In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, specifically in
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
theory, the Markov constant
of an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
is the factor for which
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and
...
can be improved for
.
History and motivation
Certain numbers can be approximated well by certain
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
; specifically, the
convergents of the continued fraction are the best
approximations by rational numbers having denominators less than a certain bound. For example, the approximation
is the best rational approximation among rational numbers with denominator up to 56. Also, some numbers can be approximated more readily than others.
Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
proved in 1840 that the least readily approximable numbers are the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
, in the sense that for every irrational number there exists infinitely many rational numbers approximating it to a certain degree of accuracy that only finitely many such
rational approximations exist for rational numbers . Specifically, he proved that for any number
there are infinitely many pairs of relatively prime numbers
such that
if and only if
is irrational.
51 years later,
Hurwitz
Hurwitz is one of the variants of a surname of Ashkenazi Jewish origin (for historical background see the Horowitz page).
Notable people with the surname include:
*Adolf Hurwitz (1859–1919), German mathematician
** Hurwitz polynomial
**Hurwitz m ...
further improved
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and
...
by a factor of , improving the right-hand side from
to
for irrational numbers:
:
The above result is best possible since the
golden ratio is irrational but if we replace by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for
.
Furthermore, he showed that among the irrational numbers, the least readily approximable numbers are those of the form
where
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
,
and
.
(These numbers are said to be ''
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
'' to
.) If we omit these numbers, just as we omitted the rational numbers in Dirichlet's theorem, then we ''can'' increase the number to 2. Again this new bound is best possible in the new setting, but this time the number
, and numbers equivalent to it, limits the bound.
If we don't allow those numbers then we ''can'' again increase the number on the right hand side of the inequality from 2 to /5,
for which the numbers equivalent to
limits the bound. The numbers generated show how well these numbers can be approximated, this can be seen as a property of the real numbers.
However, instead of considering Hurwitz's theorem (and the extensions mentioned above) as a property of the real numbers except certain special numbers, we can consider it as a property of each excluded number. Thus, the theorem can be interpreted as "numbers equivalent to
, or
are among the least readily approximable irrational numbers." This leads us to consider how accurately each number can be approximated by rationals - specifically, by how much can the factor in
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and
...
be increased to from 1 for ''that specific'' number.
Definition
Mathematically, the Markov constant of irrational
is defined as
.
If the set does not have an upper bound we define
.
Alternatively, it can be defined as
where
is defined as the closest integer to
.
Properties and results
Hurwitz's theorem implies that
for all
.
If