Kurt Hensel (1861-1941)
Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.
Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.
, 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 "\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 ]")
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 
such that 
, 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)
