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<p>L.S.</p>
<p>Wie wel eens van <b>p-adic</b> getallen heeft gehoord weet dat
rekenen met deze getallen merkwaardige resultaten oplevert:</p>
<pre><code class="!whitespace-pre hljs language-lua"> ..<span
class="hljs-number">.6666667</span>
× <span class="hljs-number">3</span>
<span class="hljs-comment">-----------</span></code></pre>
<p></p>
<p> 1</p>
<p><br>
</p>
<p>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). <br>
</p>
<p>...66666666667 is dan ook de representatie van, jawel: 1/3.</p>
<p><code class="!whitespace-pre hljs language-lua"><br>
</code></p>
<p><code class="!whitespace-pre hljs language-lua">Nog een voorbeeld
dat resulteert in 1:</code></p>
<p><code class="!whitespace-pre hljs language-lua"><br>
</code></p>
<p><code class="!whitespace-pre hljs language-lua"> ..<span
class="hljs-number">.857142857143 (= 1/7)<br>
</span></code><code class="!whitespace-pre hljs language-lua">
× <span class="hljs-number">7<br>
</span></code><code class="!whitespace-pre hljs language-lua"><span
class="hljs-comment"> -----------<br>
</span></code> 1</p>
<p></p>
<p><br>
</p>
<p>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.</p>
<p>We zitten nu met een leesprobleem:<br>
</p>
<p> 66666667</p>
<p>is uiteraard niet gelijk aan<br>
</p>
<p>...66666667</p>
<p>en</p>
<p>...66666667</p>
<p>divergeert ook niet naar oneindigheid</p>
<p>Het leesprobleem is breder: <b>De decimale notatie is ongeldig</b>
vanwege de decimale expansie.<br>
</p>
<p>' 11' stelt in de natuurlijk getallen in base-10 het getal elf
voor.</p>
<p>in p-adic (base 10) stelt</p>
<p>...11 = 1/9</p>
<p>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.<br>
</p>
<blockquote>
<p>...999999.0 = m <br>
...999990.0 = 10 m</p>
<p>9 = -9 m<br>
m = -1 <br>
</p>
</blockquote>
<p>Derhalve is ...999999.0 = -1 !</p>
<p>Het notatieprobleem omvat dus niet de grootte maar ook of het
positieve of negatieve getallen zijn. <br>
</p>
<p>Nu kan men het argument aankomen dat men geen natuurlijk getal
kan optellen bij een p-adic getal. Echter dit kan wel:</p>
<blockquote>
<p><font color="#a51d2d">ChatGPT:<br>
</font></p>
<div class="flex flex-grow flex-col max-w-full">
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data-message-id="c9107405-f7cf-4a6b-878c-fcb3f03a0a67"
class="min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&]:mt-5 overflow-x-auto">
<div
class="markdown prose w-full break-words dark:prose-invert dark">
<p><font color="#a51d2d">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.</font></p>
<p><font color="#a51d2d">Let's denote a natural number <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">n</span></span></span></span></span>
and a p-adic number <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo><msub><mi>a</mi><mi>k</mi></msub><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>…</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>0</mn></msub></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6389em; vertical-align: -0.2083em;"></span><span
class="minner">…</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3361em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight"
style="margin-right: 0.03148em;">k</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3361em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight"
style="margin-right: 0.03148em;">k</span><span class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.2083em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right: 0.1667em;"></span><span
class="minner">…</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3011em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3011em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
(in base-p notation). To add the natural number to the
p-adic number, you simply add the corresponding digits
position-wise:</font></p>
<p><font color="#a51d2d"><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mo>…</mo><msub><mi>a</mi><mi>k</mi></msub><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>…</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>0</mn></msub></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">n</span><span
class="mspace" style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6389em; vertical-align: -0.2083em;"></span><span
class="minner">…</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3361em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight"
style="margin-right: 0.03148em;">k</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3361em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight"
style="margin-right: 0.03148em;">k</span><span class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.2083em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right: 0.1667em;"></span><span
class="minner">…</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3011em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span><span
class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.3011em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span></span></font></p>
<p><font color="#a51d2d">For example, if you want to add the
natural number 123 to the 10-adic number <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo><mn>987654321</mn></mrow></semantics></math></span></span></span>,
you would add digit by digit:</font></p>
