. Applications
include business, where a negative number may often indicate a loss. Addition and
subtraction of integers still yields or results in integers. Moreover, subtracting b from
a would be the same as adding a to ¡b. Although multiplying integers still yields an
integer, dividing integers does not necessarily yield an integer. For example, 3 is not
divisible by 12, but 12 is divisible by 3. A number a is divisible by a second number
b, denoted by b # a, if a divided by b yields an integer. If b does not divide a, denoted
b 1 a, then dividing a by b results in a nonzero remainder. For example, 7 divided by
2 equals 3 with a remainder of 1, or by using fractional components, 3
1
2
, which is not
an integer. By using decimal components, one would write
7
2
3.5. To handle these
cases, one uses rational numbers.
2.3 Rational Numbers
The set Z is a subset of Q, the set of rational numbers. A rational number is a
number that one can express as a fraction
p
q
where both the numerator p and the
denominator q are integers, as long as q is not zero. (A later section will explain what
happens if q is zero.) An integer a is also a rational number since it can be expressed
as
a
1
. Applications of rational numbers include expressing a part of a whole, such
as the number of questions correct out of the total number of questions. When one
4
10. squares a rational number, or multiplies a rational number by itself, he or she ends up
with another rational number, satisfying closure under multiplication. The opposite
of squaring a rational number would be taking the square root of a number. Taking
the square root of a rational number does not always yield a rational number; one
cannot generally express that new number as a fraction with integer numerator and
denominator.
One can add, subtract, multiply, and divide rational numbers (as long as the divi-
sor is nonzero) and end up with a rational number, satisfying closure under addition,
subtraction, multiplication, and division.
2.4 Irrational Numbers
The set of irrational numbers, I, contains numbers that cannot be expressed as a
fraction with integer numerator and denominator. Common examples of irrational
numbers include
c
2, which is the side length of a square with area 2, and π, which is
the ratio of a circle’s circumference to its diameter. According to Kimberley A. Mc-
Grath and Stacey Blachford [1, “Irrational”], the first discovery of irrational numbers
dates back to sixth century B.C. by the Pythagoreans. Before this discovery, people
thought that one could always express a number as a ratio of two integers. However,
the Pythagoreans showed “that the hypotenuse of an isosceles right triangle could not
be measured exactly by any scale, no matter how fine, which would exactly measure
the legs.” In the case of taking the square root of a rational number n, if the result is
a rational number, then n is a perfect square. Otherwise, n is not a perfect square.
5
11. If n is not a perfect square, then one cannot measure the precise length of a square
with area n.
The irrational numbers are not always closed under addition, subtraction, mul-
tiplication, and nonzero division. For example,
c
2 p¡
c
2q 0,
c
2 ¡
c
2 0,
c
2
c
2 2, and
c
2
c
2
1.
2.5 Real Numbers
The set of real numbers, R, contains all the rational numbers and irrational numbers.
A real number is a number that represents a value along a number line. When
one expresses natural numbers, integers, rational numbers, or irrational numbers in
decimal notation, he or she can determine where these numbers belong on a continuous
line as well as determine which of the numbers is the largest and which is the smallest.
For example,
c
2 1.41 is greater than 1, but less than π 3.14. In this case, on a
number line, where the numbers get bigger from left to right, of these three numbers,
1 would be farthest to the left,
c
2 would be in the middle, and π would be farthest
to the right. However, operating a rational number with an irrational number results
in an irrational number.
6
12. Chapter 3
Complex Numbers
Squaring a negative number is possible; one would end up with a positive number,
satisfying closure under multiplication. This is because a negative number multiplied
by a negative number yields a positive number. Taking the square root of a nonnega-
tive real number results in a real number, satisfying closure under multiplication, but
taking the square root of a negative number does not. Consequently, mathematicians
expanded the number system to include imaginary numbers, where
c
¡1 i and
i2
¡1. The number i is referred to as the imaginary unit.
3.1 What is a Complex Number?
The set of complex numbers C contains numbers that can be expressed as a bi
where a and b are real numbers and i is the imaginary unit. Complex numbers are
commonly expressed in the way such that a is the real part while b is the imaginary
part. In the situation where b 0, the number a 0i a, which is a real number. In
7
13. the situation where a 0, the number 0 bi bi, which is called a purely imaginary
number. According to author Paul J. Nahin [2, p. 10], a complex number “is neither
purely real nor purely imaginary, but rather is a composite of the two.” Examples
of complex numbers include 7, 2i, 3 i, and 2 ¡ 2i. According to McGrath and
Blachford [1, “Complex”], these numbers are called complex numbers because the
real and the imaginary parts cannot be added or subtracted together; they need to
be left as an indicated sum or difference. In this case, one cannot add real numbers
to imaginary numbers for a more “unified” number. Complex numbers also play a
big part in the study of polynomials, as shown in the next section. Applications of
complex numbers include electrical engineering, physics, and mathematical analysis.
