MHF 3404, Notes
- 21 August
We'll start with the ancient Greeks. The history of mathematics
is a vast subject so we have to be selective. The ancient Greeks
are the first modern mathematicians. In any event,
that's how we trace our lineage.
Euclid lived in
around 300 B.C. Not the one in Virginia, the one in Egypt.
What's a Greek city doing in Egypt?
Alexander the Great
(356-323 B.C.) conquered a lot of territory. Alexandria is named after him.
Euclid wrote The Elements. Mostly we think of him in connection with
geometry, but there is a lot of number theory in The Elements also:
the Euclidean algorithm, the infinitude of the primes, perfect numbers,
and so on.
There are two things that stand out to me in the geometry of Euclid: the notion
of proof and the notion of construction (or algorithm). A proof
is an argument that something is true. Euclid required that proofs start from
things that were accepted as true and proceed step by step to the thing being proved.
The accepted things are called axioms or postulates; the steps are called deductions.
Of course the logic underlying the deductions has to be accepted also.
Two of Euclid's postulates for geometry were
These two postulates say what is true. Euclid's other three postulates
say what can be done
- Postulate 4. All right angles are equal.
- Postulate 5. If a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, the two straight
lines, if produced indefinitely, meet on that side on which the angles are
less than two right angles.
Postulate 2 says we can extend (produce) a straight line indefinitely far in
either direction. Postulate 3 says that we can draw a circle with center at
any point and with any other point on its circumference.
- Postulate 1. To draw a straight line from any point to any point.
- Postulate 2. To produce a finite straight line continuously in a
- Postulate 3. To describe a circle with any center and distance.
A Euclidean construction is one that can be done using only Postulates
1, 2 and 3. Euclid's
1 says that you can construct an equilateral triangle on a given finite straight line.
This restrictive notion of what a construction is lies behind the three famous
mathematical problems of antiquity given on page 40 of our text. Euclid's
9 says that you can bisect an angle (divide it into two equal parts). One of
the famous problems is to trisect an angle, that is, to divide it into three equal
parts. But you are only allowed to use the constructions of the first three postulates.
- 27 August
Angle trisection was shown to be impossible in 1837 by
Pierre Wantzel. He showed that you can't construct a
20 degree angle (but you can construct a 60 degree
angle). The reason is that the cosine of 20
degrees is a root of the polynomial
8X3 - 6 X2 - 1
while all the numbers you can construct by Euclidean
constructions are roots of polynomials of degree a power of
2, because the new numbers
can always be expressed in terms of square roots of
the old numbers (when you find the intersection of
a line and a circle, or two circles, you solve a
The same reasoning shows you can't duplicate a cube because
that involves constructing a number that is a root of the
polynomial X3 - 2.
Ferdinand von Lindemann showed that you can't square a
circle in 1882. He did that by showing that the number
pi is transcendental. A number is algebraic
if it is a root a polynomial with integer coefficients.
So the square root of 2, the cube root of 2, and the
cosine of 20 degrees are all algebraic. A number is
transcendental if it is not algebraic---so it is not
a root of any polynomial with integer coefficients. Every
number you get by Euclidean constructions is algebraic,
so you can't get a line of length pi, which you could
if you could square a circle of radius 1.
Lindemann had 60 Ph.D. students. The most famous was
David Hilbert who, among many other things, wrote a book
on geometry in which he made explicit many of the
assumptions that were implicit in Euclid.
Another ancient question is that of constructing regular
Carl Friedrich Gauss, possibly the greatest mathematician
of all time, showed how to construct a regular 17-gon
when he was around 20 years old (he was born in 1777).
The Greeks knew how to construct regular n-gons for
n = 3, 4, 5, 6, 8, 10, 12, 15, 16 and 20, but not 17.
You can't construct a regular n-gon for
n = 7, 9, 11, 13, 14, 18, or 19.
Go to the Mathematics
Genealogy Project and find out the mathematical ancestry of your
- 29 August
Archimedes 287-212 B.C. Lived in Syracuse which is on the island
of Sicily just off the toe of the boot that is Italy.
me a place to stand and I will move the Earth"
Heraclitus and Parmenides were presocratic philosophers who
lived around 500 B.C. (Socrates died in 399 B.C.) Heraclitus
believed that everything is changing. He said, "You can't step
into the same river twice." The idea there is that the second
time you step into it, it isn't the same river because it has
changed so much in the meantime. Parmenides, on the other hand,
believed that change and motion were illusions. Zeno's
paradoxes were meant to support Parmenides' ideas by showing
that the idea of motion was self-contradictory.
