1. A brief history of measurement
As cultures developed the need for building homes, boats, tools, etc., they needed
effective terms that could describe a notion of length. Measurement of length is
an ancient tool; just open a copy of the Bible and find hundreds of references to
measurments. The words that these societies came up with are units of measurement or units of length. We are familiar with modern units of measurement : inch,
foot, meter, mile, fathom, centimenter, etc.
Of course, modern units of measurement are, well, modern. These ancient cultures had their own units. Unfortunately, being relatively tribal in nature and
lacking advanced technology or communication, these societies had no “official”
units of measurement. Instead, they just used what almost everybody had : body
parts. They worked under the not-so-rigorous assumption that every man’s body
parts were roughly the same length. Of course, this assumption is not the case,
but it generally did not matter as long as the same “standard” was used for the
whole project. (You don’t want to end up with a boat with one side longer than
the other!)
The Greeks, as well as many other cultures, used the length of the finger as
a base measurment. The δ άκτ υλoς (finger or toe) was the width of the second
knuckle on a man’s index finger. Based, on the δ άκτ υλoς, the Greeks could then
describe other units of length. A παλαιστ ή (palm) was the width of a man’s
palm, but was generally considered δ άκτ υλoι. The πoυ̃ς (foot) was the length of
a man’s foot, but was generally considered 16 δ άκτ υλoι. (Is the length of your
foot about the width of sixteen knuckles?) From the foot, the Greeks could then
describe larger units resembling our fathom, mile, and league.
One of the most popular units was the cubit, which appeared long before Greek
culture. This was simply the length of a man’s arm from bend at the elbow
to the tip of the middle finger. This was the most common unit used in large
construction projects because they were easy to communicate. Some cultures,
such as the Egyptians tried to standardized the cubit, by introducing concept
such as the “royal cubit,” usually taken to be the king or phoaroh’s cubit.
2. How did the Greeks use measurement?
Suppose we have decided on a standard unit of measurement; we’ll just call it a
unit. In ancient Greece, the people really only had access to a ruler and a compass.
Suppose their ruler simply measured one unit. So they could find simple lengths
and angles. For example, if they needed to measure 2 units, they would just
measure two one-units. Of course, they could do that for any whole number. But
1
2
what if they need to measure on half of a unit as accurately as possible? Imagine
you have a ruler that only shows inches, nothing else. How could you measure a
fraction of an inch as exactly as possible? Think about it for a few minutes.
Consider the picture below (figure 1), depicting how to measure one half of a
unit.
s
nit
2u
A
1 unit
Figure 1
Suppose the horizontal, bottom side of the triangle has length one unit. We
would like exactly half of that unit. Well we know how to measure 2 units, so
let’s make the diagonal of the triangle have length 2 units. We will construct the
triangle so as the vertical side will have the proper length to make a right triangle.
Now, find the middle of the 2-unit side of the triangle. We can find it, because
it will just be 1 unit along the diagonal. We will now draw a vertical line from
that middle point straight down to our original orizontal line. It will it this line
at point A. Note that point A is exactly in the middle of our 1-unit side. So our
measurement will just be from one side to the point A. (Why is point A exactly
in the middle? What is the reasoning?)
There was nothing special about a diagonal of length 2 units. We could have
chosen a diagonal of length n for any whole number n. We then divide that side
into n pieces and draw vertical lines from our division points to the horizontal
1-unit side. The n − 1 places they intersect will break our length-1-unit side into
n pieces. (Why is this the case? Do some examples with various n and convince
yourself that it is true.)
3
p
, where p and
q
q are integers and q 6= 0. For simplicity, we’ll always assume our fractions are in
lowest terms. Now pick some positive rational number. For example, let’s work
with 17/10. Could the Greeks measure 17/10 units exactly? How?
Recall that a rational number is a number r of the form r =
From the example above, we know they could measure 1/10 of a unit. So
to measure 17/10 units, they would simply have to measure 17 1/10 units. Of
course, this will work for any positive rational number. (Why do we keep saying
“positive”? What would “negative length” mean?)
3. The Pythagorean theorem
One of the oldest mathematical theorems in recorded history is the famous
Pythagorean theorem. Recall that the theorem says that for a right triangle as
pictured below in figure 2, a2 + b2 = c2 .
c
b
a
Figure 2
The precise history of the Pythagorean theorem is a bit elusive, but Pythagoras’ name is attributed to the theorem (even though the man may never have
existed). Some of the first recorded documents involving the theorem mostly
involved Pythagorean triples, that is, triples of whole numbers that satisfy the theorem’s equation. For example, {3, 4, 5}, {5, 12, 13}, and {6, 8, 10} are examples of
Pythagorean triples. There is evidence that the early Egyptians and Chinese were
fascinated by these numbers and explored their properties in greater detail, but
we will focus on the Greeks’ usage of the theorem.
The theorem itself arose from specific observations. For example, consider figure
1.3 from your text. Imagine that a rope stretcher has 12 units of rope. He then
stretches the rope in a triangular pattern with 4 units going horizontally and
4
3 going vertically. This will leave the last five to form the diagonal. Notice that
42 +32 = 25 = 52 . Now suppose he picks a rope of length 24 and and sides of lengths
6, 8, and 10 (with 10 on the diagonal). Again, he notices that 62 + 82 = 100 = 102 .
Through these repeated observations, the idea of the theorem is formed.
Which proof of the theorem came first is unknown, but one of the earliest is
Euclid’s proof, which is included in his Elements. Euclid uses the picture as
figured below. The idea is to show that the area of square AGF B has the same
area as the rectangle BDLK, and that the square ACIH has the same area as
the rectangle CKLE. This would mean that the sum of the areas of the suares is
equal to the area of the large square BCED. Below is an outline for the proof,
but we will let the reader fill in the details.
