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This question already has an answer here:

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

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marked as duplicate by user21820, muaddib, Sharkos, Moishe Kohan, kingW3 Jan 29 '18 at 20:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ "This question is related to a well-known mathematical debate ". The relevance, significance, and notoriety of this so called debate is vastly overstated by novices. There is no debate or issue. If you define things to be the same then they are the same no matter how we write them out. And we did define them to be the same. $\endgroup$ – fleablood Jan 29 '18 at 1:30
  • $\begingroup$ I wrote "debate" because so many people argued about which answer is correct! We are in front of two answers 1 and 9 each answer has its own point of view which has been disscussed in the question. I don't think you read the whole question! $\endgroup$ – Yousef Essam Jan 29 '18 at 1:34
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    $\begingroup$ There's no debate what order to do the operations of 6/2(1+2). The only debate is how the \$#@% are we supposed to TYPE the difference between $\frac 6{2(1+2)}$ and $\frac 62(1+2)$ when the gold-manged KEYBOARD does not allow you to vertically place things up or down on the screen. This has NOTHING to do with math. It ONLY has to do with typing. $\endgroup$ – fleablood Jan 29 '18 at 1:34
  • $\begingroup$ Is that satisfying : 6 ÷ 2(1 +2) and this is the original form of the problem! $\endgroup$ – Yousef Essam Jan 29 '18 at 1:38
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    $\begingroup$ $b(c+d)$ is a notation for trivariate polynomial with a degree of 2. So I am dividing a monomial $a$ with a polynomial. In the latter case, however, we are dealing with three individual expressions $a$, $b$ and $c+d$. This is how I am taught. Other people may disagree. After all this is about notation, not mathematical concept. $\endgroup$ – Weijun Zhou Jan 29 '18 at 2:09
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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.

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  • $\begingroup$ I would add this is a debate about TYPING. It is not a debate about math. There is no debate how to interpret $\frac 6{2(1+2)}$ and $\frac 62(1+2)$. There is only debate about how to TYPE them. $\endgroup$ – fleablood Jan 29 '18 at 1:37
  • $\begingroup$ These two forms have no problem but the problem is in this form which I think it's mathematically correct : 6 ÷ 2(1 +2) $\endgroup$ – Yousef Essam Jan 29 '18 at 1:44
  • $\begingroup$ @fleablood I think that's pretty much what I was trying to convey. $\endgroup$ – David Jan 29 '18 at 1:59
  • $\begingroup$ Yes, I was just spelling it out. $\endgroup$ – fleablood Jan 29 '18 at 3:06
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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 $\hbar = 1.0510^{-34}\,\mathrm{Js}$.

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