The Rise of Algebra

1
The Rise of Algebra
2
Last time we saw how the ancient Greek
mathematical tradition moved out of Europe and
into Egypt as a result of Roman occupation. This
week we will see how it moved out of the
Mediterranean altogether as the influence of
Christianity grew and its leaders became
suspicious of knowledge from non-Christian (i.e.,
pagan) sources. As Europe entered the Dark Ages,
mathematics continued its growth elsewhere.
Important contributions come from India and the
Arabian peninsula.
3
In Alexandria, mathematicians treated fractions
as numbers in their own right. Previous Greek
mathematicians had considered fractions as ratios
of whole numbers, not as parts of a whole, and
used them only in describing and comparing
relationships among magnitudes. Of course
practical applications of mathematics in ancient
Greece required the treatment of fractions as
numbers, but such applications were not
considered worthy of advanced study.
Mathematicians such as Archimedes, Heron, and
Diophantus (whose contributions we will discuss
today) freely used fractions and performed
operations with them. They did not, however,
discuss the concept of fraction. Apparently
fractions were considered intuitively clear
enough to use without comment.
4
Another landmark in the history of mathematics
that came out of Alexandria was the development
of algebra as a subject independent of geometry.
Weve seen how algebra can be understood
geometrically, but weve also experienced the
limitations of this approach. Geometric
solutions of quadratic equations, for example,
are considerably more difficult than algebraic
ones. Our book does not discuss the
contributions Heron made to algebra, but he
certainly did help develop the subject. Not only
did he translate geometric procedures into
algebraic ones, but he also formulated and solved
algebraic problems without appeal to geometry
either for motivation or justification.
5
The ancient Babylonian mathematics we discussed
in Chapter 2 influenced the development of
algebra in Egypt. (We have already seen how the
base 60 number system appeared in the treatment
of trigonometry.) Alexandrian mathematicians
analyzed some of the same problems considered by
the Babylonians. They also used the Babylonian
style of presenting solutions to specific
problems rather than discussing general
procedures. Problems were often presented in the
form of a puzzle, and strangely no justification
for solutions was given. The axiomatic approach
the Greeks took to geometry did not immediately
carry over into the development of algebra.
6
The climax of Alexandrian Greek algebra can be
found in the work of Diophantus. We dont know
exactly where he originally came from, or when
exactly he lived, but we do know how long he
lived thanks to an algebra problem composed in
his honor (Problem 7.2 on page 191).
Diophantus influence on the history of
mathematics began long after his death. He wrote
his major work, the thirteen books of the
Arithmetica, as the Dark Ages descended upon
Europe, where its contents were ignored for
centuries.
7
Diophantus wrote the Arithmetica in the style of
the Rhind papyrus. It is a collection of
different problems, each solved in a unique way,
with no attempt made to collect together problems
with similar solutions, and with no deductive
proofs. Diophantus claimed to have written it as
a series of exercises for his students. In the
Arithmetica we see the first use of symbols
denoting unknown quantities to aid in problem
solving (see page 192). Because the author of
our textbook wants to present the history of
mathematics in its original form, yet doesnt
want us to get hung up on the unfamiliar Greek
letters, he suggests the use of familiar Latin
letters that correspond to the Greek ones.
8
The appearance of algebraic symbolism was a major
advancement in the history of mathematics. Over
many centuries it has become increasingly revised
and refined. A surprising feature of Diophantus
symbolism is the consideration of powers greater
than three. Until this point in the history of
math, the Greeks had no use for such powers since
they had no geometric meaning. When number is
separated from magnitude, however, powers higher
than three do make sense, and Diophantus used
them without hesitation.
9
Seven out of the thirteen books of the
Arithmetica have been lost. Of the six that
remain, the first consists mainly of problems
leading to determinate (i.e., having a unique
solution) first degree equations in one or two
unknowns. The remaining books consider
indeterminate second degree equations.
