P-adic number and First Baptist Church in the City of New York: Difference between pages

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'''The First Baptist Church in the City of New York''' is a [[Christian]] [[congregation]] based in a sanctuary built in [[1891]] at the intersection of [[Broadway (New York City)|Broadway]] and 79th Street in the [[Upper West Side]] of [[Manhattan, New York|Manhattan Island]]. FBC is a [[conservative]], independent, [[evangelistic]], mission-minded, [[Bible]]-believing church in fellowship with the [[General Association of Regular Baptist Churches]], based in [[Schaumburg, Illinois]].

{{wrongtitle|title=p-adic number}} ''(With a lower-case and preferably italicized p.)''

The '''''p''-adic number systems''' were first described by [[Kurt Hensel]] in 1897. For each [[prime number|prime]] ''p'', the ''p''-adic number system extends the ordinary arithmetic of the [[rational numbers]] in a way different from the extension of the rational number system to the [[real number|real]] and [[complex number|complex]] number systems. This is achieved by an alternative interpretation of the concept of [[absolute value]]. The ''p''-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of [[power series]] methods into [[number theory]], but their influence now extends far beyond this. For example, the field of [[P-adic analysis|''p''-adic analysis]] essentially provides an alternative form of [[calculus]].

The church holds 90-minute [[worship]] services at 11 a.m. [[Sunday]]s and midweek activities on [[Wednesday]] and [[Thursday]] [[evening]]s.

More precisely, for a given prime ''p'', the [[field (mathematics)|field]] '''Q'''_''p'' of ''p''-adic numbers is an extension of the [[rational number|rational numbers]]. If all of the fields '''Q'''_''p'' are collectively considered, we arrive at [[Helmut Hasse]]'s [[local-global principle]], which roughly states that certain equations can be solved over the rational numbers [[if and only if]] they can be solved over the [[real number|real numbers]] ''and'' over the ''p''-adic numbers for every prime ''p''. The field '''Q'''_''p'' is also given a [[topological space|topology]] derived from a [[metric space|metric]], which is itself derived from an alternative [[valuation]] on the rational numbers. This metric is [[completeness (topology)|complete]] in the sense that every [[Cauchy sequence]] converges. This is what allows the development of calculus on '''Q'''_''p'', and it is the interaction of this analytic and algebraic structure which gives the ''p''-adic number systems their power and utility.

==Revolutionary years==

== Motivation ==

The first attempt to establish a Baptist church in New York City took place in [[1711]], when William Wightman began ministering. A church was built on Gold Street but disbanded eight years later because of financial [[recession]].

If ''p'' is a fixed prime number, then any [[integer]] can be written as a ''p-adic expansion'' (writing the number in "base ''p''") in the form

:<math>\pm\sum_{i=0}^n a_i p^i</math>

where the a_i are integers in {0,...,''p'' &minus; 1}. This is expressed by saying that the integer has been "written in base ''p''". For example, the 2-adic or [[binary_numeral_system|binary]] expansion of 35 is 1&middot;2⁵ + 0&middot;2⁴ + 0&middot;2³ + 0&middot;2² + 1&middot;2¹ + 1&middot;2⁰, often written in the shorthand notation 100011₂.

Scattered Baptists organized in [[1745]] under the [[businessman]] Jeremiah Dodge and the pastor Benjamin Miller of [[Scotch Plains, New Jersey]].

The familiar approach to generalizing this description to the larger ___domain of the rationals (and, ultimately, to the reals) is to include sums of the form:

:<math>\pm\sum_{i=-\infty}^n a_i p^i</math>

A definite meaning is given to these sums based on [[Cauchy sequence]]s, using the [[absolute value]] as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...₅. In this formulation, the integers are precisely those numbers which can be represented in the form where ''a''_''i'' = 0 for all ''i'' < 0.

