[D66] Moeilijk! (correctie: plain text)

[René Oudeweg]

L.S.

Wie wel eens van p-adic getallen heeft gehoord weet dat rekenen met deze 
getallen merkwaardige resultaten oplevert:

      ...6666667
     ×         3
     -----------
               1


De ellipsis aan het begin geeft aan dat ...6666667 gelezen moet worden 
als een oneindige expansie van zessen. De meesten van ons zijn wel 
bekend met de ellipsis achter de komma: bijv 0.9999999... = 1. P-adic 
heeft een ellipsis voor de komma. Nu zou je zeggen dat '...66667.0' 
groter moet zijn dan 1 als je het getal vermenigvuldigt met 3. Nee, het 
resultaat van deze vermenigvuldiging is precies 1 (!).  Frappant niet? 
Het probleem is dat ...66666667 ook gelezen zou kunnen worden als een 
geheel getal terwijl het in feite een extensie is van de rationele 
getallen (Q).

...66666666667 is dan ook de representatie van, jawel: 1/3.


Nog een voorbeeld dat resulteert in 1:


...857142857143   (= 1/7)
×             7
     -----------
               1


Dit wordt ons niet op de basisschool of voortgezet onderwijs geleerd. 
Waarom niet? Economie gebruikt de de natuurlijke getallen N en de 
rationele getallen Q. Meestal rond men af. 12..9 euro maakt men 13 euro 
van. Met p-adic kan men geen economische handelingen berekenen (bijv. 
het tellen van vis) of de dimensies van commodities ontwerpen en 
berekenen. Alleen getaltheoretici gebruiken p-adic.

We zitten nu met een leesprobleem:

    66666667

is uiteraard niet gelijk aan

...66666667

en

...66666667

divergeert ook niet naar oneindigheid

Het leesprobleem is breder: De decimale notatie is ongeldig vanwege de 
decimale expansie.

' 11'  stelt in de natuurlijk getallen in base-10 het getal elf voor.

in p-adic (base 10) stelt

...11  = 1/9

De gehele getallen gerepresenteerd door getallen voor de komma kunnen 
onmogelijk ook een kleiner rationeel getal zijn (een breuk dus)! Kortom, 
het is een notatieprobleem van getallen omdat we nu eenmaal werken met 
een bepaalde base number (10 meestal) en geen oneindig aantal symbolen 
en glyphs voor elk natuurlijk getal hebben. Dit ongewoon notatieprobleem 
en hoe de grootte van het getal te interpreteren resulteert in de 
invalidatie van de decimale notatie.

     ...999999.0 = m
     ...999990.0 = 10 m

     9 = -9 m
     m = -1

Derhalve is ...999999.0 = -1 !

Het notatieprobleem omvat dus niet de grootte maar ook of het positieve 
of negatieve getallen zijn.

Nu kan men het argument aankomen dat men geen natuurlijk getal kan 
optellen bij een p-adic getal. Echter dit kan wel:

     ChatGPT:

     Yes, one can add a natural number to a p-adic number. In general, 
the p-adic numbers form a field, meaning that addition and 
multiplication are well-defined operations, and they satisfy the usual 
properties like associativity, commutativity, and distributivity.

     Let's denote a natural number nn and a p-adic number 
…akak−1…a1a0…ak​ak−1​…a1​a0​ (in base-p notation). To add the natural 
number to the p-adic number, you simply add the corresponding digits 
position-wise:

     n+…akak−1…a1a0n+…ak​ak−1​…a1​a0​

     For example, if you want to add the natural number 123 to the 
10-adic number …987654321, you would add digit by digit:

     123+…987654321=…987654444

     In general, the addition of a natural number to a p-adic number 
does not pose any issues, and the result is a well-defined p-adic number.


     RO:
     are there paradoxes with addition and multiplication given 
properties like associativity, commutativity, and distributivity.
     ChatGPT

     ChatGPT:

     While p-adic numbers have some unusual properties compared to real 
numbers, and their arithmetic is different, there are no inherent 
paradoxes with addition and multiplication within the context of the 
p-adic number system itself. The p-adic numbers form a field, and they 
obey the usual algebraic properties like associativity, commutativity, 
and distributivity.