<p><font color="#a51d2d"><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>123</mn><mo>+</mo><mo>…</mo><mn>987654321</mn><mo>=</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="minner">…</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord">987654444</span></span></span></span></span></font></p>
<p><font color="#a51d2d">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.</font></p>
</div>
</div>
</div>
</blockquote>
<div class="mt-1 flex justify-start gap-3 empty:hidden">
<div
class="text-gray-400 flex self-end lg:self-center justify-center lg:justify-start mt-0 -ml-1 visible"><font
color="#a51d2d"><span class="" data-state="closed"></span></font></div>
</div>
<div class="flex"><font color="#a51d2d"><span class=""
data-state="closed"><br>
</span></font></div>
<blockquote>
<div class="flex-col gap-1 md:gap-3">
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<div class=""><font color="#a51d2d">RO: </font></div>
<div class=""><font color="#a51d2d">are there paradoxes with
addition and multiplication given properties like
associativity, commutativity, and distributivity.</font></div>
</div>
</div>
</div>
<div class="px-4 py-2 justify-center text-base md:gap-6 m-auto">
<div
class="flex flex-1 text-base mx-auto gap-3 md:px-5 lg:px-1 xl:px-5 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem] group final-completion">
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<div>
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class="gizmo-shadow-stroke flex h-6 w-6 items-center justify-center overflow-hidden rounded-full">
<div
style="background-color: rgb(25, 195, 125); width: 24px; height: 24px;"
class="relative p-1 rounded-sm h-9 w-9 text-white flex items-center justify-center"><font
color="#a51d2d"><svg width="41" height="41"
viewBox="0 0 41 41" fill="none"
xmlns="http://www.w3.org/2000/svg"
class="icon-sm"><text x="-9999" y="-9999">ChatGPT</text></svg></font></div>
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<div
class="markdown prose w-full break-words dark:prose-invert dark">
<p><font color="#a51d2d">ChatGPT: </font></p>
<p><font color="#a51d2d">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.</font></p>
<p><font color="#a51d2d">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:</font></p>
<p><font color="#a51d2d"><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi
mathvariant="normal">∣</mi><mi>x</mi><mo>+</mo><mi>y</mi><msub><mi
mathvariant="normal">∣</mi><mi>p</mi></msub><mo>≤</mo><mi>max</mi><mo></mo><mo
stretchy="false">(</mo><mi
mathvariant="normal">∣</mi><mi>x</mi><msub><mi
mathvariant="normal">∣</mi><mi>p</mi></msub><mo
separator="true">,</mo><mi
mathvariant="normal">∣</mi><mi>y</mi><msub><mi
mathvariant="normal">∣</mi><mi>p</mi></msub><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord">∣</span><span
class="mord mathnormal">x</span><span
class="mspace" style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1.0361em; vertical-align: -0.2861em;"></span><span
class="mord mathnormal"
style="margin-right: 0.03588em;">y</span><span
class="mord"><span class="mord">∣</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.1514em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">p</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.2861em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right: 0.2778em;"></span><span
class="mrel">≤</span><span class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 1.0361em; vertical-align: -0.2861em;"></span><span
class="mop">max</span><span class="mopen">(</span><span
class="mord">∣</span><span
class="mord mathnormal">x</span><span
class="mord"><span class="mord">∣</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.1514em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">p</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.2861em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace"
style="margin-right: 0.1667em;"></span><span
class="mord">∣</span><span
class="mord mathnormal"
style="margin-right: 0.03588em;">y</span><span
class="mord"><span class="mord">∣</span><span
class="msupsub"><span
class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist"
style="height: 0.1514em;"><span
style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span
class="pstrut"
style="height: 2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">p</span></span></span></span><span
class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height: 0.2861em;"><span></span></span></span></span></span></span><span
class="mclose">)</span></span></span></span></span></font></p>
<p><font color="#a51d2d">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.</font></p>
<p><font color="#a51d2d">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.</font></p>
<p><font color="#a51d2d">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.</font></p>
</div>
</div>
</div>
</blockquote>
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<div class=""><font color="#a51d2d">RO: </font></div>
<div class=""><font color="#a51d2d">what are the problems of
the decimal notation and algebraic properties such as
associativity, commutativity, and distributivity givem
some axioms</font></div>
</div>
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<div
class="markdown prose w-full break-words dark:prose-invert dark">
<p><font color="#a51d2d">ChatGPT: </font></p>
<p><font color="#a51d2d">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:</font></p>
<ol>
<li>
<p><font color="#a51d2d"><strong>Repeating Decimals:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Real Numbers vs.