A church edifice was built in [[1760]] in a building, again located on Gold Street. [[John Gano]] ([[1727]]-[[1804]]), a native of [[New Jersey]], became the first full-time pastor of the congregation of twenty-seven, which by [[1762]] had grown to three hundred members and took the name "First Baptist Church in the City of New York".

As an alternative, if we extend the ''p''-adic expansions by allowing infinite sums of the form

:<math>\sum_{i=k}^{\infty} a_i p^i</math>

where ''k'' is some (not necessarily positive) integer, we obtain the [[field (mathematics)|field]] '''Q'''_''p'' of '''''p''-adic numbers'''. Those ''p''-adic numbers for which ''a''_''i'' = 0 for all ''i'' < 0 are also called the '''''p''-adic integers'''. The ''p''-adic integers form a [[subring]] of '''Q'''_''p'', denoted '''Z'''_''p''. (Note: '''Z'''_''p'' is often used to represent the set of integers modulo ''p''. If each set is needed, the latter is usually written '''Z'''/''p'''''Z''' or '''Z'''/''p''. Be sure to check the notation for any text you read.)

The church supported the [[American Revolution]] even though New York City was occupied by [[British]] forces from the [[summer]] of [[1776]] until the duration of the war. Elder Gano joined the army and was a [[chaplain]] to [[General]] [[George Washington]]. With the signing of the [[Treaty of Paris]] in [[1783]], a celebration took place in [[Newburg, New York]]. Washington called upon Gano to offer the prayer of [[thanksgiving]]. The [[Anglican]] ([[Episcopalian]]) Washington also requested that Gano [[baptize]] him because after Washington's study of the scriptures, he concluded that baptism by immersion should follow the conversion of a believer.

Intuitively, as opposed to ''p''-adic expansions which extend to the ''right'' as sums of ever smaller, increasingly negative powers of the base ''p'' (as is done for the real numbers as described above), these are numbers whose ''p''-adic expansion to the ''left'' are allowed to go on forever. For example, the ''p''-adic expansion of 1/3 in base 5 is ...1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132, ... Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a ''p''-adic integer in base 5.

On his return to New York City after the revolution, Gano found thirty-seven members who restored the church building and grew thereafter to two hundred. When the Congress offered former revolutionary soldiers land on the [[frontier]], Gano departed from New York to [[Kentucky]]. There he started several Baptist churches. He was also a founder of the Baptist-affiliated [[Brown University]] in [[Providence, Rhode Island]].

The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful - this requires the introduction of the [[p-adic metric|''p''-adic metric]]. Two different but equivalent solutions to this problem are presented below.

==Early 19th century==

In [[1802]], FBC built a new stone structure while still based on Gold Street. Under the leadership of Dr. Spencer Cone, the congregation relocated in [[1842]] to the intersection of Elizabeth and Broome streets in a gothic structure still in use by another church today. This was also the headquarters of the Baptist Home and Foreign Mission Board. When the congregation outgrew the facility, it moved to the intersection of 39th Street and [[Park Avenue]]. Under the leadership of Dr. Thomas Anderson, a gothic brown stone sanctuary was constructed as well as a separate Bible school building.

[ilan]: There is no need to limit the base to a prime number. In fact, base 10 will do just fine, and 10-adic numbers are just ordinary integers, except with possibly infinitely many digits to the left, otherwise, all rules of addition and multiplication as usual. A good exercise is to understand why the 10-adic number ...111 = -1/9. A good application of this result is the non Archimedean Zeno paradox: http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/zeno.html

==Symbolism of the sanctuary at Broadway and 79th Street==

== Constructions ==

Isaac Massey Haldeman was the pastor who thus far has served the longest tenure at FBC -- from [[1884]] to [[1933]]. Six years after his arrival, FBC relocated to the present facility, which is adjacent to a [[subway]] station.

