#### Comment Preferences

• ##### Easy....(4+ / 0-)

What is so hard in proving 6x6=36? It is pretty straight forward. Duh!

The multiplication operation in the domain of integers Z is written ×.

Let us define [[(a,b)]]⊠ as in the formal definition of integers.

That is, [[(a,b)]]⊠ is an equivalence class of ordered pairs of natural numbers under the congruence relation ⊠.
⊠ is the congruence relation defined on N×N by

(x1,y1)⊠(x2,y2)⟺x1+y2=x2+y1.

In order to streamline the notation, we will use [[a,b]] to mean [[(a,b)]]⊠, as suggested.

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of Z are the isomorphic images of the elements of equivalence classes of N×N where two tuples are equivalent if the Unique Minus between the two elements of each tuple is the same.

Thus multiplication can be formally defined on Z as the operation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the Unique Minus congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes.
It follows that:

∀a,b,c,d∈N:[[a,b]]×[[c,d]]`[[a×c+b×d,a×d+b×c]] or, more compactly, as [[ac+bd,ad+bc]].`

```This can also be defined as: n×m```+nm=m+m+⋯m

... and the validity of this is proved in Index Laws for Monoids.

It is possible to read the history of this country as one long struggle to extend the liberties established in our Constitution to everyone in America. - Molly Ivins

[ Parent ]

• ##### They only asked for 6x6, not the entire universe (0+ / 0-)

of integral arithmetic.  And it can be demonstrated by content, by addition, and by division (and therefore by subtraction). But thanks for using ∀ and ∈, as I haven't seen "for every" and "is an element of" since geometry class.  You left out "there exists" and "such that" though.  My geometry teacher made sure we learned all that notation by lots and lots of repetition of them in proofs.

• ##### I love Unicode(1+ / 0-)
Recommended by:
se portland

There are character pickers now on all operating systems that let you do this, and formula editors in all major word processing software, as well as in math software. I do this stuff in Libre Office, Character Map, and GeoGebra on Linux.

e^iπ+1 = 0

(P⊃Q)⊃((Q⊃R)⊃(P⊃R))

Unit classes: (x)(∃y)((x∈y)∧(∀z)((∼(z=x))⊃(∼(z∈y))))

You're welcome.

America—We built that!

[ Parent ]

who was harassing an English class one time with the notion that none of us could define 3. I was in the middle of Quine's Mathematical Logic at the time (one of my father's textbooks), and quoted a recursive definition in which 0 is the unit set of the empty set, and each integer is the set of sets such that removing one element leaves a member of the previous number. Take a set with three members, remove one, there are two left. QED.

Settled his hash.

Not the only time, either. He hated teaching high school, and took it out on the students with some frequency. Fortunately, he enjoyed having a student who was good enough to beat his challenges. I was glad to be able to protect the others from him as much as I could.

America—We built that!

[ Parent ]

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