Computable Numbers:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Finite Precision:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Associativity,
Commutativity, and Distributivity:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Field Axioms:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Division by Zero:</strong></font></p>
<ul>
<li><font color="#a51d2d">Division by zero is
undefined in many number systems, including the
real numbers. It can lead to inconsistencies and
undefined results in algebraic expressions.</font></li>
</ul>
</li>
<li>
<p><font color="#a51d2d"><strong>Imaginary and Complex
Numbers:</strong></font></p>
<ul>
<li><font color="#a51d2d">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.</font></li>
</ul>
</li>
</ol>
<p><font color="#a51d2d">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.</font></p>
</div>
</div>
</div>
</blockquote>
<blockquote>
<div class="flex flex-grow flex-col max-w-full">
<div data-message-author-role="assistant"
data-message-id="08cad0eb-cc03-40b8-ac4d-78b9ea14adf0"
class="min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&]:mt-5 overflow-x-auto">
<div
class="markdown prose w-full break-words dark:prose-invert dark">
<p>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.</p>
<ol>
<li>
<p><strong>Associativity:</strong></p>
<ul>
<li>
<p><strong>Definition:</strong> An operation is
associative if the grouping of elements does not
affect the result of the operation.</p>
</li>
<li>
<p><strong>In symbols:</strong> For an operation <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord">∗</span></span></span></span></span>,
it is associative if <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo
stretchy="false">(</mo><mi>a</mi><mo>∗</mo><mi>b</mi><mo
stretchy="false">)</mo><mo>∗</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>∗</mo><mo
stretchy="false">(</mo><mi>b</mi><mo>∗</mo><mi>c</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">b</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">c</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span></span></span></span></span>
holds for all elements <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>,
<span class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>,
and <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">c</span></span></span></span></span>
in the set.</p>
</li>
<li>
<p><strong>Example:</strong> Addition and
multiplication of real numbers are associative:</p>
<ul>
<li><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo
stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo
stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>+</mo><mo
stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen"><br>
(</span><span class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">b</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">c</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span></span></span></span></span>
for addition.</li>
<li><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo
stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo
stretchy="false">)</mo><mo>⋅</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mo
stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen"><br>
(</span><span class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">b</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">c</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span></span></span></span></span>
for multiplication.</li>
</ul>
</li>
</ul>
</li>
<li>
<p><strong>Commutativity:</strong></p>
<ul>
<li>
<p><strong>Definition:</strong> An operation is
commutative (or abelian) if the order of elements
does not affect the result of the operation.</p>
</li>
<li>
<p><strong>In symbols:</strong> For an operation <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord">∗</span></span></span></span></span>,
it is commutative if <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>∗</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord mathnormal"></span><span
class="mord mathnormal"><br>
a</span><span class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>
holds for all elements <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>
and <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>
in the set.</p>
</li>
<li>
<p><strong>Example:</strong> Addition is commutative
for real numbers (<span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal"><br>
a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>),
<br>
but multiplication is not (<span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi><mo
mathvariant="normal">≠</mo><mi>b</mi><mo>⋅</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord mathnormal"><br>
a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.8889em; vertical-align: -0.1944em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel"><span class="mrel"><span
class="mord vbox"><span
class="thinbox"><span class="rlap"><span
class="strut"
style="height: 0.8889em; vertical-align: -0.1944em;"></span><span
class="inner"><span
class="mord"><span
class="mrel"></span></span></span><span
class="fix"></span></span></span></span></span><span
class="mrel">=</span></span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>
in general).</p>
</li>
</ul>
</li>
<li>
<p><strong>Distributivity:</strong></p>
<ul>
<li>
<p><strong>Definition:</strong> Distributivity
relates two operations (usually addition and
multiplication) and describes how one operation
interacts with the other.</p>
</li>
<li>
<p><strong>In symbols:</strong> For operations <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord">∗</span></span></span></span></span>
