Everybody uses juxtaposition for multiplication (or dot when some separation is required) and horizontal line for division.

* and / are poor man's substitutes when all you have is ASCII.

× is used for vector products, Cartesian products and products of algebraic structures in general.

The only place I've ever seen × or ÷ being used in arithmetics was primary school. No idea why they use them.

> The only place I've ever seen × or ÷ being used in arithmetics was primary school. No idea why they use them.

Perhaps because the other prominent use is on the buttons of every single calculator in the world.

I recall using × and ÷ in primary school, before the advent of portable calculators.

I know that's mere anecdotal evidence, but I believe there do exist reliable documentary sources from that era.

Clearly, the idea to put those symbols on calculator buttons didn't come from nowhere. I think our collective point is that there exists a separate tradition for using × and ÷.

÷ is the symbol for division since it symbolises a fraction, with the two dots taking the place of the numbers. I'd say this is why, as some people have noted, : can also be used for division in some contexts: it represents a ratio, which is again just a fraction.

The way I was taught it, though, is that ratios are rather different from fractions. You might say 2:1 of tea and honey, but ⅔ tea and ⅓ honey. However, when describing only one quantity, some people will say 1:2 of honey while others might say 1:3 of honey.

In primary school in the Netherlands (and maybe elsewhere?), : is used for division.

It's actually widely used for division, but only in some contexts. (It signifies a ratio, eg. "3:2 odds".)

Right, I've seen this one too.

What then happens when you have to split a long line of math?

What then happens when you need to multiply 2 by 2?

The answer to the first question is that the following line is begun with a multiplication sign.

The answer to the second question is that bold roman type is used for vectors and often for other forms of tensor like a matrix to overload the semantics of dot and cross to not be scalar products were either of their arguments in italic.

The / is actually the - from the other symbol and the .'s are placeholders for the numbers to the left and the right of the / so those are in fact pretty close.