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Consider the expression $1\div (mn)(n^2)$. It should be understood that, in simplifying the above expression, one should get $$1\div (mn)(n^2) = 1 \div (mn) \times (n^2)=\frac{n^2}{mn}=\frac{n}{m}.$$ Note that we carry out the operation from left to right as multiplication and division are 'of the same priority'.

Now, we consider the expression $1\div mn$. My friend and I are discussing about the interpretation of this expression.

  1. My friend claims that $mn$ is an algebraic expression and therefore we should regard this as '$1$ is divided by the expression $mn$', i.e. $\dfrac{1}{mn}$.
  2. However, I make use of the same logic in the above example: Note that $m$ and $n$ are multiplied together, we should therefore carry out the operation from left to right, i.e. deal with '$1\div m$' first, then '$\times n$'.

The two calculations shall end up with different conclusions. Which one is the correct interpretation? Please advise.

Nighty
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  • If you consider multiplication and division to have the same priority, then your friend is wrong. Concatenation notation for multiplication does suggest that $mn$ has higher priority over division, so maybe your friend would prefer the notation $1 \div m \times n$? (The more general issue is that division is not an associative operation, which is why no one really writes like this.) – Alex Provost Dec 07 '17 at 13:57
  • @AlexProvost : Seems that you are implying multiplication and division may not have the same priority? Can you further elaborate on that? – Nighty Dec 07 '17 at 13:59
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    Re your friend, $a+b$ would also be considered an "algebraic expression", but that doesn't make $a+b\cdot c=(a+b)\cdot c$ ... Nevertheless, the essence of this lesson is: Avoid ambiguous notation. – Hagen von Eitzen Dec 07 '17 at 14:00
  • @LeeNL You can define whatever you want, but conventionally division $a \div b$ really means $a \times b^{-1}$, so it really is multiplication in disguise. – Alex Provost Dec 07 '17 at 14:02
  • @AlexProvost : Now I am confused. If the concatenation notation suggests a higher priority, then $1 \div (mn)$ should be the only interpretation? – Nighty Dec 07 '17 at 14:08
  • Not an answer. This is a tempest in an unnecessary teapot. In a real problem the person writing the expression will know what he or she means and write it with parentheses to enforce that meaning. Computer programmers should do the same, not rely on the language rules for order of operations. Also: the division sign is rare in mathematics after elementary school. Division is much better written with fractions. – Ethan Bolker Dec 07 '17 at 14:16
  • Possible duplicate of What is 48÷2(9+3)? – kingW3 Dec 07 '17 at 14:16
  • @LeeNL What "should be the only interpretation" is not mine to judge. But here are three things we can take out from this: a) Formally, division is a particular instance of multiplication, so it has the same priority (try it out on a calculator). b) We are not machines, and visually the concatenation notation suggests high priority, so I think most people would view $1/mn$ as $1/(mn)$. c) This is why no one really writes like this. You should write so that what you mean is clear from context. – Alex Provost Dec 07 '17 at 14:18

4 Answers4

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This is really a matter of convention, which varies over time and from place to place, but in modern mathematics precedence is generally as follows (from highest to lowest, and with those of the same precedence put on the same line) where "$∙$" denotes the positions of the subexpressions:

  • Brackets:   $(∙)$

  • Function application:   $∙(∙)$

  • Exponentiation:   $∙^∙$

  • Juxtaposition:   $∙∙$

  • Fractions:   $\dfrac∙∙$

  • Product/Summation:   $\prod_∙^∙ ∙$   $\sum_∙^∙ ∙$

  • Multiplicative operations:   $∙ \times ∙$   $∙ \div ∙$

  • Negation:   $-∙$.

  • Additive operations:   $∙+∙$   $∙-∙$

I can easily justify that juxtaposition is given higher precedence in modern mathematics. Consider that "$\prod_{k=1}^n a_k b_k \times \sum_{m=1}^n c_m d_m$" is interpreted as "$( \prod_{k=1}^n ( a_k b_k ) ) \times ( \sum_{m=1}^n ( c_m d_m ) )$" and not as "$\prod_{k=1}^n ( a_k b_k \times \sum_{m=1}^n c_m d_m )$"!

user21820
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  • Is there a source or something for this order of operations? – Benji Altman Dec 07 '17 at 14:52
  • @user21820 In that case, my first example is also wrong since we use juxtaposition for the multiplication of $(mn)$ and $(n^2)$? – Nighty Dec 07 '17 at 14:57
  • @BenjiAltman: There is no official source for such generally followed conventions. However, you can take a look at existing writings by professional mathematicians to get an idea of what they follow. Of course, there will always be differing conventions, so it's best to put brackets if there is possibility of misunderstanding. – user21820 Dec 07 '17 at 15:00
  • @LeeNL: Well I would automatically assume that $1 \div mn = 1/(mn)$. In some old books, you will very often find them using "$ab/cd$" to mean "$(ab)/(cd)$" too. I won't say it's right or wrong; just try to choose your notation to avoid confusing people. – user21820 Dec 07 '17 at 15:01
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Wring $x \div y \times z$ should be avoided as it is just a bad idea.

It doesn't matter how many people will say it has one correct interpretation, the number of people who will be confused or misinterpret that is just too big.

If you mean $(x \div y) \times z$, then just write $x \times z \div y$. The "one correct interpretation" would be this one, but it's so silly to write $x \div y \times z$ instead of $x \times z \div y$ that many people will think that's not what you mean.

If you mean $x \div (y \times z)$, then just write $\frac{x}{yz}$. If you don't want something so vertical, write $x/(yz)$ or $x(yz)^{-1}$. And if you think that's too ugly, well, you can write $x/yz$, it's less ugly although a little ambiguous.

Writing $x/yz$ for $\frac{x}{yz}$ will often be my choice. That's under the implicit convention that, unlike $\times$, juxtaposition has precedence over $\div$, or under the assumption that if I meant $xz/y$ I would have written $xz/y$. But some people will argue that it's technically wrong.

user334639
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This is where definitions become necessary. How I was taught order of operations was that division and multiplication have the same precedence, and that weather written as $ab$ or $a\cdot b$ or even $a\times b$, it's all the same thing (assuming $a$ and $b$ are numbers and not vectors or something). Based on what I just said you would be right and you would read this left to right and evaluate no mater what "feels" most natural.

That being said it is perfectly possible the book could choose to define $ab$ somewhat differently giving it higher precedence, and to back this argument try putting the string "$a/bc$" or "$a\div bc$" into Wolfram Alpha and you find it reads it as $$\frac a{bc}$$

Benji Altman
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My trick is to make the large operators × and ÷ and the small operators (mn and m/n) separate. The small operators are applied as multiplication then division.

So $ab/cd=\frac {ab}{cd}$. In essence the small operators are parts of the same 'word'.

The large operators are always applied to the first numerator,

so, $a\times b\div c \times d = \frac{abd}{c}$

This is what closest matches the intent of these operations with least brackets.

wendy.krieger
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