If $$b \neq 0 $$ then $$c \times \frac{a}{b} = \frac{c \times a}{b}$$.
But if this rule applies to $$\frac{0}{0} = 1$$ then it follows that:
$$10^9 \quad = \quad 10^9 \; \times \; \1 \quad = \quad 10^9 \; \times \; \frac{0}{0} \quad = \quad \frac{10^9 \; \times \; 0}{0} \quad = \quad \frac{0}{0} \quad = \quad 1$$.
And that doesn't make sense.
If $$\frac{0}{0} = 1$$ that doesn't change that $$f(a) = \lim_{x\to a} \frac{b x - b d}{c x - c d} = \frac{b}{c}$$ is a constant function of a and isn't equal to 1 unless $$b = c$$. So $$\frac{0}{0} = 1$$ is neither useful nor necessary to mathematics.
For the sake of further argument on this particular aspect, let's treat zero as a number, so that I can make a humble observation/suggestion which can end this "undefined" situation for the expression "0/0"....by the simple, straightforward and logically valid /preferable treatment of that "0/0" expression under the mathematical
Rules of Evaluation of Expressions; thusly...
For example, in the abovequoted exercise by you, rpenner, you could have left the "0/0" expression 'un-decomposed' (rather than decomposed to two separate multiplier entity and divisor entity) by the simple Occam's Razor expedient of putting the "0/0" in
Parentheses and treating it as a 'complete' entity all the way through.
This would avoid the trivial treatment/decomposition of that expression altogether, and hence left the "0/0=1" value intact AS "1".
I point out that any such 'decomposition' of similar
like/like 'expression' (eg, 9/9 or 10/10, or 0/0 etc) is INVALIDLY TRIVIAL when purporting to 'prove' any serious point via dependence on such a trivial and NON-occam's Razor treatment.
The only NON-trivial treatment of any "like/like" expression (which always effectively equal "1") is to
leave it alone (by putting it in parentheses right at the starting statement (and keeping it there all the way through).
My point is that your using the zeros separately as multiplier and as divisor is TRIVIAL and against the logical requirements that ANY 'like/like' expression is always to be treated AS 'unitary' all the way through, rather than 'trivially and arbitrarily decomposing' such expressions merely to suit your 'proof' treatment which is thus made trivial and illogical, especially obvious if and when the parentheses/rules are invoked to protect the integrity of that 'unitary' expression (irrespective of what nunbers are used to express that unitary 'like/like' entity).
Hence, IF zero is a number, and IF any number is used in an 'like/like' expression (such as 9/9, 10/10, 0/0 etc,), then I humbly suggest that Parentheses be always used in such cases (and especially for "0/0") so to avoid any more trivial arguments and any further "undefined" sidesteps to recognizing the actual nature/value of such an expression as "0/0".
Your further comments/opinions on this humble observation/suggestion would be greatly appreciated. Thanks.
