Wedderburn pointed out the rustic frolicsomeness of finite division rings, ascertaining that every finite division is afield.

keywords:

[Mathematics]

[Abstract Algebra]

[punctuation matters]

# Day: July 14, 2020

# OBSERVATION (nibbling)

As is well-known, an empire is weakest at its boundary. (cf: W.C. Fields’ published rant against ‘nibblers’ – nibbling probably really being a case of the cat mewing and dog having its day)

# OBSERVATION (munitions of war)

Munitions of war can serve as a metaphor for the pervasive possibility of transcendence.

# joke proof that 0! = 1

(that zero factorial equals unity)

By placing the space between the 0 and the ! on the left-hand side, we get 0 != 1, but ! before a relational symbol means the negation of the relation, and it is certainly true that 0 is not equal to 1. QED

keywords:

[Mathematics]

[Number Theory]

# OBSERVATION (battle)

The silver lining of battle is its lack of sordidness.

(cf: an anti-corruption operation conducted from a dingy warehouse (as in one of the episodes of ‘The Untouchables’) – one easily forgets the sordidness of the environment)

# concept: antonyms having the same acronym

(‘antonymous acronyms’)

for example, LOS is the acronym of both ‘lack of sharpness’ and ‘lack of sordidness’

keywords:

[Linguistics]

[Language Arts]

[wordplay]

# an explanation, by Austin Lowder, of why zero factorial equals unity

(that is, why 0! = 1)

His explanation is given in a comment by him for the article by Merriam-Webster on ‘factorial’.

Here is his explanation:

Note that 5! = 6!/6, because 6!/6 = 1 x 2 x 3 x 4 x 5 x 6 / 6 = 1 x 2 x 3 x 4 x 5 = 5!

Continuing with this process:

4! = 5!/5 = 24

3! = 4!/4 = 6

2! = 3!/3 = 2

1! = 2!/2 = 1

0! = 1!/1 = 1

That’s his explanation.

Here is the link to the article in Merriam-Webster where his comment appears.

For factorials, the fact that 0! = 1 is merely a definition, a definition made for convenience (such as making reference to binomial coefficients). What Austin Lowder provides, therefore, cannot be a PROOF, but it IS a brilliant EXPLANATION, that is a brilliant HEURISTIC, that shows why we would want to define it so. However, by ‘going to the next level’, namely, by considering the gamma function (shifted by one unit to the left), for which factorials can be defined in a way consistent with the elementary manner, it is then a THEOREM that 0! = 1.

Here is the link to the Wikipedia article on the gamma function.