Shockingly, Nahin [2, p. 187] states that physicists and engineers use complex numbers
more than mathematicians do.
3.2 Polynomials
A mathematical example where complex numbers are important is in the study of
polynomials. A polynomial is an expression of the form
anxn
an¡1xn¡1
. . . a2x2
a1x a0
for all arguments x, n a nonnegative integer, and a0, a1, a2, . . . , an are real or complex
coefficients. Polynomials can be added, subtracted, multiplied, or divided. While
adding, subtracting, and multiplying polynomials results in another polynomial, di-
vision of polynomials does not always lead to another polynomial and may result in
a remainder. For example, dividing x3
¡ 2x2
¡ 4 by x ¡ 3 equals x2
x 3 with a
8
14. remainder of 5.
The degree of a polynomial is the largest exponent, in other words, n. Math-
ematicians often classify polynomials according to the degree. If the degree of a
polynomial is one, such as x 1, then the polynomial is linear. If the degree is two,
such as x2
2x 1, then it is quadratic. If the degree is three, such as x3
2x2
x 1,
then it is cubic. One can factor a polynomial into irreducible polynomials just like one
can factor integers into prime numbers, which are also irreducible. For example, one
can factor x2
2x 1 into px 1q2
. The irreducible polynomials are either quadratic
with real coefficients or linear with complex coefficients.
The fact that polynomials can be broken down into linear terms with complex
coefficients is a consequence of the following, known as the Fundamental Theorem of
Algebra.
Theorem (Gauss). An nth-degree polynomial equation
Ppxq anxn
an¡1xn¡1
. . . a1x a0 0,
where the ai € C and n € N is a positive integer, has exactly n roots.
A number r is a root of a polynomial Ppxq if and only if
Ppxq
x ¡r
is also a polynomial. Let k ¥ 1 be the largest value with
Ppxq
px ¡rqk
a polynomial.
Then r is said to have multiplicity k. The sum of the multiplicities of all the roots
is equal to the degree of Ppxq. Generally, complex numbers make it possible to find
every root of a polynomial equation. While this theorem is easy to understand, the
9
15. proof is beyond the scope of this paper. The proof can be found in The Fundamental
Theorem of Algebra by Benjamin Fine [3].
10
16. Chapter 4
History of Complex Numbers
4.1 Heron of Alexander
The first attempt of taking the square root of a negative number dates back to the
first century A.D. when mathematician Heron of Alexandria attempted to measure
the height of a pyramid. According to Nahin [2, p. 3-4], the volume V of a truncated
pyramid is
V 1
3
hpa2
ab b2
q
where a and b are the lengths of the edges of the bottom and top squares, respectively,
and h is the height. Another relationship among the dimensions of a truncated
pyramid is
h
d
c2 ¡2
¢
a ¡b
2
2
11
17. where c is the length of the slanted edges. Applying the formula with a 28, b 4,
and c 15 to figure out h, Heron would have computed
h
d
p15q2 ¡2
¢
28 ¡4
2
2
c
¡63
but in his Stereometria, he ignored the negative sign, instead writing h
c
63.
Nahin [2, p. 4] concludes that it was not until a thousand years afterwards that a
mathematician “would even bother to take notice of such a thing—and then simply to
dismiss it as obvious nonsense—and yet five hundred years more before the square root
of a negative number would be taken seriously (but still be considered a mystery).”
4.2 Diophantus
Two centuries later, Diophantus of Alexandria encountered the square root of a neg-
ative number as well. Based on Nahin [2, p. 6], just as Euclid was known for his
Elements, Diophantus was known for his Arithmetica. In Arithmetica, Diophantus
discussed the quadratic equation
172x 336x2
24.
Using the quadratic formula yields
x 43 ¨
c
¡167
168
,
which involves the square root of a negative number. However, Diophantus wrote that
the quadratic equation was impossible, meaning that “the equation has no rational
solution because ‘half the coefficient of x multiplied into itself minus the product of
12
18. the coefficient of x2
and the units’ must make a square for a rational solution to exist,
while
¢
172
2
2
¡p336qp24q ¡668
certainly is not a square.” In other words, Diophantus did not have the mathematical
techniques to deal with the square root of negative numbers.
4.3 The First Actual Attempt
Mathematicians began a more formal attempt of understanding the square root of a
negative number when they tried solving cubic equations. A general cubic equation
takes the form
x3
a1x2
a2x a3 0.