Figure 3
(1) Draw two more lines as follows (and as pictured above) : line AD and line
CF . We now have two new triangles F BC and ABD.
(2) Show that the area of the triangle F BC has one half the area of the square
AGF B. (What is the base and height of the triangle?)
(3) Show that the area of the triangle ABD has one half the area of the rectangle BDLK.
(4) If we could show that these triangles are congruent (have the same size and
shape), then this would show that square AGF B and rectangle BDLK
have the same area. (Why?)
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(5) We will use the Side-Angle-Side property to show they are congruent. That
is, we will show that the triangles have two pairs of sides of equal length
with the angles between those sides also equal.
(6) Consider side F B. Does this have the same length of one of the sides of
triangle ABD? Which one? Why?
(7) Consider side BC. Does it have the same length of one of the sides of
triangle ABD? Which one? Why?
(8) Now look at the angles ∠F BC and ∠ABD. Show that these are equal.
(Right angles are involved.)
(9) We are now half done with the proof.
(10) Draw another two lines as follows (but not pictured above) : line AE and
line BI. Go through an identical argument to show that square ACIH
and rectangle CKLE have the same area.
4. “Absurd” numbers
So we know that the Greeks could measure any positive rational number that
they wanted. They even had the ever-useful Pythagorean theorem to assist in
measurements. But they noticed something strange (as did many other cultures
before them). Sometimes, the Pythagorean theorem gave something that they
couldn’t measure. For example, if we imagine a right triangle
with two sides of
√
length 1, we can √
determine the length of the hypotenuse, 2. Can you figure out
how to measure 2 using our method from before? (No, you can’t.) The Greeks
didn’t know quite
how to handle it either. They knew from the pythagorean
√
theorem that 2 ≈ 1.414, but all they could do is approximate it.
Well, maybe they could measure it. We haven’t exactly been rigorous in our
definition of what the Greeks could measure. We will say that a length µ is
m
measurable if the length is
units, where m and n are positive integers. So
n
a natural question
is, “are there lengths that are not measurable?” We already
√
claimed that 2 is not measurable, but let’s prove that non-measurable numbers
exists a bit more abstractly.
Refer to figure 1.4 in your text. Here we have an isosceles right triangle with
two sides of length m
(again, m and n are positive integers). We will call the
n
hypotenuse h. We want to show that h is not measurable. That is, h will be a
“number” not of the form pq . This is actually a bit surprising if we think about it.
What we are claiming is that if we take a measurable number, square it, multiply
by 2, and finally take the square root, we will end up with a non-measurable
number.
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We will prove this claim by contradiction. That is, we will assume the opposite of
what we would like to prove and then show that something absurd or contradictary
r
will come from this faulty assumption. So we assume that h = , where r and s
s
are positive integers. By the Pythagorean theorem,
r 2 m 2 m 2
m 2
=
+
=2
.
s
n
n
n
By cross-multiplying the denominators across, we get
r2 n2 = 2m2 s2 .
For simplicity, we will let x = rn and y = ms. Note that since r2 n2 = (rn)2 and
2m2 s2 = 2(ms)2 , we have that x2 = 2y 2 . Also note that x and y are still positive
integers.
We will now factor out as many 2’s as possible from x and y. What we mean
is that, with x = 24 as an example, we write x = 24 = 2 · 2 · 2 · 3 = 23 3. This is
because, we can factor out three 2’s from 24. So with our general x and y, we have
x = 2a x1 and y = 2b y1 . Here a and b, are non-negative integers. (They could be
0 if x or y is odd!) More importantly, x1 and y1 are positive odd numbers, since
we factored out all the 2’s from x and y. Finally, we will substitute these values
back into our equation to get,
(2a x1 )2 = 2(2b y1 )2 .
22a x21 = 22b+1 y12 .
Notice that 2a is an even number (maybe 0), and 2b + 1 is an odd number. So
they cannot be the same. This means that either 2a < 2b + 1 or 2b + 1 < 2a. We
will consider wach case separately.
(1) Suppose 2a > 2b + 1. Then, dividing both sides of our equation by 22b+1 ,
we get
22a−2b−1 x21 = y12 .
Note that since 2a > 2b + 1, y12 must be an even number. But remember
that y1 is an odd number, so y12 must be an odd number as well. A number
can’t be both positive and negative, so something is wrong. It must not be
the case that 2a > 2b + 1.
(2) So we must have that 2b + 1 > 2a. So similarly, we can write
x21 = 22b+1−2a y1 .
For the same reason as before, x21 must be an even number. But again, x1
is odd, so x21 is odd - another contradiction.
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So we have hit a dead end. By assuming that h was measurable, we have shown
that we end up with a number that is both even and odd. So our assumption must
be faulty. Therefore, h is not measurable.
Although the preceding proof is not Greek in origin, the risorous logic used
was a Greek trait. Earlier cultures, such as the Egyptians, did not prove their
mathematical assumptions. They would make certain observations that repeated
seemingly without end. For example, they observed that the Pythagorean theorem
always held, and they came to accept the notion as a fact. The Greeks were not
satisfied with this type of mathematics. They wanted things “proven.” In many
ways, the Greeks are the founders of modern mathematics, as it is their logical
rules of deduction that we follow today. Their society marks the begin of the shift
from inferential mathematics to deductive mathematics.
So what can we conclude? The Greeks didn’t have “enough” numbers. They, as
well as many other earlier societies, stumbled upon these non-measurable numbers.
The Greeks shied away from these kinds of numbers, and often did their best to
ignore them. Of course, we know of these numbers as the irrational numbers. This
disapproval of the irrationals was very likely a big reason why the real numbers
were never seriously considered until the 16th century.