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Why do we multiply/divide first, and then add/subtract later?

I mean, what I'm curious about is that is this a universal rule, or a man-decided rule? Also how would you decide on which to operate first?

For example, if we were to be visited by aliens, would they be using the same math? Would they also multiply first, add later?

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  • $\begingroup$ It was person-decided. I use parentheses to avoid potential ambiguity. $\endgroup$ Mar 29, 2014 at 20:56
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    $\begingroup$ All of maths is 'how man decided'. Unless you take a foolish Platonic view of mathematics. As alas so many do. $\endgroup$
    – Frank
    Mar 29, 2014 at 20:56
  • $\begingroup$ @Frank Not quite. While creating mathematics one should take care that no contradictions are produced. For example one can not just make up some arbitrary distributive law for addition and multiplication. The rule that "multiplication comes first" indeed is a choice as "addition comes first" would also have no weird consequences. $\endgroup$
    – Dirk
    Mar 29, 2014 at 21:03
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    $\begingroup$ It is an arbitrary choice, there actually are programming languages (J, for example) where operators all have the same precedence $\endgroup$ Mar 29, 2014 at 22:22
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    $\begingroup$ There is a way of writing arithmetical operations without parentheses. It's called the reverse Polish notation. It was implemented in the Hewlett-Packard calculators in the 1970's. For an overview, en.wikipedia.org/wiki/Reverse_Polish_notation $\endgroup$ Jan 22, 2017 at 0:48

3 Answers 3

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This is just a notational convention to allow one of the expressions $$(a+b)\cdot c\qquad a+(b\cdot c) $$be written without parentheses. One could have chosen the other (or to always require parentheses), but think how complicated it would make writing e.g. polynomials (unless written as product of linear factors). The same considereation holds for why $a-b-c$ stands for $(a-b)-c$ and not for $a-(b-c)$. Alternatively, one might introdue a prefix ore postfix (instead of infix) notation, aka. (reverse) Polish notation, i.e. $ab+c\cdot$ or $\cdot +abc$ vs. $abc\cdot+$ or $+a\cdot bc$.

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  • $\begingroup$ On the other hand, the other convention might make it easier to write polynomials expanded after Horner's method, so who knows what we have lost because of it? We might have a whole body of technique for dealing with Horner-expanded polynomials that doesn't exist at present. $\endgroup$
    – MJD
    Mar 29, 2014 at 21:02
  • $\begingroup$ @MJD: Horner's method alternates addition and multiplication, so wouldn't it require the same number of parentheses either way? $\endgroup$
    – user856
    Mar 29, 2014 at 21:36
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    $\begingroup$ @Rahul: If our goal were to make Horner-expanded polynomials easy to write, we could make it the convention that everything has the same precedence and is simply evaluated left to right (as we do for addition and subtraction). $\endgroup$ Mar 29, 2014 at 21:38
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    $\begingroup$ @Rahul You're right; it was an ill-considered comment. $\endgroup$
    – MJD
    Mar 29, 2014 at 21:54
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Complementary to what others said, the order of operations is also a natural consequence of rewriting expressions to avoid ambiguity.

For example, $3\times 4+1$. If this were all addition, there’d be no problem, because addition is commutative and associative. To rewrite this with only addition: $4+4+4+1$, so you’re essentially evaluating the multiplication first to make it unambiguous. Same is true for exponents and other hyperoperations like tetration. And without choosing a set order multiplication and addition would lose associativity and commutativity, eg sometimes $3\times 4+1\neq 4\times 3+1$ (which would surely lead to problems in defining other functions elsewhere). It’s completely a choice, but does make things clearer and simpler, and as others have pointed out tends to be more useful.

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The order of operations were "man-made". If there were no such thing as "Order of Operations", an expression like $3+4\times 7+2^2\div 4$ can have many different answers. One could add $3+4=7$, then times $7$ to get $49$, then plus $2^2$ to get $53$, and finally divide by $4$ to get $\dfrac{53}{4}$. But if we follow the order of operations, we would get a different answer. $$3+4\times 7+2^2\div 4=3+4\times 7+1=3+28+1=32$$ To avoid these different answers, we created the Order of Operations so that each expression has only one answer.

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