For example, one can factor x3
¡8 0 to px ¡2qpx2
2x 4q 0. Setting x ¡2 0
leads to the real solution of x 2, but when setting x2
2x 4 0, the quadratic
formula implies solutions of x ¡1 ¨
c
¡3, which involves the square root of a
negative number. According to Nahin [2, p. 8], the Franciscan friar Luca Pacioli
“declared that the solution of the cubic equation is ‘as impossible at the present state
of science as the quadrature of the circle.’” At the end of the fourteenth century,
Italian mathematician Scipione del Ferro attempted to solve the so-called depressed
cubic equation
x3
px q
13
19. where p and q are nonnegative; he determined a solution to be
x 3
d
q
2
™
q2
4
p3
27
3
d
q
2
¡
™
q2
4
p3
27
.
While this above equation yields a real result, Nahin [2, p. 11] uses calculus and a
graph to show that del Ferro’s depressed cubic equation always has one real root and
two complex roots.
4.4 A Secret Revealed
Although del Ferro found a solution to the depressed cubic equation, he wanted to
keep it a secret, but it was not to be. Based on Nahin [2, p. 15], this is an example
where “once a problem is known to have a solution, others quickly find it, too—a
phenomenon related, [he thinks], to sports records, e.g., within months of Roger
Bannister breaking the four-minute mile it seemed as though every good runner in
the world started doing it.” As it turned out, an Italian mathematician Gerolamo
Cardano found out this secret. He wrote it in the Ars magna, which, according
to Suzanne Michele Bourgoin and Paula K. Byers, [4, “Cardano”], “contained the
theories of algebraic equations as they were known at that time.” Eventually, he
presented a complicated formula to solve the depressed cubic equation. Cardano’s
formula can be found in “Cubic Formula” in Wolfram MathWorld by Eric Weisstein
[5, “Cubic”].
14
20. 4.5 Bombelli
Italian mathematician Rafael Bombelli in his Algebra of 1572 was the first to truly
understand imaginary numbers. Bombelli wrote the equation
x3
15x 4,
which has the solution x 4. However, using Cardano’s formula,
x 3
˜
2
c
¡121 3
˜
2 ¡
c
¡121.
At least one knows that Bombelli found a working solution. According to Nahin [2, p.
19], Bombelli “wrote in his Algebra, ‘It was a wild thought in the judgment of many;
and I too for a long time was of the same opinion.’”
4.6 The Term Imaginary
More importantly, researcher Robin Hartshorne [6, p. 493] describes that for an equa-
tion of degree n with fewer than n roots, one can imagine the non-real roots, which is
how French mathematician Ren´e Descartes formulated the word imaginary. Although
imaginary numbers refer to impossible quantities, Swiss mathematician Leonhard Eu-
ler pointed out that one could still make useful calculations and derive useful results
with them. Imaginary numbers played an important role in trigonometric functions,
logarithms, exponentials, and infinite series. While Gauss introduced the term com-
plex number in his theory of biquadratic residues, Hartshorne [6, p. 493] concludes
that the geometrical representations of Wessel and Jean-Robert Argand as well as
15
21. the later developments of Cauchy and Hamilton “put the theory on a solid logical
foundation.” Going back to
Ppxq anxn
an¡1xn¡1
. . . a1x a0 0,
from the Fundamental Theorem of Algebra, based on Nahin [2, p. 227], Argand, not
Gauss, pointed out that the a’s can possibly be complex numbers.
4.7 A Geometrical Interpretation of Complex Numbers
Although Bombelli found a working solution for
c
¡1, there still needed to be a
geometrical interpretation. Based on Nahin [2, p. 31-34], Ren´e Descartes constructed
the square root of a line segment in his 1637 La Geometrie. However, his constructions
only dealt with the square root of positive lengths, not negative lengths. Descartes
considered the construction of the square root of negative lengths, but he stated the
square root of a negative number geometrically meant “the sheer impossibility of
doing a geometric construction.”
Norwegian-Danish Caspar Wessel was the first mathematician to describe com-
plex numbers geometrically using the complex plane. According to Hartshorne [6,
p. 494], Wessel attempted to “develop an analytic representation, with addition and
multiplication, of lines with direction (which we would call vectors) in the plane, and
he [wished] to apply this to the solution of plane polygons.” In other words, Wessel
wanted to use lines and planes to represent complex numbers. As a cartographer,
Wessel also spent several years measuring distances and creating maps, which moti-
vated this goal. Hartshorne [6, p. 494] then states that Wessel “finds that a line of
16
22. unit length, perpendicular to a line 1 taken as unity, has the property that 2
¡1.
