In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term partitio numerorum.
Many of Euler’s results in number theory involved divergent series. He was courageous in manipulating these but had remarkable insight and, almost invariably, his findings, although not rigorously established, were valid.
Partitions
In number theory, a partition of a positive integer 
Continue reading ‘Number Partitions: Euler’s Astonishing Insight’
. This was known to Galileo. However, with the usual ordering,
is a proper subset of 
, it must be smaller than 
, Greek for hand. If chirality is absent, we have an achiral object.
be a Banach Space
. Another triangle, with sides 
and 
is stacked upon the first. This process continues, giving hypotenuse lengths 
for all 
-th entry in row 
(read 
and 
having the same underlying set 
[Image from Wikimedia Commons].
.
is the group of isometries of 
, we have the group 
, comprising all combinations of translations, rotations and reflections of the plane.
to diameter 
the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that 
is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post 
. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.
. More generally, for an N-gon the ratio is easily shown to be 
. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum 
is the angle through which the disk has rotated. The centre of the disk is at 
.
be the radius of the circle and 
the distance from the axis to the centre of the circle, with 
.
as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result
. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,
can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.
. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:
are the
, and ratios of these, the positive rational numbers 
. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.
, which include rationals, irrationals like 
.
of points in the plane 
(Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle? The answer is given by Jung’s Theorem.
for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).
we can usually find 
, whereas we may not be able to find 
.
ThatsMaths 