10
Section 7.4 gives us a flavor of the types of
problems Diophantus considered and the kinds of
solutions he gave.
Problem 7.3 on page 193 comes from Book I. How
would we solve it? Did Diophantus use the same
technique?
Note how Diophantus expressed one unknown in
terms of another. Many of his solutions consist
of transforming a problem into one that he has
already explained how to solve.
Problem 7.5 comes from Book II. How would we
state the problem?
Lets look at the solution Diophantus provides.
11
Of all Diophantus problems, this one is the most
famous. Why?
This is a Latin translation of the Arithmetica
which includes comments by Pierre de Fermat.
It is impossible to separate a cube into two
cubes, or a fourth power into two fourth powers,
or in general, any power higher than the second
into two likepowers. I have discovered a truly
marvelous proof of this, which this margin is too
narrow to contain.
12
Fermat was one of the two most prominent
mathematicians of the 17th century (the other was
Descartes). His work paved the way for
differential calculus and he is considered the
founder of modern number theory. He refused to
publish his work, but his friends and relatives
preserved it. Fermat had written notes in his
copy of Diophantus Arithmetica, and his son
published a version of this book complete with
his fathers notes. For over three centuries the
worlds best mathematicians failed to discover
the marvelous proof of what became known as
Fermats Last Theorem (the last to be proved,
that is). None were successful until 1994 when
Andrew Wiles finally announced a proof over one
hundred pages long.
13
Next time for homework I will ask you to rewrite
both Problem 7.6 (page 194) and its solution and
Lemma 7.1 (page 195) and its proof including all
missing details and using modern notation.
Next lets work through the solution of Problem
7.8. I will rewrite the problem and the solution
using modern notation and including missing
details.
Compare the length of our proof to that given by
Diophantus.
14
The range of equations that Diophantus could
reduce and solve is impressive. We dont know
how he discovered his methods, but we do know
they are considerably different from anything
already known at the time. Although Diophantus
did consider fractions to be numbers, as opposed
to strictly ratios, he did not recognize complex,
irrational, or even negative numbers. With
Problem 7.9 on page 196, for example, we see
Diophantus dismiss a negative solution to a
problem as absurd.
15
Diophantus was not the only mathematician to
avoid irrational, complex, and negative numbers.
Reluctance to fully accept these quantities
continued until the 19th century! Why?
Numbers arose from counting things. Four sheep,
four children, and four pigs all have something
in common. This common property is described by
the number four. Counting was refined with the
introduction of fractions, but still numbers
described quantity.
In this sense even zero is a difficult concept.
If numbers represent quantity, then zero must not
be a number. In Chapter 8 we will see how zero
began life as a place holder, a separate symbol
used in positional number systems to indicate the
absence of number in a particular position.
16
In the 9th century, Indian mathematicians made a
conceptual leap that marked a major advancement
in the history of mathematics. They began to
treat the place holding symbol as an quantity on
par with the counting numbers. By creating rules
for adding, subtracting, multiplying, and
dividing by zero, they showed how it could be
treated as a number even though it didnt count
anything. This revolutionary idea took a long
time to gain acceptance, and even in the 16th and
17th centuries some prominent mathematicians were
still reluctant to accept zero as a root of an
equation.
17
Given the reluctance to accept zero as a number,
it should come as no surprise that the use of
negative, irrational, and complex numbers was
firmly resisted.
Negative numbers come about naturally as
solutions to equations. For example, what
equation is suggested by the following problem?
I am 7 years old and my sister is 2. When will I
be exactly twice as old as my sister?
The equation is (x 7) 2(2 x) and the
solution is x 3. What happens if we replace
the ages 7 and 2 by 18 and 11?
The equation is (x 18) 2(11 x) and the
solution is x 4.
18
Lets look at Problem 7.9 again. This is the one
in which Diophantus dismissed a negative
solution. At first it can be hard to figure out
where the problem comes in, so Id like to fill
in the details of the solution.