This implies that
c
¡1. Although Wessel never used the terms complex nor
imaginary in his texts, this geometric analysis motivated the introduction of complex
numbers. Amateur mathematician Jean-Robert Argand also described complex num-
bers geometrically in a paper in 1806. Based on Hartshorne [6, p. 495], this “depends
on two ‘principles’ that are established only by analogy, and so must be treated as
‘hypothesis’ to be justified later.”
4.8 Famous Uses of Complex Numbers
French mathematician Augustin-Louis Cauchy launched the theory of functions of
complex variables between 1825 and 1831. According to Nahin [2, p. 188], this allows
one to “calculate the values of a seemingly endless number of incredibly odd, strange,
and downright wonderully mysterious-looking definite real integrals.” For example
» V
¡V
cos x
1 x2
dx π
e
.
According to Bourgoin and Byers [4, “Cauchy”], Cauchy also published numerous
works after 1838 including a remarkably modern representation of complex numbers
in terms of polynomial congruences.
Putting complex numbers to use, Euler formulated
eiθ
cos θ i sin θ.
Using this formula, French mathematician Abraham de Moivre derived that
cos nθ i sin nθ pcos θ i sin θqn
einθ
.
17
23. Substituting 1π for nθ, this yields the amazing result
eiπ
¡1 or eiπ
1 0,
which involves five mathematical constants 1, 0, e, π, and i. Based on McGrath and
Blachford [1, “Complex”], these mathematicians as well as the Bernoullis and others
“gave formal recognition to imaginary and complex numbers as legitimate numbers.”
18
24. Chapter 5
Euclidean Geometry
5.1 Introduction
In geometry, mathematicians define parallel lines as lines that do not intersect. In
other words, they do not share a common point. One first studies parallel lines by
visualizing the two non-intersecting lines from above, in which no matter how one
extends the lines, the lines are equidistant from each other and do not intersect. This
is shown in Figure 5.1. A common example of parallel lines are the rails of a railroad
track. When looking from above, the rails are equidistant from each other and do
not intersect.
This is true in Euclidean geometry, where, according to Coxeter [7, p. v], one
constructs figures using ruler and compass, using measurements to compare these
figures. Additionally, in Euclidean geometry, one thinks of a line as something straight
that stretches endlessly while one thinks of a plane as a flat surface, again stretching
endlessly. Interestingly, Coxeter [7, p. 2] states that Euclidean geometry “seems at
19
25. Figure 5.1: In Euclidean geometry, parallel lines are equidistant and do not intersect
regardless of how one extends these lines.
first to have very little connection with the familiar derivation of the name geometry
as ‘earth measurement.’” This is because the earth is round, while Euclidean geometry
would assume that the earth is flat.
5.2 Do parallel lines actually intersect?
However, according to Coxeter [7, p. 2], one can imagine the possibility that if one
“could extend [parallel lines] for millions or billions of miles [one] might find the lines
getting closer or farther apart.” In this case, when looking along the railroad tracks
from the ground, it appears as if the rails intersect way out in the distance, as shown
in Figure 5.2. The idea is that lines may appear one way from one perspective but
appear differently from another perspective, leading to the idea of projective geometry.
One can think of projective geometry as an extension of Euclidean geometry, where
some geometric properties remain the same. For example, a closed curve remains
closed in projective geometry. However, angles and sizes may change when viewed
20
26. Figure 5.2: Despite being parallel, the rails seem to intersect way out in the distance.
from different perspectives in projective geometry.
21
27. Chapter 6
Projective Geometry
Projective geometry deals with geometric properties that are invariant regardless
of projective transformation. In other words, these geometric properties still apply
regardless of the perspective in which one views something. Think about which
geometric properties remain when one looks at something from different points of
view. According to Coxeter [7, p. 3], projective geometry “waives the customary
distinction between a circle, an ellipse, a parabola, and a hyperbola; these curves are
simply conics, all alike.” A conic section deals with the part of a cone that intersects
with a plane. McGrath and Blachford [1, “Projective”] includes examples that deal
with how something appears on a slide and how it appears on the screen such as the
intersection of two lines appearing on both the slide and the screen as well as the
different lengths of a shadow during the different times of a day. If three points lie on
a line on a slide, the three points will lie on the same line on the screen. If two lines
intersect on the slide, they will also intersect on the screen. If a point appears above
a second point on the slide, it will never appear below the second point on the screen.
22
28. To generalize, projection does not change collinearity, intersection, and order, but it
may change sizes and angles.
