In Chapter 1 we talked about how tallying or keeping track of quantity tokens or pebbles might be en

1
Chapter 3
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In Chapter 1 we talked about how tallying or
keeping track of quantity tokens or pebbles might
be enough to deal with the quantitative needs of
nomadic life. We saw how permanent settlements
created the need for more efficient ways of
tallying, for number words to ease communication
about tally symbols, and for more sophisticated
mathematical ideas such as area, volume, slope,
and proportion. The ancient Egyptian additive
number systems provided a natural link from
concrete objects to the abstract symbols used to
represent quantity. The idea that one mark
stands for one object is very natural, as is the
idea of replacing a group of marks with another
symbol for easier recording of large numbers.
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The Egyptians managed large numbers by inventing
new symbols. In Chapter 2 we saw how the
Babylonians managed large numbers by introducing
positional notation.
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The Babylonians used base 60. When one works
with very large numbers it is convenient to have
a large base. Base 60 eases working with
fractions since 60 has so many divisors. Position
al representation of numbers is conceptually
difficult, as we are reminded when we try to work
in an unfamiliar base. Where does base 60
survive in our lives today? What other bases do
we use?
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In Chapter 3 we turn to the mathematics of
ancient Greece. Some instructors would begin a
course on the history of mathematics in Greece
without ever mentioning Egypt, Mesopotamia, or
other places in the world that had number
systems, arithmetic, and geometry. Why?
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In Chapter 3 we turn to the mathematics of
ancient Greece. Some instructors would begin a
course on the history of mathematics in Greece
without ever mentioning Egypt, Mesopotamia, or
other places in the world that had number
systems, arithmetic, and geometry. Why? In
ancient Greece people first began to consider
numbers as entities in their own right, as nouns
rather than adjectives. The Babylonians had
neither a place holding zero symbol nor the
equivalent of a decimal point because the context
of a computation provided the information they
needed. Numbers described quantities of real
objects and did not exist independently. In
Greece for the first time philosophers
investigated the properties of numbers themselves
and deduced their results from first principles.
This was the first mathematics. Before Greece
all was arithmetic.
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The ancient Greeks consciously realized that
numbers and geometrical figures are abstractions.
They exist only in the mind and have no physical
presence. Although our knowledge of Greek
mathematics comes from sources that are not
completely reliable (there are no original
manuscripts), one can reasonably attribute the
recognition of the abstract nature of
mathematical objects to Pythagoras. This
realization did, however, take some time to
achieve. In the beginning Pythagoras and his
followers, the Pythagoreans, viewed numbers as
tiny spheres.
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The most natural thing for humans to do with a
new collection of objects is to classify them.
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Pages 40 42 show the triangular numbers, square
numbers, pentagonal numbers, hexagonal numbers,
and heptagonal numbers. What other ways of
classifying numbers can you think of?
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The Pythagorean theorem is one of the most famous
in all of mathematics. What does it claim?
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The Pythagorean theorem is one of the most famous
in all of mathematics. What does it claim?
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The Pythagorean theorem is one of the most famous
in all of mathematics. What does it
claim? This result may or may not have
been deduced by Pythagoras. In either case
Pythagoras wouldnt have been the first to have
done so. What is known with relative confidence
is that the Pythagoreans had a system for
generating numbers a, b, and c such that a2 b2
c2.
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What method for constructing Pythagorean triples
suggested by Example 3.4? By Example 3.5? Can
you construct Pythagorean triples starting with
any number you choose? The homework problems ask
you to show how Pythagoras might have arrived at
these observation by considering the gnomons of
square numbers. Gnomons of various numbers are
pictured on pages 41 42.
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Following the discussion of Pythagorean triples
in the text is an introduction to commensurable
and incommensurable numbers. To say two numbers
are commensurable means they have a greatest
common divisor. The algorithm used to find the
greatest common divisor of two numbers is called
the Euclidean Algorithm because Euclid included
it in the Elements. The central idea of this
algorithm can also be applied to line segments
(numbers viewed as lengths) or more generally to
other geometric figures. The original
demonstration that incommensurable, i.e.,
irrational numbers exist may have depended on the
Euclidean algorithm. It may also have depended
on the properties of even and odd numbers. A
possible reconstruction of such a result appears
in Section 3.3.1.