Diophantus did know how to expand products of the
form (x a)(x b) which means he knew a
negative times a negative should be a positive
and a negative times a positive should be a
negative.
This realization complicated conceptual
understanding of imaginary numbers. If a
negative times a negative is a positive, then v
1 cant be negative, since v1 v1 1. On the
other hand, it cant be positive either, since
any positive times a positive is a positive.
19
Section 7.5 describes the end of scholarship and
the beginning of the Dark Ages in Europe. During
this period Pappus tried to revitalize interest
in theoretical mathematics. For homework I will
ask you to interpret some of his findings. I
wont ask for proofs, but I will ask for modern
translations of proposition statements.
Pappus Collection
20
The first two sections of Chapter 8 describe the
origins of Chinese mathematics. If you skim
through them you will notice some interesting and
impressive results. It is, however, unclear how
much influence Chinese mathematics had on the
development of mathematics as a whole. For this
reason we will not spend time in class on the
details. Indian mathematics, on the other
hand, did have a considerable impact on the
development of mathematics as we know it.
Sections 8.3 and 8.4 describe the Indian
contribution to the history of mathematics.
21
At the time Greek mathematics was beginning to
develop, there was already a local tradition of
mathematics in India. Astronomy provided the
inspiration for many of the first problems
studied, and later Indian mathematicians became
interested in studying mathematics for its own
sake. As was the case with Greek mathematics,
many of the ancient texts have been lost. Of
those that remain, Aryabhatas 6th century work
is earliest. The most important mathematicians
of the 7th century were Brahmagupta and Bhaskara,
who were among the first people to recognize and
work with negative quantities. Another important
Bhaskara lived in the 12th century. In almost
all cases, the mathematical texts we have are
portions of books on astronomy.
22
The best known invention of Indian mathematicians
is the base 10 number system. They used numerals
for the numbers one through nine that clearly
gave rise to the ones we use today. They also
used a small dot to indicate the absence of a
number in a given position. This dot evolved
into our zero symbol.
The methods Indians developed for doing
arithmetic with nine numerals and a place holder
forced people to operate with the place holder in
the same way they operated with numbers. This
helped ease the conceptual leap of treating zero
as a number.
23
Indian mathematicians also made contributions to
trigonometry. They probably learned chord
measurement from the work of Hipparchus, a
predecessor of Ptolemy. While the Greeks focused
on the chord of an angle, the Indians realized
half the chord on twice the angle was more useful.
The called the length jya-ardha, Sanskrit for
half-chord, or jya, for short. The Arabs
translated jya as jiba. When Europeans
translated the Arabic into Latin, they mistook
jiba for jaib, which means cove or bay, so
they choose the Latin word sinus which meant
bay.
24
Our term sine came from a mistranslation of a
translation. How about cosine? Every so often
one needs the sine of the complement of an angle,
the sinus complementi, or co. sinus for short.
Indian mathematicians were also interested in
algebra and some aspects of combinatorics. They
had methods for computing square and cube roots,
and could compute the sum of an arithmetic
progression. They also solved quadratic
equations in essentially the same way we do now,
though they didnt use our efficient symbolism.
25
Indian mathematicians could also solve equations
in several variables. To cut down on the number
of possible solutions to such equations, they
would impose constraints such as allowing only
whole numbers.
Aryabhata and Brahmagupta could solve linear
equations of the form ax by c where a, b, and
c are whole numbers. Brahmagupta also studied
harder problems of this kind, such as finding
integers x and y for which 92x2 1 y2. The
second Bhaskara generalized these ideas. He
found a method for finding whole number solutions
to nx2 b y2.
26
Unlike the Greeks, Indian mathematicians did not
prove their results deductively, nor did they
describe how they arrived at their solutions.
When Europe emerged from the Dark Ages,
mathematicians became interested in new
developments from the east. They learned Indian
mathematics through contact with the Arabic
mathematical tradition, the subject of Chapter 9.