23
29. Chapter 7
History of Projective Geometry
7.1 Beginning
According to Coxeter [7, p. 3], geometers such as Archimedes, Apollonius, Euclid,
and Menaechmus studied conics in the third and fourth centuries B.C., but in the
third century A.D., Pappus of Alexandria was the first to discover true geometric
properties of a projective nature. French mathematician Jean-Victor Poncelet used
purely projective reasoning to prove many of Pappus’s theorems in the nineteenth
century. According to Weisstein [5, “Pappus’s”], Pappus states one of the oldest of
projective theorems, which is illustrated in Figure 7.1.
Theorem (Pappus). If A, B, and C are three points on one line, D, E, and F are
three points on another line, and AE meets BD at X, AF meets CD at Y, and BF
meets CE at Z, then the three points X, Y, and Z are collinear.
This theorem is also a special case of a theorem by French mathematician Blaise
24
30. Figure 7.1
Pascal, discussed in a later section.
According to McGrath and Blachford [1, “Projective”], the need for projective
geometry “began with Renaissance artists who wanted to portray a scene as someone
actually on the scene might see it.” Consequently, artists needed some perspective
principles to make the imagined scenes look real. To do this, Coxeter [7, p. 2-3] states
that Italian architect Filippo Brunelleschi in 1425 discussed the perspective theory,
and Italian author Leon Battista Alberti combined this theory into a treatise years
later. Because of the application of projective geometry in fine arts, it is common to
begin in a three-dimensional space, but there is much to say when studying a single
plane.
7.2 The Point at Infinity
In the sixteenth and seventeenth centuries, German astronomer Johann Kepler and
French architect Girard Desargues developed an important concept in projective ge-
ometry: the point at infinity. The railroad example mentioned earlier illustrates this
25
31. concept, where the rails appear to intersect at a point infinitely far from the observer.
Based on McGrath and Blachford in [1, “Projective”], while Kepler introduced this
idea, Desargues made a systematic use of it. Weisstein [5, “Desargues’”] states De-
sargues’ theorem, which is shown in Figure 7.2.
Theorem (Desargues). If the three straight lines joining the corresponding vertices
of two triangles ABC and A’B’C’ all meet in a point (the perspector), then the three
intersections of pairs of corresponding sides lie on a straight line (the perspectrix).
Equivalently, if two triangles are perspective from a point, they are perspective from
a line.
Figure 7.2: Desargues’ Theorem. If the point O exists, then R, S, and T are collinear.
Continuing with Coxeter [7, p. 3], these geometers “declared that a parabola has
two foci, one of which is infinitely distant in both of two opposite directions, and that
any point on the curve is joined to this ‘blind focus’ by a line parallel to the axis.” As
it turned out, Desargues would make an important justification about parallel lines.
26
32. Back to Coxeter [7, p. 3], Desargues stated that parallel lines “have a common end
at an infinite distance.”
From this, Poncelet was able to formulate this so-called projective space from
ordinary space, where parallel planes intersect at a “line at infinity.” Back to Coxeter
[7, p. 3], this concept “serves to justify our assumption that, in a plane, any two lines
meet; for, if the lines have no ordinary point in common we say that they meet in a
point at infinity.” This is the point where one can value the extra points as much as
the ordinary points.
7.3 The Influence of Desargues
According to researcher Alexis Conrad [8], Desargues was one of the most important
figures in projective geometry, using projection to prove several theorems on conics.
Desargues influenced Blaise Pascal and Philippe de La Hire as they both took a strong
interest in the properties of conic sections in the seventeenth century. According to
Weisstein [5, “Pascal’s”], Pascal proved the following theorem, a variation on an
earlier theorem by French mathematician Charles Julien Brianchon, illustrated in
Figure 7.3.
Theorem (Pascal). Given a (not necessarily regular, or even convex) hexagon in-
scribed in a conic section, the three pairs of the continuations of opposite sides meet
on a straight line, called the Pascal line.
This theorem is also a generalization of Pappus’s theorem stated in an earlier
section. Additionally, Conrad [8] reported that La Hire wrote Sectiones Conicae and
27
33. Figure 7.3: The Pascal Line
studied the work of Appollonios, proving almost all of the theorems by Appollonios.
What is interesting is that these underappreciated works did not become popular
until the nineteenth century.
