New answers tagged

1

I have been taught both methods. By different teacher, and at different age, although I do not remember in which method first. Today I am using both indifferently, sometimes changing of method in the middle of the subtraction. I noted a tiny preference for the Austrian method, but I think it is purely esthetic, because most of the time the choice is totally ...


1

Both these methods suffer from the same problem: they work from right to left, despite our reading numbers from left to right. That's why children have trouble with long sums. I discovered as a small child that I could always be the first kid in class to put my hand up when the teacher wrote a subtraction (or addition) sum on the blackboard, by working from ...


2

At first I thought comments would suffice, instead of an answer, because there have been a couple good answers already. However, it appears that there are a number of issues at hand, some of them regarding math education rather than mathematical notation. Nevertheless, let me first make a couple notes about notation though, as part of the subject. I ...


3

This would be better as a comment but I don't have sufficient reputation. Daniel R. Collins asked for an example of a mathematical text writing something like $a/bc$ to mean $a/(bc)$, and it might be useful as a counterexample to the strong claims that a mathematician would never write such a thing. On page 2 of Geometry Revisited by Coxeter and Greitzer, ...


2

Part of the difficulty is that order of operations is taught wrong. The basic rule of order of operations is this: "Operations are performed from left to right unless..." The first unless is the existence of parentheses. Now here's where another factor (pun unintentional) comes into play: we write in lines. Because of that, something has to come first ...


3

In the order of operations, one multiplication does not take precedence over another; all multiplications and divisions are performed from left to right. A so-called "concatenation" with parentheses a(b) has the exact same meaning as "a times b". The parenthesis operation only takes precedence over other operations to evaluate what is inside. After that, ...


3

I think (other than the fact that it's pretty much a deliberately ambiguous question) the thing that is missing is the concept of things being single terms, which ought to be evaluated first. I would interpret the expression as 1, because I'd consider 2(2 + 2) to be a single term that should be evaluated first before the answer is substituted back into the ...


2

This is one of the cases where a linguist has to admit that we don't actually make the rules for a language. We discover them, and codify them. And there's always exceptions like I before E Except when your foreign neighbor Keith receives eight beige counterfeit sleighs from feisty caffeinated weightlifters English is weird. PEMDAS is a linguistic ...


7

Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority? The conventional order-of-operations in textbook math (or any math) simply don't prioritize ...


3

The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification. I'm going to suggest a variant of this which you might consider a fourth option: The $\div$ operator is at best confusing if not ambiguous, and should not be used. For a long time I actually thought it was a ...


5

My disagreement lies in the dismissal of the parentheses, as my explanation would be that $8÷2(x+x)$ would, as a first step, simplify to $8÷2(2x)$ and then $8÷4x$ It seems to me that you are: ignoring the Left To Right rule, not making the implied multiplication between $2$ and $(2+2)$


6

I am with you on this one. I feel like concatenation (implied multiplication) is of higher precedence than explicit division. For me $8:2x$ means "8 divided by 2 x'es" - $8:(2x)$ not $4x$. Replacing $x$ with $(2+2)$ shouldn't change anything. But the formal answer is that it's undefined. There is no C for concatenation in PEMDAS. For me it should be PECMDAS....


2

In a comment you write "multiplying what's in parentheses should take priority"... But the only expression within parentheses is the sum: $(2+2)$. Hence we have $$8\div 2(4) = 8 \div 2 \times 4.\tag 1 $$ I think you are confusing concatenation of $2$ with $(2+2)$ as meaning that All of $2(2+2) = 2 \times (2+2)$ must be evaluated as the dividend of $8$. ...


14

If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right). But an expression such as $48\div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying. If one interprets the parentheses ...


20

The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous. If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than ...


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