The FBC building was designed by George M. Kaiser, who also planned the [[Apollo Theater]]. A balcony was added in [[1903]]. This created a seating capacity of one thousand. Two unequal towers over the corner entrance to the main auditorium are examples of biblical symbolism. The taller tower represents [[Jesus Christ]] as the Head of the Church. The lower tower, which appears incomplete, was designed to represent the church, while will remain unfulfilled until the return of Christ. Two shorter towers represent the [[Old Testament]] and the [[New Testament]].

=== Analytic approach ===

A large rose window facing Broadway pictures Christ as the center of the New Testament chuch. He is in the large inner circle. The star depicts Him as the Bright and Morning Star. The crown shows Him as King of Kings. The frame of sun portrays Him as the Sun of Righteousness. The five upper circles depict the writers of the New Testament [[Epistles]], while the botton four circles represent [[Matthew]], [[Mark]], [[Luke]], and [[John]], the [[Gospel]] authors.

The [[real number]]s can be defined as [[equivalence class]]es of [[Cauchy sequence]]s of [[rational number]]s; this allows us to, for example, write 1 as 1.000... = 0.999... . However, the definition of a Cauchy sequence relies on the [[metric space|metric]] chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the [[Euclidean metric]].

==FBC faith and practice==

For a given prime ''p'', we define the ''p-adic metric'' in '''Q''' as follows:

The five front steps of the sanctuary conform to the teaching that [[salvation]] is by the grace of God alone through Jesus. FBC teaches the "faith once delivered to the saints":

for any non-zero rational number ''x'', there is a unique integer ''n'' allowing us to write ''x'' = ''p''^''n''(''a''/''b''), where neither of the integers ''a'' and ''b'' is [[divisor|divisible]] by ''p''. Unless the numerator or denominator of ''x'' contains a factor of ''p'', ''n'' will be 0. Now define |''x''|_''p'' = ''p''^&minus;''n''. We also define |0|_''p'' = 0.

(1) [[Sovereignty]] of the [[Trinity]] ([[God]] as Father, Son, and [[Holy Spirit]])

For example with ''x'' = 63/550 = 2^&minus;1 3² 5^&minus;2 7 11^&minus;1<br>

:<math>|x|_2=2</math>

:<math>|x|_3=1/9</math>

:<math>|x|_5=25</math>

:<math>|x|_7=1/7</math>

:<math>|x|_{11}=11</math>

:<math>|x|_{\mbox{any other prime}}=1</math>

(2) Inspiration and inerrancy of scripture

This definition of |''x''|_''p'' has the effect that high powers of ''p'' become "small".

(3) The Virgin Birth of Christ

It can be proved that each [[Norm_(mathematics)|norm]] on '''Q''' is equivalent either to the Euclidean norm or to one of the ''p''-adic norms for some prime ''p''. The ''p''-adic norm defines a metric d_''p'' on '''Q''' by setting

:<math>d_p(x,y)=|x-y|_p</math>

The field '''Q'''_''p'' of ''p''-adic numbers can then be defined as the [[completion (topology)|completion]] of the metric space ('''Q''',d_''p''); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains '''Q'''.

(4) The sinlessness of Jesus

It can be shown that in '''Q'''_''p'', every element ''x'' may be written in a unique way as

:<math>\sum_{i=k}^{\infty} a_i p^i</math>

where ''k'' is some integer and each ''a''_''i'' is in {0,...,''p'' &minus; 1}. This series [[limit (mathematics)|converges]] to ''x'' with respect to the metric d_''p''.

(5) Christ's vicarious atonement at [[Calvary]]

=== Algebraic approach ===

(6) Bodily [[resurrection]] and [[ascension]] of Christ

In the algebraic approach, we first define the ring of ''p''-adic integers, and then construct the field of quotients of this ring to get the field of ''p''-adic numbers.

We(7) startThe with thepre-tribulation [[inverse limitrapture]] of the rings church

'''Z'''/''pⁿ'''''Z''' (see [[modular arithmetic]]): a ''p''-adic integer is then a sequence

(''a_n'')_''n''&ge;1 such that ''a_n'' is in

'''Z'''/''pⁿ'''''Z''', and if ''n'' < ''m'',

''a_n'' = ''a_m'' (mod ''pⁿ'').