Incommensurable magnitudes. The magnitudes we have in
mind here are lengths. We say that a length u
can be used to measure a length a if we can
cut a up into a bunch of pieces each of which has
length u. We say that two lengths a and b
are commensurable if we can find a length u that
measures both a and b. The Pythagoreans thought
that all lengths should be commensurable, but it
turned out that the side of an isosceles right triangle is
not commensurable with its hypotenuse. The Pythagorean
theorem relates this to the irrationality of the square root
of 2. Indeed, if a = mu and b = nu,
then a² + a² = b²,
2m² = m² + m² = n²
which shows that 2 = (m/n)² so
the square root of 2 is equal to m/n, a rational number.
- 31 August
[The difficulties in the study of the infinite arise because] we
attempt, with our finite minds, to discuss the infinite, assigning
to it those properties which we give to the finite and limited;
but this ... is wrong, for we cannot speak of infinite quantities
as being the one greater or less than or equal to another.
Galileo, Dialogues Concerning Two New Sciences 1638
A discussion of Galileo's paradox and Georg Cantor can be found in
Here is Carl Friedrich Gauss on completed infinity: As to your
proof, I must protest most vehemently against your use of the infinite
as something consumated, as this is never permitted in mathematics.
The infinite is but a figure of speech . . . .
For more discussion about infinity and its history see
can't get there from here.
- 3 September
Here are the other books by Dirk J. Struik that are in the
WebLuis catalog of the
State University System of Florida.
- The origins of American science (New England). 1957
First published as "Yankee Science in the Making" 1948
- Lectures on classical differential geometry. 1950
- Lectures on analytic and projective geometry. 1953
- A source book in mathematics, 1200-1800. 1969
- Birth of the Communist manifesto, with full text of the
Manifesto, all prefaces by Marx and Engels, early drafts
by Engels and other supplementary material. 1971
I used his differential geometry book as a student. It's a pretty
little book. Differential geometry is a combination of analysis
(calculus) and geometry. Struik's Marxist leanings got him into
trouble with the infamous House UnAmerican Activities Committee
in 1951. He refused to answer their questions, invoking the
Fifth Amendment. He was suspended from MIT for several years.
I think you can see the influence of Marxism from time to
time in our text in some of the offhand remarks.
Struik died last year at the age of 106.
Square root of 2. There are two parts to the argument that
the square root of 2 is irrational (cannot be written as a
fraction, that is, as a quotient of two integers).
But even numbers all have the common factor 2.
So these two facts give us a contradiction if the square of
p/q is 2. On the one hand,
p and q must both be even; on the other we can
assume they have no common factor. That is presumably what
Aristotle was thinking about when he said that you end up with
a number that is both even and odd: one of p and q
must be odd, because they have no common factor, yet both must be
- If the square of a rational number
p/q is 2, then both p and q are even.
This uses properties of even and odd numbers under multiplication
(the square of an even number is even, the square of an odd number
- Every fraction can be written in
lowest terms, that is, with p and q having
no common factor.
- 4 September
The theory of proportions is ascribed (by Archimedes) to
It is presented in Book V of Euclid's Elements.
Euclid uses this notion in Book VI when he proves things like
Proposition 5: If two triangles have their sides proportional,
then the angles opposite corresponding sides will be equal.
What does it mean for the sides to be proportional? This is a
relationship among four sides, a, a',
b', and b'. It is expressed by saying, for example,
a is to a' as b is to b'.
Or that the ratio of a to a' is the same
as the ratio of b to b'. What is a ratio?
Euclid says, in Definition 3,
- A ratio is a sort of relation in respect of size
between magnitudes of the same kind.
Two lengths are "magnitudes of the same kind", as are two areas.
The notion of ratio is pretty muddy here, but it is clarified
in Definition 5 where Euclid says what it means for two ratios
to be equal. It really makes no difference what a ratio is,
what counts is when they are equal. In fact, if you understand when
two ratios are equal, you will have a handle on what a ratio is.
