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Numbers and Their Application - Lesson 11
Theorems: Pythagorean, Fermat's Last, etc.
Lesson Overview
One of the most important discoveries in antiquity was that not only did
32 + 42 = 52, but also, if such a triple
could be found, these were the side lengths of a right triangle.
(A right triangle contains one 90° or right angle.) Several
cultures (Chinese, Babylonians, Egyptians, and Greeks) may have
independently made this discovery, but due to our historic European slant
and records preservation, this has been known as the Pythagorean Theorem.
However, the Greeks went further, developing geometry not only for practical
purposes, but also in abstraction and for its logical structure.
The Pythagorean Theorem is one of the most important facts learned in Geometry.
|
A triangle with sides a, b, and c (longest) is a
right triangle if and only if a2 + b2
= c2. |
Hence we know how the sides are related if it is a right triangle. We
can also prove the triangle to be a right triangle if its sides have this
relationship—the converse situation.
There are over three hundred different proofs of the Pythagorean Theorem.
One of the common proofs uses a square within a square
(see figure below). Each side of
the inner square has length c. Each corner of the inner square
intersects the sides of the outer square. The four triangles formed by the
intersection are all congruent. Therefore each side of the outer square
is made up of two segments, a and b.
![[figure of square with side length c rotated within
square with side length a+b]](pythag2.gif)
In order to find the distance of c in terms of a and b,
we use the fact that the area of the outer square is the same as the sum of
the area of the four triangles and the inner square.
The rest is algebraic manipulation.
(a + b)2 =
c2 + 4(1/2)ab.
Expanding, we get: a2 + 2ab +
b2 = c2 + 2ab.
After subtracting
2ab from both sides, we conclude that c2 =
a2 + b2. Q.E.D.
|
A pythagorean triple is a set of three integers a, b,
c such that a2 + b2 =
c2. |
|
A primitive pythagorean triple is a pythagorean triple such that
GCF(a,b) = 1. |
Common pythagorean triple are: 3, 4, 5; 5, 12, 13;
7, 24, 25; 9, 40, 41; and 6, 8, 10. All but this last triple are
primitive. The last is called a multiple. Note: it follows that
if GCF(a,b) = n, then n is also a factor of c.
Notice how 32 = 4 + 5; 52 = 12 + 13, .... This is a
characteristic of a general class of primitive pythagorean triples
involving squares and two
consecutive integers and was illustrated in homework 3,
problem 6.
Pythagorean triples such as 8, 15, 17 do not have this characteristic.
A regular polygon has all sides equal (equilateral)
and all angles equal (equiangular). In a triangle these cannot
occur independently. The resulting triangle with sides in the ratio 1:1:1
and angles of 60°-60°-60° is discussed, in part, below.
The three most important right triangles are: the 3-4-5; the isosceles right
(45°-45°-90°); and the 30°-60°-90° triangle.
The 3-4-5 triangle has angle measures of about 37°-53°-90°.
Watch especially for these special angles and triangles.
The isosceles (2 or more sides equal)
right (having a 90° angle) triangle can be thought of as having
legs (the shorter sides of a right triangle) of length 1.
Thus the hypotenuse (the longest side of a right triangle) is the square root of 12 +
12 or ![[square root of two]](sqrt2.gif)
.
The 30°-60°-90° triangle can be
thought of as a
bisected equilateral triangle.
Thus one side might be 1, the hypotenuse then is 2 and the
other side must satisfy 12 + x2 = 22,
or x2 = 3, thus
x=![[square root of three]](sqrt3.gif)
.
These side length ratios must be memorized and will be seen often in trigonometry
which is the study of triangle measure, but primarily involves triangle side
length ratios.
Note: if a2 + b2 < c2,
the triangle is obtuse (contains an angle more than 90°).
If a2 + b2 > c2,
the triangle is acute (all three angles are less then 90°).