(8) The pre-millennial return and reign of Christ.

Every natural number ''m'' defines such a sequence (''m'' mod ''pⁿ''), and can therefore be regarded as a ''p''-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence {1, 3, 3, 3, 3, 35, 35, 35, ...}.

In the church's Gano Chapel are [[painting]]s of Gano baptizing Washington and of Gano praying in thanksgiving for the British surrender. They are copies of the originals located at [[William Jewell College]] in [[Liberty, Missouri]]. The college collection includes Gano's sword, which was a gift from General Washington, who had received it from the French General [[Marquis de Lafayette]].

Note that pointwise addition and multiplication of such sequences is well defined, since addition and multiplication commute with the ''mod'' operator, see [[modular arithmetic]]. Also, every sequence (''a_n'') where the first element is not 0 has an inverse: since in that case, for every ''n'', ''a_n'' and ''p'' are [[coprime]], and so ''a_n'' and ''pⁿ'' are relatively prime. Therefore, each ''a_n'' has an inverse mod ''pⁿ'', and the sequence of these inverses, (''b_n''), is the sought inverse of (''a_n'').

==List of FBC pastors since 1884==

Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3² + 1*3³ + 0*3⁴ + ... The [[partial sum]]s of this latter series are the elements of the given series.

FBC has had eighteen pastors, two of which were chaplains of the [[United States Congress]]. Two were also college presidents. Two founded colleges. Collectively, the pastors have written more than three hundred books, mostly on the Christian religion. Many have been leaders of the Baptist denomination.

The ring of ''p''-adic integers has no zero divisors, so we can take the [[quotient field]] to get the field '''Q'''_''p'' of ''p''-adic numbers. Note that in this quotient field, every number can be uniquely written as ''p^&minus;nu'' with a [[natural number]] ''n'' and a ''p''-adic integer ''u''.

[[Robert C. Gage]] ([[2000]] - [[September 9]], [[2007]]). The current assistant pastor is Matthew A. Carpenter.

== Properties ==

Richard Daniel Burke ([[1976]]-[[1998]])

The set of ''p''-adic integers is [[uncountable]].

William Fliedner, Jr. ([[1972]]-[[1975]])

The ''p''-adic numbers contain the rational numbers '''Q''' and form a field of [[characteristic (algebra)|characteristic]] 0. This field cannot be turned into an [[ordered field]].

Carl E. Elgena ([[May 21, 1917]] - [[December 26]], [[2006]]) served as pastor from [[1965]]-[[1968]]). His last residence was in [[Bear, Delaware|Bear]] in [[New Castle County, Delaware|NewCastle County]] in [[Delaware]].

The [[topology]] of the set of ''p''-adic integers is that of a [[Cantor set]]; the topology of the set of ''p''-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of ''p''-adic integers is [[compact]] while the space of ''p''-adic numbers is not; it is only [[locally compact]].

As [[metric space]]s, both the ''p''-adic integers and the ''p''-adic numbers are [[completeness (topology)|complete]].

Peter Hoogendam ([[January 5]], [[1910]] - [[December 12]], [[2001]]) served as pastor from [[1956]]-1965). His last residence was [[Chino, California|Chino]] in [[San Bernardino County, California|San Bernardino County]] in [[California]].

The real numbers have only a single proper [[algebraic extension]], the [[complex number]]s; in other words, this quadratic extension is already [[algebraically closed field|algebraically closed]]. By contrast, the [[algebraic closure]] of the ''p''-adic numbers has infinite degree. Furthermore, '''Q'''_''p'' has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, the algebraic closure of '''Q'''_''p'' is not (metrically) complete. Its (metric) completion is called '''&Omega;'''_''p''. Here an end is reached, as '''&Omega;'''_''p'' is algebraically closed.