A ratio is given by two magnitudes. A simple ratio occurs when
the first magnitude is twice as big as the second. That ratio is
sometimes written 2:1. A more complicated ratio is when twice
the first magnitude is equal to three times the second. That ratio
is written 3:2. Why not 2:3? Well, for one thing, the first
magnitude is clearly bigger. Notice that if we break
the first magnitude up into three equal pieces, and the second into
two equal pieces, then all the pieces have the same size. If the
first magnitude is 3 inches long, then the second will be 2
Here is Euclid's Definition 6:
- Magnitudes which have the same ratio are called proportional.
So to talk of proportional magnitudes, we need two ratios, hence
The theory of proportions was developed in order to deal with
incommensurable magnitudes. These are magnitudes whose ratios
cannot be expressed, like 2:1 or 3:2, in terms of the ratios of
Here is a rephrasing of Euclid's Definition 5. What does it mean
to say, of four magnitudes
a, a', b, and b',
a is to a' as b is to b'?
It means, among other things, that if
7a is greater than 5a', then
7b is greater than 5b'. The same has to be true
for any other whole numbers we use instead of 7 and 5.
7a and 7b are equimultiples
of a and b, in the language of Definition 5
(they are both gotten by multiplying by 7).
We know what it means for a magnitude to be 7 times as large
as another magnitude, and we know what it means for one magnitude
to be larger than another.
Definition 5 reduces the question of when two ratios are equal to
questions about when one magnitude is greater than another.
- 16 September
Archimedes believed his greatest work was on The sphere and the
cylinder. He wanted that inscribed on his tombstone. He showed
that the volume of a sphere is equal to 2/3 of the volume of the
circumscribed cylinder, and that the area of a sphere is also equal
to 2/3 the area of the circumscribed cylinder (including the top and
bottom of the cylinder).
Heron of Alexandria lived around 50 A.D. The name is sometimes
spelled "Hero" (Similarly, Plato is sometimes spelled "Platon").
He is famous for a formula, given in Book I of his Metrica,
that shows how to compute the area A of a triangle if you
know the lengths a, b, and c, of its sides.
Heron's formula is
A2 = s(s - a)(s - b)(s - c)
where s is the semiperimeter (a + b + c)/2.
Problem II-8 in Arithmetica by Diophantus of Alexandria, who
lived around 150 A.D., is
To divide a given square number into two squares
Diophantus illustrates how to do this by doing for the square 16.
He wants to write 16 = x2 +
(16 - x2), where 16 - x2 is
a square. He suggests trying to find x such that
16 - x2 = (2x - 4)2
If you can do that, then 16 - x2 will be a square
and you will have solved the problem. Expanding the right hand
side of the desired equation gives
16 - x2 = 4x2 + 16 - 16x
Canceling the 16's and adding x2 to both sides
gives 5x2 = 16, so x = 16/5. Note that
2x - 4 = 32/5 - 20/5 = 12/5, so we
16 = (16/5)2 + (12/5)2
Try the same trick with 3x - 4, and with 5x - 4,
instead of 2x - 4.
Pierre de Fermat, around 1637, wrote in the margin of his copy
of Arithmetica that
- It is impossible to write a cube as a sum of two cubes, a
fourth power as a sum of two fourth powers, and, in general,
any power beyond the second as a sum of two similar powers.
For this, I have discovered a truly wonderful proof, but the
margin is too small to contain it.
This is the famous Fermat's last theorem, so called because
it was the last theorem to be proved of the many that Fermat had
claimed that he could prove but did not furnish the proof. It was
not proved until 1993 (by Andrew Wiles).
In The Greek Anthology, around 500 A.D., Metrodorus
proposed the following problem about Diophantus:
- His boyhood lasted 1/6 of his life. He married after 1/7
more. His beard grew after 1/12 more. His son was born
5 years later. The son lived to half his father's age. The
father died 4 years after the son. How long did Diophantus
- 19 September
Diophantine equations are equations for which we seek
integer solutions. Diophantus was looking for rational
solutions to the equation
16 = x2 + y2.
That is the same as looking for integers a, b, and c
satisfying the equation
16c2 = a2 + b2.
At the international conference of mathematicians in Paris in 1900,
David Hilbert gave a list of 23 problems for mathematicians of the
20-th century to work on. Hilbert's tenth problem was
"Unknown quantities" are the unknowns, "rational integral numerical
coefficients" simply means that the coefficients are integers, and
"rational integers" are just integers. What Hilbert is looking for
here is essentially a computer program that will look at such
an equation and figure out whether it has a solution (in integers).