     However, the p-adic numbers can be counterintuitive when compared 
to the real numbers. One notable difference is that the p-adic norm (a 
measure of size) behaves differently. In particular, the p-adic norm 
satisfies the ultrametric inequality:

     ∣x+y∣p≤max⁡(∣x∣p,∣y∣p)∣x+y∣p​≤max(∣x∣p​,∣y∣p​)

     This means that in the p-adic numbers, the distance between two 
numbers is determined more by the larger of the two numbers rather than 
the sum of their absolute values.

     These differences can lead to interesting and sometimes unexpected 
results. For example, some sequences of rational numbers that converge 
in the real numbers may diverge in the p-adic numbers, and vice versa.

     While these differences might seem paradoxical at first, they arise 
from the unique structure of the p-adic numbers and do not violate the 
fundamental properties of the p-adic arithmetic. They are simply a 
reflection of the fact that the p-adic numbers are a different 
completion of the rational numbers compared to the real numbers.


     RO:
     what are the problems of the decimal notation and algebraic 
properties such as associativity, commutativity, and distributivity 
givem some axioms

     ChatGPT:

     The decimal notation and the algebraic properties you 
mentioned—associativity, commutativity, and distributivity—are 
foundational concepts in number systems. However, the decimal notation 
and these algebraic properties do not come without their own challenges 
or limitations. Here are some considerations:

         Repeating Decimals:
             The representation of certain fractions in decimal notation 
results in repeating or non-terminating decimals. While these 
representations are accurate, they can make arithmetic operations less 
straightforward.

         Real Numbers vs. Computable Numbers:
             Not all real numbers can be represented precisely in 
decimal notation, particularly irrational numbers like π or the square 
root of 2. This limitation can lead to rounding errors in computations.

         Finite Precision:
             In practical applications, computers have finite precision. 
Representing real numbers with infinite precision in a finite amount of 
memory can lead to rounding and approximation errors.

         Associativity, Commutativity, and Distributivity:
             While these properties hold for most familiar number 
systems, there are mathematical structures where they may not. For 
example, non-commutative algebraic structures like matrices or 
non-associative structures can violate these properties.

         Field Axioms:
             The properties of associativity, commutativity, and 
distributivity are part of the field axioms that define the structure of 
fields. However, not all algebraic systems satisfy these axioms. For 
instance, the integers under subtraction do not form a field because 
subtraction is not commutative.

         Division by Zero:
             Division by zero is undefined in many number systems, 
including the real numbers. It can lead to inconsistencies and undefined 
results in algebraic expressions.

         Imaginary and Complex Numbers:
             The extension of real numbers to complex numbers involves 
the introduction of the imaginary unit 'i' and the consideration of 
complex conjugates. This departure from real numbers might seem 
unintuitive at first.

     In summary, while the decimal notation and standard algebraic 
properties are foundational and widely applicable, they do have 
limitations and may not fully capture the complexities of all 
mathematical structures. Mathematicians explore various number systems 
and algebraic structures to address specific needs and to study 
mathematical phenomena more deeply.

     Associativity, commutativity, and distributivity are fundamental 
properties that govern operations in algebraic structures, such as sets 
equipped with binary operations. These properties are crucial in 
defining and understanding mathematical structures, including groups, 
rings, and fields.

         Associativity:

             Definition: An operation is associative if the grouping of 
elements does not affect the result of the operation.

             In symbols: For an operation ∗∗, it is associative if 
(a∗b)∗c=a∗(b∗c)(a∗b)∗c=a∗(b∗c) holds for all elements aa, bb, and cc in 
the set.

             Example: Addition and multiplication of real numbers are 
associative:
                 (a+b)+c=a+(b+c)
                  (a+b)+c=a+(b+c) for addition.
                 (a⋅b)⋅c=a⋅(b⋅c)
                 (a⋅b)⋅c=a⋅(b⋅c) for multiplication.