7.4 Difficulties of Understanding
Conrad [8] states that the works by Desargues were difficult to understand, especially
with the terminologies, such as the use of “palm” by Desargues to describe a straight
line. Another challenge in the understanding of seventeenth century mathematical
literature is that mathematicians’ interests were focused on applying mathematics to
science and technology, including the newly discovered calculus. More importantly,
people viewed projective geometry centuries ago differently than they do today. A
theme is present in these developments: mathematicians and geometers at first viewed
geometry as something more analytic before they started using projective geometry
28
35. Chapter 8
Algebraic Structures
If a 0 a, a¡0 a, and a¤0 0, what would
a
0
equal? If mathematicians invented
complex numbers to allow the square root of a negative number and projective geom-
etry to allow the intersection of parallel lines, what should mathematicians invent to
allow division by zero? To answer this question, one needs to know about algebraic
structures. An algebraic structure contains a set and at least one binary operation. If
operating on any two elements in a set yields a third element that also belongs in the
set, then the set is closed under that operation. Examples of an algebraic structure
include pZ, q, the set of integers under the operation of addition and pZ, ¢q, the set
of integers under the operation of multiplication. Adding two integers results in an
integer and multiplying two integers results in an integer. However, pR, {q, the set of
real numbers under division, is not closed, since division by zero is undefined.
30
36. 8.1 Groups
Let G be a set together with a binary operation that assigns to each ordered pair
pa, bq of elements of G an element in G denoted by ab. Then G is a group under this
operation if the following four properties are satisfied.
1. Closure: For every a and b in G, ab is also in G.
2. Associativity: pabqc apbcq for every a, b, and c in G.
3. Identity: There exists an element e in G such that ae ea a for every a in
G.
4. Inverses: For every element a in G, there exists an element b in G such that
ab ba e.
An example of a group is pZ, q, the set of integers under addition. For all a and b in
Z, a b is also in Z, satisfying closure. For all a, b, and c in Z, pa bq c a pb cq,
satisfying associativity. The number 0 is the identity since a 0 0 a a for all
a in Z. The number ¡a is the inverse of a in Z since a p¡aq p¡aq a 0.
8.2 Rings
A ring R is a set closed under two binary operations, “addition” (denoted by a b)
and “multiplication” (denoted by ab), such that for all a, b, and c in R:
1. pa bq c a pb cq. (Addition is associative)
31
37. 2. An additive identity 0 exists. There is an element 0 in R such that a 0 a
for every a in R.
3. For every element a in R, there exists an element –a in R such that a p¡aq 0.
(Additive inverse exists)
4. a b b a. (Addition is commutative)
5. apbcq pabqc. (Multiplication is associative)
6. apb cq ab ac and pb cqa ba ca. (Multiplication distributes over
addition)
Note that properties 1-3 imply that every ring is a group under addition. The set
of integers, Z, under normal addition and multiplication, is not only a group under
addition as mentioned above, but also contains the multiplicative properties of a ring.
For all a, b, and c in Z, pabqc apbcq, implying that multiplication is associative.
Additionally, apb cq ab ac and pb cqa ba ca for all a, b, and c in Z, implying
that multiplication distributes over addition.
In a ring, however, multiplication need not be commutative. The most common
example of a ring where multiplication is not commutative is the set of n¢n matrices
with real entries under matrix multiplication. Some rings do have a multiplicative
identity, or a unity. For example, 1 is the unity in Z. Ring elements need not have
multiplicative inverses. For example, Z has elements such as 2 without a multiplicative
inverse. If an element x in a ring R has a multiplicative inverse, then x is called a
unit.
32
39. of a field. Alternately, there is no solution to the equation 0x 1. It would be nice
if every element has a multiplicative inverse as every element has an additive inverse.
However, the lack of a multiplicative inverse for 0 results in the field not being closed
under division by zero.
8.4 Remarks
As we move from groups to rings to fields, we see that the structures become more
intricate. The relationship among these structures is summarized in Figure 8.1
34
40. Figure 8.1: This a a summary of the major algebraic structures. A field has all of
the properties of a ring, which has all the properties of a group. Examples of a field
include the set of complex numbers, real numbers, and rational numbers while the
set of integers belongs to a ring.
35
41. Chapter 9
Wheels
9.1 Definition
A wheel, denoted W, is a set with two binary operations and ¤ and four special
elements, 0, 1, V, and u such that for all x, y, and z in W, the following axioms
hold [9].
1. Commutativity
x y y x.
x ¤y y ¤x.
2. Associativity
px yq z x py zq.
px ¤yq¤z x ¤py ¤zq.
3. Distributivity
36
42. If z $ V, then px yq¤z x ¤z y ¤z.
4. Neutral Elements
x 0 x.
x ¤1 x.
5. Inverse Elements
If x ‚ tV, uu, then there exists an additive inverse denoted ¡x € W such
that x p¡xq 0.
If x ‚ t0, Vu, then there exists a multiplicative inverse
1
x
€ W such that
x ¤ 1
x
1.
6. Definition of V, u .
V 1
0
.
u 0 ¤V.