- Given a diophantine equation with any number of unknown
quantities and with rational integral numerical coefficients:
To devise a process according to which it can be
determined by a finite number of operations whether
the equation is solvable in rational integers.
This problem was settled in 1970 when Yuri Matiyasevich, a
Russian mathematician, showed that such a program could not exist.
Pythagorean triples. These are solutions to the diophantine
x2 + y2 = z2.
They are called Pythagorean because of the Pythagorean
theorem which relates the sides of a right triangle in exactly
this way. When Diophantus wrote 16 as the sum of two squares (of
rational numbers) he constructed the Pythagorean triple 12, 16, 20.
Do you see why? This triple is 4 times the triple 3, 4, 5, the
smallest and most famous Pythagorean triple. If the numbers in
the triple have no common factor, the triple is said to be
primitive. So 3, 4, 5 is primitive but 12, 16, 20 is not.
Clearly the primitive triples are the most interesting ones.
Al-Khwarizmi. His full name is: Abu Ja'far Muhammad ibn Musa Al-Khwarizmi.
The "ibn Musa" here means "son of Moses". "Abu" means "father of". So
Al-Khwarizmi was Muhammad, the father of Ja'far and the son of Moses.
"Al" is "the" in arabic,
like "el" in Spanish. "Al-Khwarizmi" is analogous to "El Greco", the
Spanish Painter who was Greek, hence the name "El Greco" which
means "the Greek". Al-Khwarizmi came from a region called Khwarizm.
In Hisab al-jabr w'al-muqabala you can see two instances
of the word "al". The "w" before "al-muqabala" means "and".
- 21 September
The continuum hypothesis. The first problem that
Hilbert posed at the 1900 Paris conference was to prove Cantor's
continuum hypothesis: "Every system of
infinitely many real number is either equivalent to the assemblage
of natural integers, 1, 2, 3, ... or to the assemblage of all
real numbers and therefore to the continuum, that is, to the
points of a line."
Today we would probably replace Hilbert's words "system" and "assemblage"
by the word "set", which means the same thing. We still use the
word "equivalent" the way Hilbert, following Cantor, is using
it here: two sets are
equivalent if there is a one-to-one correspondence between
them. Recall that Galileo had observed that the set of positive
integers is equivalent, in this sense, to the set of squares of
In some sense, the continuum hypothesis was settled in the twentieth
century. In another sense, it remains wide open. In the 1930's
Kurt Goedel showed that you couldn't disprove the
continuum hypothesis using the commonly accepted axioms of set
theory. This was quite impressive because all reasoning about
sets was done in accordance with those axioms, and pretty much
still is. In the 1960's, Paul Cohen showed that you couldn't
prove the continuum hypothesis using the
axioms of set theory. Thus we have an independence result:
the continuum hypothesis is independent of the axioms of set
theory---you can neither prove it nor disprove it.
So is the continuum hypothesis true or not? We can't settle that
question without introducing more axioms for set theory, but
no one has proposed any axioms for set theory that have achieved
any degree of acceptance since Goedel's result in the 1930's. An
alternative view is that there is no fact of the matter: it just
isn't the case that the continuum hypothesis is either true or
false. That seems to be the position we are drifting into,
although it is contrary to the way people have always viewed
- 23 September
Preparation for the test on Friday, September 28.
syllabus says, "You should be able to distinguish Euclid from Euler,
both as to time and place, and as to their contributions to
mathematics." Of course that doesn't mean just Euclid and Euler.
You are supposed to be able to do that with all the major players
that we have discussed. For the test, these players are:
Pierre de Fermat,
Carl Friedrich Gauss,
You should also be able to explain the ideas that we have
encountered. For the test, be prepared for:
algebraic and transcendental numbers,
the continuum hypothesis,
the golden ratio,
the harmonic series.
potential and completed infinity,
reductio ad absurdum,
squaring the circle,
the theory of proportions,
- 24 September
Infinity is an interesting site that mentions a lot of the
people and topics concerning infinity that have come up in class.
Kurt Gödel, 1906-1978, was born in what is now Brno in the
Czech Republic. Brno is not that far from Vienna, where Gödel
was a student and a professor. In 1931 he published what was to
become one of the landmark mathematics papers of the twentieth
century. He proved that for any system of axioms for mathematics,
there will always be statements that you can neither prove nor
disprove. In fact, there will be true statements that you cannot
prove. This is his incompleteness theorem. Much has been
made of this theorem. Some people use it to argue that we
are smarter than computers because we can see that the unprovable
true statement is true, but the computers presumably cannot.