A quick introduction to a semester of trigonometry can be summarized as
follows. Three items taken two at a time can be done six different ways
(3P2=3!/(3-2)!=6/1=6). One trigonometric definition
involves ratios (two numbers) of the three sides of a right triangle.
For sake of future reference, we will identify the triangle as ABC with
right angle C. This is a very standard convension.
Side c is then the hypotenuse and is opposite angle C, etc.
In relation to angle A, a is its opposite side and b is its
adjacent side (adjacent means to lie nearby).
|
sin A=opposite/hypotenuse
cos A=adjacent/hypotenuse
tan A=opposite/adjacent
|
sin is the normal abbreviation for sine which comes from the
Latin word for curve which came from a Sanskrit word
meaning bowstring. cos is the normal abbreviation for cosine
where the prefix co- has the usual meaning of together or
partner. tan
is the normal abbreviation for tangent from Latin meaning to touch
which has a more general geometric meaning of the intersection of two
geometric figures at a point. These relationships are often remembered via
the mnenomic SOH CAH TOA. One can readily see that tan A=sin A/cos A.
The remaining three trigonometric functions: secant or sec A=1/cos A;
cosecant or csc A=1/sin A; and cotangent or
cot A=1/tan A are less frequently used and usually don't even appear on
calculators. Remember, there is only one cofunction in each reciprocal
relationship.
It is important to note that a rather confusing notation
is historically used for the inverse trigonometric functions.
sin-1 x refers not to the reciprocal of sin A,
but rather to the inverse function.
That is sin-1 x is an angle whose sin is equal to x.
However, sin2x means (sin (x))2 and
must be entered as such on your calculator.
The table below follows directly from these special triangles and trigonometric
definitions.
tan 90° is ill-defined since cos 90°=0 (or the adjacent side is zero)
and division by zero is not allowed. More will be presented on the trigonometric
function definitions after the cartesian coordinate system and
transcendental numbers have been introduced in the
next few lessons.
| Angle (°)
| Angle (Radians)
| Sine
| Cosine
| Tangent
| | 0° | 0 | 0 | 1 | 0
|
| 30° | ![[pi]](pi.gif) /6 | 1/2
| /2
| /3
|
| 36°52'11.63..." | 0.64350... | 3/5 | 4/5 | 3/4
|
| 45° | /4 | /2
| /2 | 1
|
| 60° | /3 | /2
| 1/2
|
|
| 53°7'48.36..." | 0.92729... | 4/5 | 3/5 | 4/3
|
| 90° | /2 | 1 | 0 | ill-defined |
|
The most important applications of the Pythagorean Theorem is for finding
the distance between points in a plane. See the
next lesson for the formal development of
the cartesian coordinate system.
Consider the points (1,2) and (4,6). Since our x and y axes
are orthogonal (as in at right angles or mutually perpendicular),
it should be clear that the distance
between them is the square root of (4-1)2 + (6-2)2 =
32 + 42 = 9 + 16 = 25,
which is 5. In general, the distance between two points
(x1, y1) and (x2,
y2) is:
Points 1 and 2 may be interchanged with no affect since the squaring
operation forces the result positive.
Integers were the first numbers to be discovered and studied.
As a result, considerable efforts went into finding integer solutions
to some problems. Diophantus of Alexandria, a Greek, lived about
AD 250, wrote a treatise introducing symbolism
whose indeterminate equations are solved with rational values.
Consider the problem of finding triangular numbers
which are also square. We already know the formulae for both and
can set them equal: n(n+1)/2 = x2 or
n(n+1) = 2x2. 0, 1, 36, 1225, ... are solutions when (n,x)
{(0,0), (1,1), (8,6), (49,35),...}.
Such analysis can be quite difficult and might involve expressing square roots
as continued fractions, etc. and sparked the
early interest of many mathematicians.
Fermat considered extensions to the Pythagorean Theorem and wondered if there
existed any natural numbers such that
xn + yn =
zn for n > 2. This became know as
Fermat's Last Theorem and was solved in the negative only in recent years.