and <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∘</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord">∘</span></span></span></span></span>,
distributivity implies that <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><br>
<br>
<br>
a</mi><mo>∗</mo><mo stretchy="false">(</mo><mi>b</mi><mo>∘</mo><mi>c</mi><mo
stretchy="false">)</mo><mo>=</mo><mo
stretchy="false">(</mo><mi>a</mi><mo>∗</mo><mi>b</mi><mo
stretchy="false">)</mo><mo>∘</mo><mo
stretchy="false">(</mo><mi>a</mi><mo>∗</mo><mi>c</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4653em;"></span><span
class="mord mathnormal"><br>
a</span><span class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∘</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">b</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∘</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">∗</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span></span></span></span></span>.</p>
</li>
<li>
<p><strong>Example:</strong> Distributivity holds
for multiplication over addition in real numbers:
<span class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>⋅</mo><mo
stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo
stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord mathnormal"><br>
a</span><span class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span
class="mord mathnormal">c</span><span
class="mclose">)</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4445em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">⋅</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">c</span></span></span></span></span>.</p>
</li>
</ul>
</li>
</ol>
<p>These properties play a crucial role in algebraic
structures. For instance:</p>
<ul>
<li>A group is a set equipped with an associative
operation and an identity element.</li>
<li>A ring is a set equipped with two operations (usually
addition and multiplication) that satisfy associativity,
distributivity, and the existence of an additive
identity.</li>
<li>A field is a ring in which multiplication is also
commutative, and every nonzero element has a
multiplicative inverse.</li>
</ul>
<p>Understanding these properties is essential for working
with algebraic structures and ensuring the consistency of
mathematical operations.</p>
<p><br>
</p>
<div class="flex flex-grow flex-col max-w-full">
<div data-message-author-role="assistant"
data-message-id="a5bf21ba-6101-425d-800c-5631aa0b3947"
class="min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&]:mt-5 overflow-x-auto">
<div
class="markdown prose w-full break-words dark:prose-invert dark">
<p>Subtraction is not commutative, meaning that the
order of the numbers matters. In general, <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>
is not equal to <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>.</p>
<p>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 <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>,
it is equivalent to adding the additive inverse of <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>
to <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>.</p>
<p>Mathematically, <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>
can be defined as <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mo
stretchy="false">(</mo><mo>−</mo><mi>b</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord">−</span><span
class="mord mathnormal">b</span><span
class="mclose">)</span></span></span></span></span>,
where <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord">−</span><span
class="mord mathnormal">b</span></span></span></span></span>
is the additive inverse of <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>.
So, when you reverse the order and have <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span></span></span></span></span>,
it is equivalent to <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>+</mo><mo
stretchy="false">(</mo><mo>−</mo><mi>a</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord">−</span><span
class="mord mathnormal">a</span><span
class="mclose">)</span></span></span></span></span>.</p>
<p>In general, <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mo
stretchy="false">(</mo><mo>−</mo><mi>b</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord">−</span><span
class="mord mathnormal">b</span><span
class="mclose">)</span></span></span></span></span>
is not the same as <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>+</mo><mo
stretchy="false">(</mo><mo>−</mo><mi>a</mi><mo
stretchy="false">)</mo></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord mathnormal">b</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">+</span><span class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span
class="mord">−</span><span
class="mord mathnormal">a</span><span
class="mclose">)</span></span></span></span></span>.
The additive inverses <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7778em; vertical-align: -0.0833em;"></span><span
class="mord">−</span><span
class="mord mathnormal">b</span></span></span></span></span>
and <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>a</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.6667em; vertical-align: -0.0833em;"></span><span
class="mord">−</span><span
class="mord mathnormal">a</span></span></span></span></span>
might not be the same, unless <span
class="math math-inline"><span class="katex"><span
class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.4306em;"></span><span
class="mord mathnormal">a</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6944em;"></span><span
class="mord mathnormal">b</span></span></span></span></span>.
