[D66] Moeilijk! (On p-adic numbers and the invalidation of the decimal notation system of numbers)
René Oudeweg
roudeweg at gmail.com
Fri Feb 2 09:39:59 CET 2024
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…akak−1…a1a0(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+…akak−1…a1a0
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:
1.
*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.
2.
*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.
3.
*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.
4.
*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.
5.
*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.
6.
*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.
7.
*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.
1.
*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:
o (a+b)+c=a+(b+c)
(a+b)+c=a+(b+c) for addition.
o (a⋅b)⋅c=a⋅(b⋅c)
(a⋅b)⋅c=a⋅(b⋅c) for multiplication.
2.
*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).
3.
*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:
1.
*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.
2.
*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
2.
|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
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