7. Laws for u.
x¤ uu.
x uu.
¡ uu.
1
u u.
8. Laws for V.
If x ‚ tV, uu, then x V V.
37
43. V V u.
If x ‚ t0, uu, then x ¤V V.
¡V V.
1
V 0.
9. Non-triviality
0 $ 1.
Generally, dividing by zero results in two extra elements, V 1
0
and u 0
0
.
Remember that two fractions,
a
b
and
c
d
, are equivalent if ad bc. The two elements
V and u are not the same because in 1 ¤0 0 ¤0, cancelling out the zeros results in
1 0, but clearly 1 $ 0. Therefore, a wheel has only some, but not all, properties
of a group, a ring, and a field. For instance, in a wheel, the inverse property does
not hold if x € t0, V, uu but in a group, the inverse property must hold for all x.
Additionally, the distributive property in a wheel does not hold if z V but in a
ring, the distributive property must hold for all z.
9.2 History of Division by Zero
According to Romig [10, p. 387-389], while Indian mathematician Brahmagupta
in 628 A.D. gave no quotient for division by zero, another Indian mathematician
Bh¯ascara in 1152 defined the resulting quotient as infinite, but did not use any sym-
bols. In 1657, John Wallis was the first mathematician to use V 1
0
for infinity.
However, Wallis and Bishop Berkeley declared that there is no such number as zero.
38
44. Additionally, Wallis and John Craig, who declared zero to be an infinitesimal in 1716,
mistakenly considered negative fractions greater than infinity. Isaac Newton defined
the quotient resulting from division by zero as the infinite area under a hyperbola.
Most importantly, Martin Ohm of Berlin in 1828 absolutely excluded zero as a di-
visor. In 1831, De Morgan states that the lack of proper symbols for
1
0
caused its
various interpretations. In 1832, Wolfgang Bolyai de Bolya of Hungary asserted using
the idea of zero defined as 1 ¡1 0 that
1
z
approaches infinity as z approaches 0. In
1864, William Walton of Trinity College, Cambridge leaves the denominator blank in
1
and uses the symbol 0 to mean the infinitesimal. Finally, in 1877, Rudolf Lipschitz
explained that one cannot let
F1
m1
0 when dividing
F
m
by
F1
m1
because no fractions
give
F
m
when multiplied by zero.
This history shows how V 1
0
is one of the cases that hold if division by zero
is allowed. However, it does not talk about the history of u 0
0
. Rudolf Lipschitz’s
explanation follows that the inverse property does not hold if division by zero is
allowed.
9.3 Polynomials over Wheels
If R is a ring, then the set Rrxsof polynomials with coefficients in R, called the polyno-
mials over R, is also a ring. Consider three polynomials, fpxq, gpxq, and hpxq in Rrxs.
In terms of commutativity, fpxq gpxq gpxq fpxqand fpxqgpxq gpxqfpxq, both of
which are in Rrxs. In terms of associativity, pfpxq gpxqq hpxq fpxq pgpxq hpxqq
and pfpxqgpxqqhpxq fpxqpgpxqhpxqq, both of which are in Rrxs. The numbers 0 and
39
45. 1 are the additive identity and the multiplicative identity, respectively, while ¡fpxq
is the additive inverse. In terms of distributivity over addition, fpxqpgpxq hpxqq
fpxqgpxq fpxqhpxq.
The situation is quite different when we form a polynomial using coefficients from
a wheel.
Theorem. If we consider the set Wrxs of polynomials with coefficients from a wheel
W, then this set is neither a wheel nor a ring.
Proof. Suppose that the leading coefficient in a polynomial is V. Using the definition
supplied above, x ¤V V if x ‚ t0, uu. With that in mind, if the polynomial has V
as its leading coefficient while all of the other coefficients are not V or u, then the
polynomial equals V if the indeterminate variable x ‚ t0, uu. For example,
Vx3
4x2
4x 5 V
if x ‚ t0, uu. If x € t0, uu, then
Vx3
4x2
4x 5 u
since V ¤ 0 u, x¤ uu, and x uu. If more than one coefficient of a polynomial
is V, then the polynomial is equal to u because x ¤V V or x ¤V u and V V
V uu uu. Since operating u under addition or multiplication still yields u,
if a polynomial has any instances of u, then the polynomial equals u. If we attempted
to find what x equals when setting Vx3
4x2
4x 5 0, then we would either
end up with V 0 or u 0, which are not true in a ring or a wheel, making the set
Wrxs neither a wheel nor a ring.
