Difference between Omitting the multiplication sign and keeping it

Question

What is the difference between 2*a and 2a? What is the difference between 2(3+4) and 2*(3+4)?

We all know that omitting the multiplication sign still means multiplication so nothing changed!

This question is related to a well-known mathematical debate as follows: 6/2(1+2) If we use the rule of order of operation which states that : Parentheses first, exponents next, multiplication and division from left to right and finally addition and subtraction from left to right The result will be 9 However, some would say that 6/2(1+2) is like 6/(2(1+2))

Having mentioned that they would say what is the result of 9a^2/3a?

I know this question might be "duplicate" or whatever... But I really searched a lot and didn't find a satisfying answer.

Is 3a the same as 3*a or (3*a)?

So 9a^2/3a = (9*a^2) / (3*a) Or 9*a^2/3*a

Answer 1

However, some would say that 6 / 2(1 + 2) is like 6 / (2(1 + 2))

Indeed they would, and this mistake is encouraged by the way YOU have written it, with the extra spaces around the division symbol. So the really important lesson is:

do not write things in a way which encourages misunderstanding. Any misinterpretation which arises from this sort of thing is your fault, not the reader's fault. Include "unnecessary" brackets if it helps to make your meaning clear, for example, (6/2)(1+2) or 6/(2(1+2))

As so often happens, this issue is expressed far better than I could do it in xkcd.

Answer 2

Generally the multiplication sign is dropped for convenience. I suspect part of the reason for it in physics (which is where a lot of mathematical notation got its start) is that you can multiply anything together, but other operations have requirements on units. So you end up doing a lot more multiplying.

The main exception to this rule is when multiplying explicit numbers or units. I think it's obvious why $2\times 3 = 6$ is preferred to $23 = 6$ and $\hbar = 1.05 \times10^{-34}\,\mathrm{J\cdot s}$ is preferred to $\hbar = 1.0510^{-34}\,\mathrm{Js}$.