In 1940, Gödel came to the Institute for Advanced Studies
in Princeton where he remained until his death. In 1940, in
a Princeton monograph, he showed that you couldn't disprove
Cantor's continuum hypothesis. For more about Gödel see
- 10 October. A history of the cycloid, with related links,
is given at
- 11 October.
of a Sphere the author is using Cavalieri's principle to
find the volume of a hemisphere. On the left is the hemisphere;
on the right is the cylinder that would just contain that
hemisphere. Inside the cylinder is an upside down cone. The
green disk on the left is a cross section of the hemisphere.
The orange ring on the right is a cross section of what's
left if you cut the cone out of the cylinder. There is an
unstated argument to the effect that the green disk has the
same area as the orange ring, so, by Cavalieri's principle,
the volume of the hemisphere is equal to the volume of the
cylinder that lies outside the cone. You can drag the disk
and the ring up and down.
- 18 October.
Newton's memorandum of what he accomplished
during the plague in London.
From Descartes' Geometry, 1637:
Galileo (who was going blind at the time) responded:
- Geometry should not include lines that are like strings
in that they are sometimes straight and sometimes curved, since
the ratio between straight and curved lines are not known and
I believe cannot be discovered by human minds
- Who is so blind as not to see that if there are two equal
straight lines, one of which is then bent into a curve, that
curve will be equal to the straight line?
Christopher Wren, the great English architect who designed Saint
Paul's Cathedral, showed in 1658 that the length of the cycloid
is four times the diameter of the generating circle.
(All of this material is found in Calculus Gems by
George Simmons, McGraw-Hill 1992.)
In his Method of Fluxions (1671), Newton gave the fundamental
task of calculus as: "The relation of the fluents being given, to
find the relation of their fluxions [and conversely]".
The fluents are the (flowing) quantities and the fluxions
are their rates of change. This task is what is nowadays called a
related rates problem.
Analyst, by George Berkeley, is a criticism of Newton's method
of fluxions, in particular of Lemma II of Principia Mathematica (see
Sections 9 and 10).
- 19 October. Newton's Lemma II.
The Genitum that Newton is talking about is an expression
A2B3/5C -1D -2/3
consisting of a product of variables raised to various powers.
The powers can be fractions, either positive or negative. These
variables, and the Genitum itself, are "variable and indetermined".
We still use the nouns variable and indeterminate
in the same
way Newton is using them here. He goes on to say that they are
"increasing or decreasing as it were by a perpetual motion or
flux". That emphasizes their variable nature.
Now he talks about the "Moment" associated with each variable.
The easiest way to translate "Moment" is by "differential". So,
in Leibnizian notation, the moment of A is dA.
Newton himself denotes the moment of A by a, a
notation that becomes awkward after a while (all variables have
to be denoted by capital letters).
In the statement of the lemma, Newton uses the phrase "drawn
into". That can be translated by "multiplied by".
The "generating sides" are the basic variables---so the sides of
A2B3/5C -1D -2/3
are A, B, C, and D.
"The indices of the powers of those sides" are the exponents
2, 3/5, -1, and -2/3. Newton explains what the coefficient
of a generating side is in the last sentence of the first
paragraph: "the quantity which arises by applying the Genitum
to that side". What does it mean to apply the Genitum to a side?
It means to form the quotient "Genitum over side", or
G/S. So applying the Genitum above to the side
A2B3/5C -1D -2/3/C =
A2B3/5C -2D -2/3
Note that Newton says "It will be the same thing, if, instead
of moments, we use ... the Velocities of the increments and
decrements (... fluxions ...)". What that means is that,
in the lemma, we
could use dA/dt, the velocity of A,
instead of dA, the differential of A. Just divide
everything by dt.
Look at his examples in the second paragraph and see if you
can calculate the moments using the instructions given in the
statement of the lemma. Or maybe I should say, see if you can
figure out what those instructions are by looking at the
- 22 October. Comments the October 19 assignment.
For the Java applet, I would like to have seen the observation
that the rectangles are the same in all the (approximate)
parallelograms, and there are the same number of them, so the
total areas are the obviously same.