40
46. 9.4 Division Algorithm
Theorem 16.2 in Joseph Gallian’s [11, p. 296] Contemporary Abstract Algebra states
something fundamental about polynomials. Because any field F is also a ring, then
Frxs is also a ring of polynomials.
Theorem (Gallian). Let F be a field and let fpxq and gpxq € Frxs with gpxq $ 0.
Then there exist unique polynomials qpxq and rpxq in Frxs such that fpxq gpxqqpxq
rpxq and either rpxq 0 or deg prpxqq deg pgpxqq.
For example, dividing x3
¡ 2x2
¡ 4 by x ¡ 3 equals x2
x 3 with a remainder
of 5, so
x3
¡2x2
¡4 px ¡3qpx2
x 3q 5.
However, the division algorithm does not hold in the case of a wheel, where we
allow division by zero.
Theorem. Let W be a wheel. Then the division algorithm does not hold in Wrxs.
Proof. Two different examples will be given, where the division algorithm fails to
hold in Wrxs.
Assume the division algorithm holds. By following Gallian’s proof [11, p. 296-
297] of Theorem 16.2, assume fpxq gpxqqpxq rpxq. Let fpxq anxn
. . . a0
and let gpxq bmxm
. . . b0. When long dividing gpxq into fpxq, let f1pxq
fpxq ¡ anb¡1
m xn¡m
gpxq. Suppose fpxq V, gpxq V, an V, and bm V. As
mentioned earlier, two different polynomials can both be equal to V if both of them
have V in only one of its coefficients and x ‚ t0, uu.
41
47. According to Setzer [9], b¡1
m 1
bm
1
V 0. anb¡1
m V ¤ 0 u. Since x¤ uu,
anb¡1
m xn¡m
u. Then f1pxq V¤ uu. Going back to the division algorithm
fpxq gpxqqpxq rpxq, we let qpxq anb¡1
m xn¡m
and let rpxq f1pxq. We think that
when we apply the division algorithm after doing long division, we should return to
the original polynomial. But
gpxqqpxq rpxq V¤ u uu uu$ fpxq.
This contradicts the assumption that fpxq gpxqqpxq rpxq. Therefore, the assump-
tion that the division algorithm holds is not true.
For the second example, let an V and bm ‚ t0, V, uu meaning that fpxq V
and gpxq ‚ t0, V, uu. This means that fpxq and gpxq are unique. anb¡1
m V¤ 1
bm
V
so that anb¡1
m xn¡m
V, provided that x ‚ t0, uu. Then f1pxq V¡V V p¡Vq
V V u. Going back to the division algorithm, we let
gpxqqpxq rpxq gpxq¤V uu$ fpxq.
Again, this example contradicts the assumption that fpxq gpxqqpxq rpxq.
As it turns out, division by zero is not entirely without consequences. Expanding
the number system to include V 1
0
and u 0
0
can be done consistently. However,
new challenges arise when using these new numbers as coefficients in polynomials.
42
48. Chapter 10
Conclusion
It is hard to tell which rules of mathematics will mathematicians will extend. The
most telling sign is that mathematicians will always be searching ways to push bound-
aries. By allowing complex numbers, one is now able to take the square root of a
negative number. Projective geometry allows one to look at parallel lines from a per-
spective in which they intersect at a point at infinity. However, changing the rules of
mathematics involves tradeoffs. As shown, allowing division by zero in the algebraic
structure of a wheel makes the division algorithm not always true. In a diverse soci-
ety that requires mathematicians to think abstractly, they will use their quantitative
talents to fill in the uncomfortable gaps that lie in their logical systems.
43
49. Bibliography
[1] K. A. McGrath, S. Blachford, The Gale Encyclopedia of Science, Gale Group,
2001.
[2] P. J. Nahin, An imaginary tale: The story of
c
¡1, 1998.
[3] B. Fine, The fundamental theorem of algebra, Springer, 1997.
[4] S. M. Bourgoin, P. K. Byers, Encyclopedia of World Biography, Gale/Cengage
Learning, 1998.
[5] E. Weisstein, Wolfram mathworld, 2007.
[6] R. Hartshorne. History of complex numbers: Between algebra and geometry.
HISTORIA MATHEMATICA 2006, 33, 493–496.
[7] H. S. M. Coxeter, Projective geometry, Springer, 2003.
[8] A. Conrad, Projective Geometry, 2000. http://www.math.rutgers.edu/
~cherlin/History/Papers2000/conrad.html.
[9] A. Setzer. December 7, 1997. 1997.
44
50. [10] H. Romig. Discussions: Early History of Division by Zero. American Mathemat-
ical Monthly 1924, 387–389.
[11] J. Gallian, Contemporary abstract algebra, Cengage Learning, 2009.
45
