This is easily proven with elementary mathematics using a counting argument, as follows:

Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem.

axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form

Another theorem that is commonly proved with a double counting argument states that every undirected graph contains an even number of vertices of odd degree.

The inquisitorial nature of the proceedings, with assessors riding around and counting windows, aroused strong opposition, and many refused to pay, making the constitutional argument that this tax was not being levied in proportion to population.

In general, if the counting formula involves a division, a similar double counting argument ( if it exists ) gives the most straightforward combinatorial proof of the identity, but double counting arguments are not limited to situations where the formula is of this form.

proves the Lubell – Yamamoto – Meshalkin inequality by a double counting argument in which he counts the permutations of U in two different ways.

However proving such a fact can be difficult with a standard counting argument.

The main idea of the proof of the theorem is to use a counting argument, to show that any failure of local injectivity ( twin patterns ) leads to an orphan pattern, and vice versa.

Conversely, an argument involving König's infinity lemma shows that any non-surjective rule must have an orphan, and ( letting be the size of a bounding box of this orphan ) a very similar counting argument shows that the number of patterns that fit within an square and do not contain an orphan is too small to provide a distinct successor to every starting pattern within an square, from which it follows that some two of the possible starting patterns are twins.

In cellular automata defined over tessellations of the hyperbolic plane, or of higher dimensional hyperbolic spaces, the counting argument in the proof of the Garden of Eden theorem does not work, because it depends implicitly on the property of Euclidean spaces that the boundary of a region grows less quickly than its volume as a function of the radius.

Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing.

: A counting argument shows that

The above relation can be derived very easily through a counting argument as follows:

This follows from the fact that for each composite n, 3 / 4s of all possible a values are witnesses ; a simple counting argument similar to the one in the proof that BPP in P / poly below shows that there exists a suitable list of a values for every input size, although finding it may be expensive.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true ; for instance, he shows that the equation x < sup > 4 </ sup > = 4x − 3, although incomplete, has four solutions ( counting multiplicities ): 1 ( twice ), − 1 + i √< span style =" text-decoration: overline "> 2 </ span >, and − 1 − i √< span style =" text-decoration: overline "> 2 </ span >.

The famous etching by William Hogarth shows Lovat awaiting execution in The Tower, counting with his fingers the various Clans that he had brought to his cause and battle to support the Stuart claim to the throne.

In fact, for any such solution, r < sup > ℓ </ sup > Y ( θ, φ ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic ( see below ), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.

The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points.

Some research shows that dolphins among other animals understand concepts such as numerical continuity ( but not necessarily counting ).

To show the calculation of a state's delegation, the following example shows the size of the Texas delegation ( based on the current political makeup and not counting additional delegates due to the intervening Census ; Texas is a GOP stronghold ):

In TV's early years, there was a saturation of boxing programs, as many as six prime-time network shows in one week, not even counting the myriad local shows.

Kasem would spend noon ( 11: 30 a. m. in 1990 ) until 12: 30 a. m. on New Year's Eve Day counting down the 25 " most classic " episodes of the TV shows currently airing on Nick at Nite determined by viewers at home, revealing the # 1 episode at midnight.

Lygo also holds the record for the highest individual show score — not counting special shows — on 100 %.

In a language with noun classifiers, a noun may or may not be accompanied by a noun classifier, which shows a conceptual classification of the referent of a noun ( not the noun itself ) and is commonly used when counting.

In 1952 and 1953 the races counting towards the World Championship were to Formula 2 rules so drivers of these cars could take part in those events as the table below shows.

In 2008 the band did a 1980s shows counting comedy tour called " Manie Manie ".

* The appearance of an awkward Anne Bancroft, who admitted to her dislike of chat shows, and appeared before the audience counting up to ten in order to calm her nerves.

* The character Adrian Monk of the TV series Monk shows arithmomania as part of his obsessive compulsive disorder, including counting poles along a footpath, and counting the number of phobias he has.

The eyeteeth ( third from the middle on top, counting each front tooth as the first ) beginning to protrude like fangs.

At the end of the session, each voter tossed one of these into a large clay jar which was afterwards cracked open for the counting of the ballots.

Josephus suggested a method of collective suicide: they drew lots and killed each other, one by one, counting to every third person.

Not only has Secretariat's record time stood for 39 years and counting, but in the race itself, he did something unique in Triple Crown races: each successive quarter, his times were faster.

In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.

Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set.

Base 10 appears to come from counting one's fingers, base 20 from the fingers and toes, base 8 from counting the spaces between the fingers ( attested in California ), and base 12 from counting the knuckles ( 3 each for the four fingers ).

Another method of counting ratings ballots is to find the median score of each candidate, and elect the candidate with the highest median score.

In runoff voting, the counting of votes in each round is simple and occurs in the same way as under the plurality system.

As a garbage collection algorithm, reference counting tracks, for each object, a count of the number of references to it held by other objects.

By the nature of reference counting, each of these garbage components must contain at least one cycle.

In weighted reference counting, we assign each reference a weight, and each object tracks not the number of references referring to it, but the total weight of the references referring to it.

* Because garbage-collection is only done on built-in types, reference counting can be efficiently integrated into the library routines used to manipulate each datatype, keeping the overhead needed for updating of reference counts low.

Naive counting leads to 19, 683 possible board layouts ( 3 < sup > 9 </ sup > since each of the nine spaces can be X, O or blank ), and 362, 880 ( i. e. 9!

Another example of a unary counting system clustered in counts of five is the Chinese, Japanese and Korean custom of writing the Chinese character, Korean Hanja character, or Japanese kanji character 正 which takes 5 strokes to write, one stroke each time something is added.

During each frame the counting rate of the detector is sampled twice at intervals separated by 37 seconds.

They take into account of a detailed counting of family structure, not only with two biological parents or stepparent families, but also with types of single-parent families with each other ( mother-only, father-only, never-married, and other ).

A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.

Cash and prizes are awarded immediately ( with the cash value of each prize counting toward the contestant's total, and a new prize replacing the old one in the same square ).

When counting up the material for each side, typical values for pieces are 1 point for a pawn, 3 points for a knight or bishop, 5 points for a rook, and 9 points for a queen.

To count the number of electrons at each kinetic energy value, with the minimum of error, XPS must be performed under ultra-high vacuum ( UHV ) conditions because electron counting detectors in XPS instruments are typically one meter away from the material irradiated with X-rays.