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Lets work through the demonstration that the
diagonal of a square is incommensurable with its
side on the top of page 48.
16
A
B
D
C
17
A
B
D
C
18
A
B
b
D
C
a
19
A
B
b
a2 a2 b2 b2 2a2
a
D
C
a
20
A
B
b
a2 a2 b2 b2 2a2 b is even
a
D
C
a
21
A
B
b
a2 a2 b2 b2 2a2 b is even a is odd
a
D
C
a
22
A
B
b
E
a2 a2 b2 b2 2a2 b is even a is odd
a
D
C
a
23
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is odd
a
c
D
C
a
24
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is odd
a
c
D
C
a
25
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is odd
a
c
c
D
C
a
26
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is
odd c2 c2 a2 a2 2c2
a
c
c
D
C
a
27
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is
odd c2 c2 a2 a2 2c2 a is even
a
c
c
D
C
a
28
A
B
c
E
a2 a2 b2 b2 2a2 b is even a is
odd c2 c2 a2 a2 2c2 a is even
a
c
c
D
C
a
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In the beginning there was a lot of resistance to
the idea of incommensurable numbers since they
contradicted Pythagorean number mysticism.
Once incommensurable numbers were acknowledged,
however, they had to be measured. Section 3.3.2
describes a procedure for approximating
irrational numbers. Example 3.9 shows how this
works for .
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In the beginning there was a lot of resistance to
the idea of incommensurable numbers since they
contradicted Pythagorean number mysticism.
Once incommensurable numbers were acknowledged,
however, they had to be measured. Section 3.3.2
describes a procedure for approximating
irrational numbers. Example 3.9 shows how this
works for . Why does this method work? To
answer this question, we use the fact that all
pairs of values side s and diagonal d have the
property that the square on d is either 1 less or
1 more than twice the square on s. We can easily
check this for the first few values of d and s.
Why is does the property remain true for as many
values as we care to consider?
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Section 3.3.3 on page 51 provides motivation for
the ancient Greek focus on geometry. Lets see
what it says.
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Chapter 5 focuses on Euclidean geometry, i.e.,
those results which appear in the thirteen
volumes of the book Elements compiled and written
by Euclid. Chapter 4 surveys the geometry known
before Euclids time. Section 4.1 discusses some
of the results deduced by the Pythagoreans. Secti
on 4.2 introduces the three classical problems of
Greek geometry, which are listed on page 65.
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On page 66 we read how Hippias of Elis was able
to trisect an angle using a curve called the
quadratrix. This curve was later used to square
the circle. At first this seems to contradict
what we learn in abstract algebra courses it is
impossible to square a circle or trisect an
angle. How can we resolve this apparent conflict?
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On page 66 we read how Hippias of Elis was able
to trisect an angle using a curve called the
quadratrix. This curve was later used to square
the circle. At first this seems to contradict
what we learn in abstract algebra courses it is
impossible to square a circle or trisect an
angle. How can we resolve this apparent
conflict? The use of the quadratrix to trisect
an angle requires one to first trisect a line
segment. This link shows several ways to do this.
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Squaring the circle with only a compass and
straightedge is impossible. It is, however,
possible to square certain figures with curved
sides using only these instruments. Section
4.2.6 shows how Hippocrates of Chios did this for
various lunes. First we consider how to square
triangles, quadrilaterals, pentagons, and other
figures with straight sides.
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Given rectangle BCDE, extend BE and mark F so
that DE and EF have the same length. Bisect BF
and call the midpoint G. Draw the semicircle
with center G and radius BG. Extend DE and label
as H its intersection with the semicircle. The
square EHLK on side EH will have the same area as
the original rectangle BCDE. This is true since
the area of BCDE is BE DE BE EF (a
b)(a b) a2 b2 c2.
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To square a triangle, first find a rectangle with
the same area as the triangle, then use the
method on the last slide to square the rectangle.
Other figures with straight sides can be squared
by first breaking them down into triangles.
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A lune is a region bounded by intersecting
circles (see page 69). To square particular
lunes Hippocrates needed two preliminary results.
How should we interpret Conjectures 4.7 and
4.8? Which lune is Hippocrates able to square in
Proposition 4.5? Lets look at how he did
it. The remaining sections in Chapter 4
highlight the contributions of Plato, Aristotle,
and others to the history of mathematics.