Specifically, Fermat conjectured this equation to be false. His notes are in
the margin of his copy of Diophantus' Arithmetica where he remarked
about 1637: "I have discovered a truly remarkable proof but this margin is too
small to contain it." This was, of course, written in Latin, since that is
what European scientists and mathematicians communicated in until Isaac
Newton's book Optiks was published in 1704 in the vernacular
(language native to the region, as in English).
Fermat clearly proved his theorem for n=4.
It is also clear that to prove it for all prime n is sufficient.
Euler produced an incomplete proof for n=3 in 1770
which was completed by later mathematicians.
Legendre proved it for n=5 in 1823.
Lamé proved it for n=7 in 1839. In 1850 Kummer
proved it for all n's which did not divide the numerators of
Bernoulli numbers.
One early proof failed because
prime factorization is not unique over the complex numbers.
Andrew Wiles in 1993 gave a three day series of lectures where he
stunned the world on the last day by completing a
proof of something
which implied FLT (Fermat's Last Theorem).
Although it required a little patching up over the course of the next year or
so, it is now well accepted. However, at 300 pages and dependant on
recent advances in mathematics, it seems
doubtful Fermat ever had a proof, but his margin certainly was too small!
Consider a three dimensional application of Pythagorean Theorem. In a box
with dimensions 3×4×12, it is clear the longest (body) diagonal is 13
(52 + 122 = 169 = 132).
There are 3 different lengths of diagonals on the faces:
In a perfect cuboid (box or parallelopiped), all seven of these numbers: three lengths,
three face diagonals, and one body diagonal would be integers.
This seems like a another
potential EXPO project and two homework problems will give two of the
three types of close encounters known. January 2005 I received
an e-mail from a Gale Greenlee claiming to have proved it impossible.
The
Fermat-Catalan Conjecture
is a generalization of Fermat's Last Theorem.
It asks if with x, y, and z as relatively prime integers,
can the equation:
xp + yq = zr,
with 1/p + 1/q + 1/r < 1 be satisfied.
p, q, and r are also integers.
Here are the only known solutions:
| x | y
| z | p
| q | r
| | 1 | 2 | 3 | 7 | 3 | 2
|
| 2 | 7 | 3 | 5 | 2 | 4
|
| 7 | 13 | 2 | 3 | 2 | 9
|
| 2 | 17 | 71 | 7 | 3 | 2
|
| 3 | 11 | 122 | 5 | 4 | 2
|
| 17 | 76271 | 210663928 | 7 | 3 | 2
|
| 1414 | 2213459 | 65 | 3 | 2 | 7
|
| 9262 | 15312283 | 113 | 3 | 2 | 7
|
| 43 | 96222 | 30042907 | 8 | 3 | 2
|
| 33 | 1549034 | 15613 | 8 | 2 | 3
|
For the first row, 17 + 23 = 1 + 8 = 9 =32
with 1/7 + 1/3 + 1/2 = 41/42 < 1.
The second row has 25 + 72 = 32 + 49 = 81 = 34
with 1/5 + 1/2 + 1/4 = 19/20 < 1.
Several students in 1997–98 attempted 25000 bonus points for
finding another solution and some continued their research in
2000–01 as an EXPO projects or college research.
Christian Goldbach lived in Russia 1690–1764. His mathematical work
includes what has become known as Goldbach's Conjecture which states:
every even number greater than 2 can be expressed as the sum of 2 primes, not
necessarily distinct. No counterexample has ever been found, but a
complete proof has eluded mathematicians since 1742.
However, during the summer of 2003 two groups, one Chinese, one Iranian,
both claimed proof. I reject the
Chinese proof out of hand.
They may have proved something similar, but not Goldbach's Conjecture.
They assume one is prime—elsewise, it is elegant.
You be the judge of the Iranian proof.
Example: 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53.
