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 
-th entry in row 
(read 
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 
and transcendental numbers like 
.
for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).
. The number of atoms in the universe is estimated to be about 
. When we consider permutations of large sets, even more breadth-taking numbers emerge.
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the 
can be expanded as a continued fraction:![\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x+%3D+a_0+%2B+%5Ccfrac%7B1%7D%7B+a_1+%2B+%5Ccfrac%7B1%7D%7B+a_2+%2B+%5Ccfrac%7B1%7D%7B+a_3+%2B+%5Cdotsb+%2B+%5Ccfrac%7B1%7D%7Ba_n%7D+%7D+%7D%7D+%3D+%5B+a_0+%3B+a_1+%2C+a_2+%2C+a_3+%2C+%5Cdots+%2C+a_n+%5D+&bg=ffffff&fg=000000&s=0&c=20201002)
are integers, all positive except perhaps 
. If 
we add it to 
; then the expansion is unique.
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see 
, but equality is violated for this enormous value and, intermittently, for larger values of 
is chosen randomly with probability 
of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.
![Multiplication table for octonions, of the formz=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]](https://thatsmaths.files.wordpress.com/2014/10/octonions.jpg)
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