Arthur Whiting ([[1950]]-[[1955]])

The field '''&Omega;'''_''p'' is isomorphic to the field '''C''' of complex numbers, so we may regard '''&Omega;'''_''p'' as the complex numbers endowed with an exotic metric. It should be noted that the existence of such a field isomorphism relies on the [[axiom of choice]], and no explicit isomorphism can be given.

William L. Pettingill ([[1948]]-[[1950]]

The ''p''-adic numbers contain the ''n''th [[cyclotomic field]] if and only if ''n'' divides ''p'' &minus; 1. For instance, the ''n''th cyclotomic field is a subfield of '''Q'''₁₃ iff ''n'' = 1, 2, 3, 4, 6, or 12.

Arthur Williams ([[1941]]-[[1947]])

The number ''[[e (mathematical constant)|e]]'', defined as the sum of reciprocals of [[factorial]]s, is not a member of any ''p''-adic field; but ''e^p'' is a ''p''-adic number for all ''p'' except 2, for which one must take at least the fourth power. Thus ''e'' is a member of the algebraic closure of ''p''-adic numbers for all ''p''.

William H. Rogers ([[1934]]-[[1940]]

Over the reals, the only functions whose [[derivative]] is zero are the constant functions. This is not true over '''Q'''_''p''. For instance, the function

I.M. Haldeman (1884-1933)

:''f'': '''Q'''_''p'' &rarr; '''Q'''_''p'', ''f''(''x'') = (1/|''x''|_''p'')² for ''x'' &ne; 0, ''f''(0) = 0,

There were interim periods with guest ministers when FBC was without a pastor.

has zero derivative everywhere but is not even [[locally constant function|locally constant]] at 0.

==References==

Given any elements ''r''_&infin;, ''r''₂, ''r''₃, ''r''₅, ''r''₇, ... where ''r''_''p'' is in '''Q'''_''p'' (and '''Q'''_&infin; stands for '''R'''), it is possible to find a sequence (''x''_''n'') in '''Q''' such that for all ''p'' (including &infin;), the limit of ''x''_''n'' in '''Q'''_''p'' is ''r''_''p''.

"The History and Architectural Symbolism of The First Baptist Church in the City of New York, 79th at Broadway", First Baptist Church brochure

== Generalizations and related concepts ==

http://www.firstbaptist-nyc.org/

The reals and the ''p''-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [[algebraic number field]]s, in an analogous way. This will be described now.

[[Category:Baptist churches in the United States|First Baptist Church in the City of New York]]

Suppose ''D'' is a [[Dedekind ___domain]] and ''E'' is its [[quotient field]]. The non-zero [[prime ideal]]s of ''D'' are also called ''finite places'' or ''finite primes'' of ''E''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a fractional ideal and can be uniquely factored as a product of positive and negative powers of finite primes of ''E''. If ''P'' is such a finite prime, we write ord_''P''(''x'') for the exponent of ''P'' in this factorization, and define

[[Category:General Association of Regular Baptist Churches|First Baptist Church in the City of New York]]

:<math>|x|_P = (NP)^{-\operatorname{ord}_P(x)}</math>

[[Category:1762 establishments|First Baptist Church in the City of New York]]

where ''NP'' denotes the (finite) cardinality of ''D''/''P''. Completing with respect to this norm |.|_''P'' then yields a field ''E''_''P'', the proper generalization of the field of ''p''-adic numbers to this setting.

[[Category:Churches in New York City|First Baptist Church in the City of New York]]

[[Category:Places of worship in New York City|First Baptist Church in the City of New York]]

Often, one needs to simultaneously keep track of all the above mentioned completions, which are seen as encoding "local" information. This is accomplished by [[adele ring]]s and [[idele group]]s.

==See also==

* [[Mahler's theorem]]

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[[Category:Field theory]]

[[Category:Number theory]]

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