Therefore, subtraction is not commutative except
when the two numbers are equal.</p>
<p>For example:</p>
<ul>
<li><span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>−</mo><mn>2</mn></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7278em; vertical-align: -0.0833em;"></span><span
class="mord">3</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="mord">2</span></span></span></span></span>
is not equal to <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>3</mn></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7278em; vertical-align: -0.0833em;"></span><span
class="mord">2</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="mord">3</span></span></span></span></span>,
as <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mo>=</mo><mn>1</mn></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7278em; vertical-align: -0.0833em;"></span><span
class="mord">3</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="mord">2</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="mord">1</span></span></span></span></span>
and <span class="math math-inline"><span
class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></span><span
class="katex-html" aria-hidden="true"><span
class="base"><span class="strut"
style="height: 0.7278em; vertical-align: -0.0833em;"></span><span
class="mord">2</span><span
class="mspace"
style="margin-right: 0.2222em;"></span><span
class="mbin">−</span><span
class="mspace"
style="margin-right: 0.2222em;"></span></span><span
class="base"><span class="strut"
style="height: 0.6444em;"></span><span
class="mord">3</span><span
class="mspace"
style="margin-right: 0.2778em;"></span><span
class="mrel">=</span><span
class="mspace"
style="margin-right: 0.2778em;"></span></span><span
class="base"><span class="strut"
style="height: 0.7278em; vertical-align: -0.0833em;"></span><span
class="mord">−</span><span class="mord">1</span></span></span></span></span>.</li>
</ul>
<p>It's important to note that addition is
commutative, but subtraction is not.</p>
</div>
</div>
</div>
<div class="mt-1 flex justify-start gap-3 empty:hidden">
<div
class="text-gray-400 flex self-end lg:self-center justify-center lg:justify-start mt-0 -ml-1 visible"><span
class="" data-state="closed"></span></div>
</div>
<div class="flex"><span class="" data-state="closed"><br>
</span></div>
<div class="flex"><span class="" data-state="closed"><br>
</span></div>
<p></p>
</div>
</div>
</div>
</blockquote>
<p></p>
<p><font color="#a51d2d">R.O.</font></p>
<p><font color="#a51d2d">--</font></p>
<p>ChatGPT:</p>
<p><br>
</p>
<blockquote>
<p>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.</p>
<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:</p>
<ul>
<li>If x ≠ 0, then |x|ₚ = p⁻ᵏ, where k is the highest power of p
that divides x.</li>
<li>If x = 0, then |0|ₚ = 0.</li>
</ul>
<p>Let's look at some examples of p-adic numbers and their
arithmetic, using the 2-adic numbers (p = 2) for simplicity:</p>
<ol>
<li>
<p><strong>Example of 2-adic number:</strong>
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.</p>
</li>
<li>
<p><strong>Addition in 2-adic numbers:</strong>
If you want to add two 2-adic numbers, you can perform the
addition digit by digit. For example:</p>
<pre><div class="dark bg-black rounded-md"><div
class="flex items-center relative text-token-text-secondary bg-token-surface-primary px-4 py-2 text-xs font-sans justify-between rounded-t-md"><span>scss</span><span
class="" data-state="closed"></span></div></div></pre>
</li>
</ol>
<ol>
<li value="2">
<pre><div class="dark bg-black rounded-md"><div
class="p-4 overflow-y-auto"><code
class="!whitespace-pre hljs language-scss"><span
class="hljs-number">1</span> + <span class="hljs-number">2</span> + <span
class="hljs-number">4</span> + <span class="hljs-number">8</span> + <span
class="hljs-number">16</span> + ... (<span
class="hljs-number">2</span>-adic representation of -<span
class="hljs-number">1</span>)
</code></div></div></pre>
</li>
</ol>
<ul>
<li>1 + 2 + 4 + 8 + 16 + ... (2-adic representation of -1)</li>
</ul>
<hr>
<p><br>
</p>
</blockquote>
<a class="moz-txt-link-freetext" href="https://en.wikipedia.org/wiki/P-adic_number">https://en.wikipedia.org/wiki/P-adic_number</a><br>
<p></p>
</body>
</html>