Calculate 8 ÷ 2a where a = 4. Then,

Calculate 8 ÷ 2 × a where a = 4.

See how in the first form a is implied to be part of the fraction where in the second it isn’t?

It's not implied, it's explicitly because of the definition of Terms.

P.S. now substitute a=2 and you'll see why it matters.

A dot • could be between 2 and a and it would still follow the first example

No, it wouldn't. Inserting a dot (multiplication) makes it the same as your second example. i.e. 3 Terms, not 2 Terms.

In vector multiplication, dot and cross products produce different results.

This isn't vector multiplication. This is Year 7 algebra.

[–] Rentlar@lemmy.ca 1 points 7 months ago* (last edited 7 months ago) (1 children)

We agree that the two situations are separate. You can call them terms or whatever, since we're multiplying and dividing the idea of terms seems irrelevant here to me but I suppose for pedantry's sake we can entertain it. It doesn't matter what a is, but the first result is 4 ÷ a the second result is 4a.

I use the dot as an expression of the same term rather than separate. This is matter of my notational convention, due to dot multiplication providing a single scalar result where cross multiplication resulting in a vector product. You can write 10 paragraphs telling me that my convention is wrong and why but I don't really care that much, to save you some time.

ETA: When I handwrite out non-Vector equations, you will rarely see me write a • or a × sign inside of a fraction anyway, it's placement is generally clear enough for me, and I will put brackets where it's confusing.

[–] SmartmanApps@programming.dev 1 points 7 months ago

We agree that the two situations are separate

Ok, that's a start.

but the first result is 4 ÷ a the second result is 4a

Exactly! So when a=2 then 4÷a=2, and 4a=8, which isn't the same thing. Welcome to why 2a and 2xa (and therefore also 2.a) aren't the same thing.

I use the dot as an expression of the same term rather than separate.

But that is incorrect. A dot is used for multiplication. i.e. it separates terms. If you use a . for 2.a, then you are writing the same thing as 2xa, not the same thing as 2a.

This is matter of my notational convention

Well, that's fine enough if you keep it to yourself, but don't use it in anything anyone else is going to read, or you're going to run into the issues I just pointed out