Figuring out how to make this look pretty took far too long. First crack at LaTeX.
might even be right this time
ā¦very last line should come out 182 though.
lmfaoā¦ have to say a,b,c are in the set of non-negative integers or integers >= 0 and <=9
apparently /mathbb N in proofs mean natural including 0. Iāll take it.
https://proofwiki.org/wiki/Symbols:N
I tried to post my latest contribution to the encyclopedia of the inane with a preamble of āI think I got this one rightā but I read it and it was was not. Here I present 2. You do not realize how stupid you are until you think about the number 2. Thatās not a deep statement. Really, if you spend any time at all thinking about the number Two you have a screw loose. I probably spent 2 hours on it. How many ways is that stupid; It took me a few hours to get right (I hope), . I spent a couple of hours on a proof that 2*x = x+x . The first time I thought I was ready to post it was still wrong. My table of contents is still broken and I donāt know what kind of grouping to use for examples. Havenāt quite sorted out the hierarchy of things in my head between sections subsections subsubsections theorems proofs and examples. Did 11 as well but it is still broken and I dont want to take a picture. I will probably finish 11 tomorrow and play with 5 or 9 and try to fix my table of contents. LaTeX reminds me a lot of emacs. Incredibly powerful but requiring a full ship of monkeys to operate.
ā¦ thinking about this some more I suspect this is a definition or a proposition.
I am stating something as a fact without proof here.
2 * b = b + b
proves nothing. Iām not sure that it is a tautology.
Is Multiplication defined by Addition? In one sense 2b = b + b would certainly make you think so but 123987410 * 1123423546 makes you think that maybe it isnāt because I certainly do not add to get my second answer. There is not enough coffee in the entire world for this level of inane.
Nice! You have the feel of the thing.
Comments:
Give the pattern a name. Right now itās Fountain.4.1
2nd line after āGivenā should be labeled Theorem/Prove/Show - this is the statement you are going to prove.
Thereās a bit of confusion in your notation.
a|b*a|c = a*(a+b+c)+b*c
on the lhs (left hand side) a & b are as stipulated single digits but on the rhs a is a multiple of 10 as you show when you work the examples. Stick with the | notation or go full algebra and write in 10a on the rhs.
b<=9 allows for numbers like -12345 perhaps |b|<9 ?
Thereās no reason to restrict d & c to <10 or in any way. The proof doesnāt require it at all and the pattern works eve for negative numbers or large numbers. You used the positional notation
a|b to mark off ones and tens, so they can have any value without a problem
2734 = (3|-3) * 3|4 = 30(30-3+4) -12 = 918
138 = 1|3* 1|-2 = 10(10+3-2)-6 = 104
If you have repeating digits in any configuration but diagonal thereās a generalization.
ax * ay , xa * xy, xxay the the cross product is always going to be x(a+y)
Personally I donāt care for the expansion in this form a*(a+b+c)+b*c. Itās perfectly legitimate but it muddles together different powers of 10 which is a main source of error for me. This shows, that even in the matter of mathematical proofs thereās a lot of room for taste & style. Your proofs should answer your questions.
Prove 2x=x+x
You canāt. Itās not proveable. Itās an axiom. It is or very nearly one of the defining Axioms of the construction of the Integers. Itās just above the bare metal that holds the numbers together. You might as well try to prove from first principles that a chess bishop moves diagonally. You canāt itās a basic rule. Numbers are just like chess pieces. They are defined entirely by set of arbitrary Axioms or rules and the rest is working out the implications.
1+1=2 by definition!
2 is the name of the number when you +1 starting from one.
a*(b+1) = ab + a
Axiom 2 from the Construction of the Natural Numbers, Unit basedlet a = x & b = 1
x(1+1)=x*1+x
and given that 1+1=2 we can write:
x*2 = x + x QED
Thatās only slightly different than the statement of the Axiom 2. You are up against the foundation.
Nevertheless, I am most impressed that you asked that question. And itās the right kind of question to ask - itās revealing - you found the bedrock. Most people have given no thought to what a number actually is.
Found out by accident that if you right click on a math expression in Wikipedia, it offers the option of copying Lex expression
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Springer Verlag has some great templates and style guides both for articles and manuscripts.
Perfect for keeping notebooks tidy I suspect. Instructions for Authors: Manuscript Guidelines | Springer ā International Publisher
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Iām thinking that Iām not making it hard enough for myself yetā¦ Build a linux vm to host LaTex. Install Emacs to run org-mode to integrate task management into LaTex. Create a notebook using the Springer Verlag Book Template to work on trivial algebraic proofs while playing with mentally calculating numbers. Seems too simple. I have decided that I should probably bring my stenography machine down into the dungeon and transcribe LaTeX, Math Symbols, and text using plover http://www.openstenoproject.org/ and this keyboard
to key text composed of sounded out syllables and phrases with the need to create custom dictionaries for both LaTeX and emacs command strings to the appropriate multi-finger chord on the stenotype.
I have not quite decided whether I should do this all with aid of Jamesonās Catholic Whiskey or the newly legalized cannabis now available in Canuckistan.
My emacs is dreadfully rusty and I havenāt practiced stenography in a year. Iām going upstairs for a glass.
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Groups, Rings, Fields
Properties of Numbers.
ā¦ debugging tex ā¦
Group
A collection of objects G, together with one operation, \oplus which has the following properties:
Associativity
\text{For an operation, } \oplus \text{ how objects are grouped does not affect the result. }\\
a \oplus ( b \oplus c ) = (a \oplus b ) \oplus c
Identity
\text{There is an element, } \textit{e} \in G \text{ s.t. } \textit{e} \oplus g = g \oplus \textit{e} \, \forall \, g \in G
Inverse
\forall \, g \in G \, \exists \, g^-1 \in G \text{ s.t. } g \oplus g^-1 = g^-1 \oplus g = \textit{e}