5,014 research outputs found
The Olympic medals ranks, lexicographic ordering and numerical infinities
Several ways used to rank countries with respect to medals won during
Olympic Games are discussed. In particular, it is shown that the unofficial
rank used by the Olympic Committee is the only rank that does not allow
one to use a numerical counter for ranking – this rank uses the lexicographic
ordering to rank countries: one gold medal is more precious than any number
of silver medals and one silver medal is more precious than any number of
bronze medals. How can we quantify what do these words, more precious,
mean? Can we introduce a counter that for any possible number of medals
would allow us to compute a numerical rank of a country using the number
of gold, silver, and bronze medals in such a way that the higher resulting
number would put the country in the higher position in the rank? Here we
show that it is impossible to solve this problem using the positional numeral
system with any finite base. Then we demonstrate that this problem can be
easily solved by applying numerical computations with recently developed
actual infinite numbers. These computations can be done on a new kind of
a computer – the recently patented Infinity Computer. Its working software
prototype is described briefly and examples of computations are given. It is
shown that the new way of counting can be used in all situations where the
lexicographic ordering is required
Evaluating the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge
Very often traditional approaches studying dynamics of self-similarity
processes are not able to give their quantitative characteristics at infinity
and, as a consequence, use limits to overcome this difficulty. For example, it
is well know that the limit area of Sierpinski's carpet and volume of Menger's
sponge are equal to zero. It is shown in this paper that recently introduced
infinite and infinitesimal numbers allow us to use exact expressions instead of
limits and to calculate exact infinitesimal values of areas and volumes at
various points at infinity even if the chosen moment of the observation is
infinitely faraway on the time axis from the starting point. It is interesting
that traditional results that can be obtained without the usage of infinite and
infinitesimal numbers can be produced just as finite approximations of the new
ones
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area
The Koch snowflake is one of the first fractals that were mathematically
described. It is interesting because it has an infinite perimeter in the limit
but its limit area is finite. In this paper, a recently proposed computational
methodology allowing one to execute numerical computations with infinities
and infinitesimals is applied to study the Koch snowflake at infinity. Numerical
computations with actual infinite and infinitesimal numbers can be
executed on the Infinity Computer being a new supercomputer patented in
USA and EU. It is revealed in the paper that at infinity the snowflake is not
unique, i.e., different snowflakes can be distinguished for different infinite
numbers of steps executed during the process of their generation. It is then
shown that for any given infinite number n of steps it becomes possible to
calculate the exact infinite number, Nn, of sides of the snowflake, the exact
infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn,
of the Koch snowflake as the result of multiplication of the infinite Nn by
the infinitesimal Ln. It is established that for different infinite n and k the
infinite perimeters Pn and Pk are also different and the difference can be infinite.
It is shown that the finite areas An and Ak of the snowflakes can be
also calculated exactly (up to infinitesimals) for different infinite n and k and
the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed
starting from different initial conditions are also studied and their
quantitative characteristics at infinity are computed
- …