Here is a simple problem in probability.
(1) Pick a number k between 1 and 9. Assume all digits are equally likely.
(2) Pick a number m in the range from 1 to k.
What is the probability distribution for the number m?
A graph of the probability distribution is shown in the figure here.
Probability distribution for a decimal digit selected in a two-step process.
Can you derive a formula for this probability distribution?
Can you generalise it to the range from 1 to 10^n?
Can you relate this problem to Benford’s Law [described here]?
Solution, and more on Benford’s Law, next week.
in reduced form (
and 
coprime), a Ford circle is a circle with center at 
and radius 
. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.
and 
. This immediately yields the well-known sequence
is its modulus or absolute value 
. We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric 
and, using it, we can define the usual topology on the real numbers 
.
) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.
are dense in the real numbers 
is 
. We cannot list them in ascending order, because there is no least rational number greater than 
.
![\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BZ%7D%5Bi%5D+%5Cequiv+%5C%7B+m+%2B+i+n+%3A+m%2C+n+%5Cin+%5Cmathbb%7BZ%7D+%5C%7D+%5C%2C.+&bg=ffffff&fg=000000&s=0&c=20201002)
![{\mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
forms a two-dimensional lattice in the complex plane. For any element ![{g \in \mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
we consider the four numbers 
as associates. The associates of 
are known as units: 
.
is a way of writing 
. 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 
![\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.
ThatsMaths 