The a in Rouse Ball's example is an ordinary real number,
not infinitesimal or infinite. It is the ratio of the two sides
of the right triangle.
The ordinate at a point of the base is the line extending
from that point to the hypotenuse. We are thinking of the
(area of the) triangle as being composed of all those lines.
The reason (1/2)na is "inconsiderable compared with"
(1/2)n2a is because the quotient is
1/n which is an infinitesimally small number.
The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ....
Those of the form 4n + 1 are 5, 13, 17, 29, .... The number
29 can be written as the sum of the two squares 25 and 4. Part
of this problem was to show that there is no other way to write
it as a sum of two squares. ("can be expressed once, and only
once, as the sum of two squares"). How do you show that?
One way is successively to subtract squares from 29 and see that
you only get a square when you subtract 4 or 25. So 29 - 1 = 28,
not a square; 29 - 9 = 20, not a square; 29 - 16 = 13, not a
square. You can stop when the squares you are trying to subtract
are bigger than 29. Actually, you can stop when the squares are
bigger than 29/2 (why?).
If the number is not prime, then you might be able to write it
as the sum of two squares in two different ways. For example,
65 = 64 + 1 = 49 + 16. Here is a sort of explanation of that
phenomenon. The number 65 is 5 times 13. We can write
5 = (2 + i)(2 - i)
because 5 is the sum of two squares. Similarly
13 = (3 + 2i)(3 - 2i)
We get the two ways of writing 65 by multiplying a factor from 13
and a factor of 5. So (2 + i)(3 + 2i) =
4 + 7i gives us 16 + 49, while (2 + i)(3 - 2i) =
8 - i gives us 64 + 1.
I think maybe 65 is the smallest number that can be written as
the sum of two squares in two different ways (if you don't count
25 = 0 + 25 = 9 + 16). There is a famous
about Hardy's visiting Ramanujan in the hospital and the smallest
number that can be written as the sum of two cubes in two
A converse of Fermat's theorem is that an
odd number which is the sum of two squares must be of the form
4n + 1. This follows easily from the fact that, modulo 4,
the only squares are 0 and 1, so the only sums of two squares are
0, 1, and 2. So an odd number that is the sum of two squares
must be equal to 1 modulo 4. Fermat's theorem is much more
difficult to prove than this.
- 24 October.
The first calculus book was
des infiniment petits (Infinitesimal analysis) by the
de l'Hospital in 1696.
It was in use for almost 100 years.
It is here that the famous "L'Hôpital's rule" appears.
Johann (John) Bernoulli later claimed it as his own.
L'Hospital had presumably paid Bernoulli for the right to
use Bernoulli's results. In addition, he gave credit to
him in the preface to his book:
"I am obliged to the gentlemen Bernoulli for their many bright ideas; particularly to the
younger Mr. Bernoulli who is now a professor in Groningen."
Here is another, more complete, translation of that passage from
the preface to L'Hôpital's book: "I must own myself very much
obliged to the labours of Messieurs Bernoulli, but particularly to
those of the present Professor at Groeningen, as having made free
with their Discoveries as well as those of Mr Leibniz: So that
whatever they please to claim as their own I frankly return to them."
The first four propositions in the book are
- d(a + x + y - z) = dx + dy - dz
- d(xy) = y dx + x dy
- d(x/y) = (y dx - x dy)/yy
- d(xr) = rxr-1 where r
can be any positive or negative integer or rational
A nice history of the cycloid can be found in
Helen of geometers including mention of the roles
of Newton, Leibniz and de l'Hospital.
- 29 October
Preparation for the test on Friday, November 2.
Know the players:
Know the topics:
Analyzing the cycloid,
Solving cubic, quartic, and quintic equations,
- 31 October
Here is a
you can play with. Happy Halloween!
- 9 November
The Legendre Symbol. I will write it here as
( a/p), although it is usually seen more
The fraction line is not a fraction line---it does not mean division.
The a is an arbitrary integer, the p is a prime. The
Legendre symbol is equal to 0 if a
is divisible by p, is equal to 1 if a is not divisible
by p and is equal to some square modulo p, and is equal
to -1 if a is not equal to any square modulo p.
If ( a/p) = 1, we say that a
is a quadratic residue modulo p. If
( a/p) = -1, we say that a
is a quadratic nonresidue modulo p.
