# And now for something truly trivial

Figuring out how to make this look pretty took far too long. First crack at LaTeX.
might even be right this time

…very last line should come out 182 though.

lmfao… 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. I’ll take it.
https://proofwiki.org/wiki/Symbols:N

… 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. I’m 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 isn’t 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.

Give the pattern a name. Right now it’s Fountain.4.1

2nd line after ‘Given’ should be labeled Theorem/Prove/Show - this is the statement you are going to prove.

There’s 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 ?

There’s no reason to restrict d & c to <10 or in any way. The proof doesn’t 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
13
8 = 1|3* 1|-2 = 10(10+3-2)-6 = 104

If you have repeating digits in any configuration but diagonal there’s a generalization.

ax * ay , xa * xy, xxay the the cross product is always going to be x(a+y)

Personally I don’t care for the expansion in this form a*(a+b+c)+b*c. It’s 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 there’s a lot of room for taste & style. Your proofs should answer your questions.

Prove 2x=x+x

You can’t. It’s not proveable. It’s an axiom. It is or very nearly one of the defining Axioms of the construction of the Integers. It’s 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 can’t it’s 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 based

natural numbers Axiom C

let a = x & b = 1

x(1+1)=x*1+x

and given that 1+1=2 we can write:

x*2 = x + x QED

That’s 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 it’s the right kind of question to ask - it’s 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. https://www.springer.com/gp/authors-editors/book-authors-editors/resources-guidelines/book-manuscript-guidelines

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I’m thinking that I’m not making it hard enough for myself yet… 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 Jameson’s Catholic Whiskey or the newly legalized cannabis now available in Canuckistan.

My emacs is dreadfully rusty and I haven’t practiced stenography in a year. I’m going upstairs for a glass.

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Groups, Rings, Fields
Properties of Numbers.
… debugging tex …

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}