         Commutativity:

             Definition: An operation is commutative (or abelian) if the 
order of elements does not affect the result of the operation.

             In symbols: For an operation ∗∗, it is commutative if a∗b=b∗a
             a∗b=b∗a holds for all elements aa and bb in the set.

             Example: Addition is commutative for real numbers (a+b=b+a
 
                                         a+b=b+a),
             but multiplication is not (a⋅b≠b⋅a
                                                            a⋅b=b⋅a in 
general).

         Distributivity:

             Definition: Distributivity relates two operations (usually 
addition and multiplication) and describes how one operation interacts 
with the other.

             In symbols: For operations ∗∗ and ∘∘, distributivity 
implies that


             a∗(b∘c)=(a∗b)∘(a∗c)
             a∗(b∘c)=(a∗b)∘(a∗c).

             Example: Distributivity holds for multiplication over 
addition in real numbers: a⋅(b+c)=a⋅b+a⋅c
             a⋅(b+c)=a⋅b+a⋅c.

     These properties play a crucial role in algebraic structures. For 
instance:

         A group is a set equipped with an associative operation and an 
identity element.
         A ring is a set equipped with two operations (usually addition 
and multiplication) that satisfy associativity, distributivity, and the 
existence of an additive identity.
         A field is a ring in which multiplication is also commutative, 
and every nonzero element has a multiplicative inverse.

     Understanding these properties is essential for working with 
algebraic structures and ensuring the consistency of mathematical 
operations.


     Subtraction is not commutative, meaning that the order of the 
numbers matters. In general, a−ba−b is not equal to b−ab−a.

     The reason for this lack of commutativity in subtraction lies in 
the definition of subtraction itself. Subtraction is essentially the 
inverse operation of addition. If you consider the expression a−ba−b, it 
is equivalent to adding the additive inverse of bb to aa.

     Mathematically, a−ba−b can be defined as a+(−b)a+(−b), where −b−b 
is the additive inverse of bb. So, when you reverse the order and have 
b−ab−a, it is equivalent to b+(−a)b+(−a).

     In general, a+(−b)a+(−b) is not the same as b+(−a)b+(−a). The 
additive inverses −b−b and −a−a might not be the same, unless a=ba=b. 
Therefore, subtraction is not commutative except when the two numbers 
are equal.

     For example:

         3−23−2 is not equal to 2−32−3, as 3−2=13−2=1 and 2−3=−12−3=−1.

     It's important to note that addition is commutative, but 
subtraction is not.


R.O.

--

ChatGPT:


     P-adic numbers are a type of number system that extends the 
rational numbers. They are often denoted by ℚₚ, where p is a prime 
number. The p-adic numbers were introduced by Kurt Hensel in 1897. The 
key idea is to define a metric that measures the size of a number in 
relation to its divisibility by powers of p.

     The p-adic numbers are equipped with a p-adic valuation, which 
measures the highest power of p that divides a given rational number. 
The p-adic norm, denoted by |x|ₚ, is defined as follows:

         If x ≠ 0, then |x|ₚ = p⁻ᵏ, where k is the highest power of p 
that divides x.
         If x = 0, then |0|ₚ = 0.

     Let's look at some examples of p-adic numbers and their arithmetic, 
using the 2-adic numbers (p = 2) for simplicity:

         Example of 2-adic number: Let's consider the sequence of 
rational numbers {1, 2, 4, 8, 16, ...}. In the 2-adic numbers, these 
converge to a single 2-adic number. For instance, the limit of this 
sequence is represented by the 2-adic number 1 + 2 + 4 + 8 + 16 + ... = -1.

         Addition in 2-adic numbers: If you want to add two 2-adic 
numbers, you can perform the addition digit by digit. For example:

         scss

         1 + 2 + 4 + 8 + 16 + ...    (2-adic representation of -1)

         1 + 2 + 4 + 8 + 16 + ... (2-adic representation of -1)


https://en.wikipedia.org/wiki/P-adic_number