So ( 2/7) = 1 because 2 = 32 mod 7,
while ( 2/5) = -1 because the squares modulo 5
are 12 = 1, 22 = 4, 32 = 4, and
42 = 1. Because every nonzero square is the square of two
different numbers, half of the numbers 1, 2, ..., p-1 are
quadratic residues, and half are quadratic nonresidues.
symbol is like the Legendre symbol except that p can be
any odd number. The Jacobi symbol takes on the values 0, 1, and -1,
just like the Legendre symbol, and it is equal to the Legendre symbol
when p is a prime. But if p is not a prime, it doesn't
tell you whether or not a is a quadratic residue. It is used
in the computation of Legendre symbols.
- 14 November
Line has an applet illustrating the circumcenter, centroid, and
orthocenter of a triangle. You can grab a vertex and change the shape
of the triangle. See also
Euler Line of a Triangle, another interactive applet which draws
in more lines (the altitudes, medians, and perpendicular bisectors of
the sides) and provides more explanation.
concerning the Petersburg Game has excerpts from the letters
between Nicolas Bernoulli, Montmort, Cramer, and Daniel Bernoulli
concerning this game.
Nicolas Bernoulli told Montmort about the game in a letter
of 9 September 1713. He asked Daniel Bernoulli his opinion of it
in a letter of 27 October 1728. In 1713 Montmort published the game
in his Essay d'analyse sur les jeux de hazard (analysis of
games of chance).
and the Grand Jan has a picture of the 1713 edition of Montmort's
book, and a description of some of its contents.
The following passage is from Daniel Bernoulli's paper,
"Specimen theoriae novae de mensura sortis", which appeared in 1738
in Commentarii Academiae Scientiarum Imperialis Petropolitanae
[Papers of the Imperial Academy of Sciences in Petersburg] 5
The translation, "Exposition of a new theory of the measurement
of risk" appeared in Econometrica 22 (1954) 123-136.
My most honorable cousin the celebrated Nicolaus Bernoulli,
Professor utriusque iuris at the University of Basle, once submitted
five problems to the highly distinguished mathematician Montmort.
These problems are reproduced in the work L'analyse sur les jeux
de hazard de M. de Montmort, p. 402. The last of these
problems runs as follows: Peter tosses a coin and continues to do
so until it should land "heads" when it comes to the ground. He agrees
to give Paul one ducat if he gets "heads" on the very first throw,
two ducats if he gets it on the second, four if on the third, eight if
on the fourth, and so on, so that with each additional throw the
number of ducats he must pay is doubled. Suppose we seek to determine
the value of Paul's expectation. My aforementioned cousin
discussed this problem in a letter to me asking for my opinion.
Although the standard calculation shows that the value of Paul's
expectation is infinitely great, it has, he said, to be admitted
that any fairly reasonable man would sell his chance, with great
pleasure, for twenty ducats.
- 19 November
Cover of Instituzioni
analitiche ad uso della gioventu' italiana by Maria Gaetana Agnesi.
Also some of the pages of the text (click on "Table of Contents").
story of noneuclidean geometry
Ferdinand Karl Schweikart [1780-1859], memorandum to Gauss, 1818.
There are two kinds of geometry--a geometry in the strict sense--the
Euclidean; and an astral geometry. Triangles in the latter have the property
that the sum of their three angles is not equal to two right angles. This
being assumed, we can prove rigorously:
Papers, References, Internet Sites. Has some good links to sites
about noneuclidean geometry. From a geometry course taught by
Richard Delaware at the University of Missouri - Kansas City.
If this constant were for us the radius of the earth, so that every line
drawn in the universe from one fixed star to another, distant 90 degrees
from the first, would be tangent to the surface of the earth), it would
be infinitely great in comparison with the spaces which occur in daily
- That the sum of the three angles of a triangle is less than two right angles;
- that the sum becomes ever less, the greater the area of the triangle;
- that the altitude of an isosceles right-angled triangle continually grows,
as the sides increase, but it can never become greater than a certain length,
which I call the constant.
The Euclidean geometry holds only on the assumption that the constant
is infinite. Only in this case is it true that the three angles of every
triangle are equal to two right angles.
abstraction, amateur English translation. Original French version:
Préface Scientifique à la théorie de Galois
- 21 November
French mathematician by Tom Petsinis is a recent novel
about Evariste Galois, written in the first person as if by
Galois himself. It looks easy to read and mentions lots of
our people: Cardano, Tartaglia, Lagrange, Descartes, Pascal,
Archimedes, Euclid, Abel, Mersenne, Cauchy, Euler, Diophantus,
etc. It is available in paperback.
From a letter in 1799 from Gauss to the father of János
Bolyai (who wasn't born yet):
In my own work theron I myself have advanced far (though my
other wholly heterogeneous employments leave me little time
therefor) but the way, which I have hit upon, leads
not so much to the goal, which one wishes, as much more to
making doubtful the truth of geometry.
excerpts from letters of Gauss; in particular, on noneuclidean geometry. The last
one is to
1776-1831 (Paris), a mathematician who used
the name Antoine-August Le Blanc in the tradition of women writers who
assumed men's names so that their work would not be rejected
out of hand. Gauss goes on to say
Indeed I have come upon much, which with most no doubt would
pass for a proof, but which in my eyes proves as good as
For example, if one could prove, that a rectilineal triangle
is possible, whose content may be greater, than any given
surface, then I am in condition, to prove with perfect rigor
Most would indeed let that pass as an axiom; I not; it might
well be possible, that, how far apart soever one took the
three vertices of the triangle in space, yet the content was
always under a given limit.
I have more such theorems, but in none do I find anything
The scientific notes with which your letters are so richly filled
have given me a thousand pleasures. I have studied them with attention
and I admire the ease with which you penetrate all branches of
arithmetic, and the wisdom with which you generalize and perfect.
Arithmetic here means number theory. Sophie Germain worked
on Fermat's last theorem, which she divided into two cases, called
Case 1 and Case 2 to this day. Her ideas and methods influenced
research on this problem for almost 200 years. She has a class of
prime numbers named after her: a prime p is called a
Sophie Germain prime if 2p + 1 is also prime. So
5 is a Sophie Germain prime because 11 is also prime, but 7 is not
a Sophie Germain prime because 15 is not a prime. For more on Sophie
eliminates Euclid's flaw. For a picture of the cover of the book, see
ab omni naevo vindicatus.
- 26 November
Notes 6 of Bill Cherowitzo, University of Colorado at Denver, on
noneuclidean geometry, from his course "Higher Geometry I".
- 27 November
Our last topic will be Gödel's incompleteness theorem, possibly
the most interesting mathematical result of the twentieth century.
on Gödel's incompleteness theorem
innocence, an article by Keith Devlin for the general public about
the incompleteness theorem.
8 in 21st-century science talks about Hilbert, Turing,
the paradox of the liar, and incompleteness. Pictures of Hilbert and
Gödel, and a couple of Escher drawings.
- 28 November
in the Movies, a guide to major motion pictures with scenes of
real mathematics. Be sure to see "A beautiful mind", the movie
about the mathematician John Nash who won the Nobel prize in
economics, when it comes out this winter.
- 29 November
Student Perception of Teaching (SPOT) assessment
forms will be passed out and completed in class on Friday, November 30.
The final examination, Friday, December 7 from 1:15 to 3:45, will be like the other two tests, but longer:
Know the players:
William Rowan Hamilton,
Know the topics:
algebraic and transcendental numbers;
potential and completed infinity;
reductio ad absurdum;
Saint Petersburg paradox;
solving cubic, quartic, and quintic equations;
squaring the circle;
theory of proportions;
witch of Agnesi;
- 30 November
Some other notable women mathematicians
- Julia Robinson, 1919-1985. Contributed to the solving of Hilbert's
tenth problem. Was president of the American Mathematical Society.
Her sister, Constance Reid, wrote a biography of
her: Julia: A life in mathematics. See Julia
- Sofia Kovalevskaya, 1850-1891. Born in Moskow. Got her
doctorate in 1874 at Gottingen. The Association of Women Mathematicians
is sponsoring Sonia Kovalevsky High School Mathematics Days
with grants to colleges for workshops, talks, and problem
solving competitions for high school women students and their
teachers. See Kovalevskaya.
- Emmy Noether, 1882-1935. One of the founders of modern
algebra. Hilbert, when he had difficulty getting her a position
at Gottingen because she was a woman, asked the question, "Is
this a university or a bathhouse?" She taught courses there
which were listed under Hilbert's name. See Emmy Noether: A
tribute to her life and work by Brewer and Smith (that's
FAU's own Jim Brewer). See also Emmy