modellingman wrote:XFool wrote:6 / 2(2+1) = ?
As written, this is clearly ambiguous. If taken as a straight line only expression: 6 / 2 multiplied by (2+1) the 'answer' obviously is 9. But, if taken as 6 divided by 2(2+1) it would be one.
From the spacing given in the OP, this seems an entirely reasonable interpretation. A previous post points out that the expression was originally given online with a ÷ sign and different spacing which
may clarify the interpretation as an expression all on one line.
Embolding added.
This is clearly the nub of it and I disagree. There's no ambiguity and it is not a reasonable interpretation to separate the "2(2+1)" and evaluate it before the division operation. Further, "/" is simply an alternative to "÷" and the meaning of an expression does not change if one is substituted for the other. The expression is all on one line because its author placed all its characters on the same line.
I still disagree! Why?
I am not a mathematician, as a self described mathematician you describe all this purely in terms of a process solely involving a set of formal rules. In other words, as purely mathematical! Fair enough, but is that really adequate here?
Mathematics (as I understand it) is a set of purely formal rules, so the expression (if well formed) should be correctly evaluated by any correct mathematical 'machine'. Interestingly, the only two mathematical machines I have tried this on both give 'wrong' answers.
My Windows Scientific Calculator evaluates on the fly and so gives two separate answers of 3. Firstly, 3 for 6/2 (correct as per your description) and then a completely separate answer of 3 for (2+1). But, without an explicit symbol to multiply the first part of the expression by the second part of the expression it just treats them as two separate sums. My hand calculator takes in the whole expression (no spaces) and evaluates it to 1. With an explicit multiply symbol added it evaluates to 9.
You may well say they are both wrong, in which case you will have to take it up with their programmers. An expression evaluated by a human containing a digit '7' may be correct, but what if it had been given to a mathematical machine, and the '7' had been written in the continental style; perhaps the machine wouldn't be able to evaluate it according to the rules? What if Roman numerals had been used?
What is going on here involves a person interpreting what another person
intended by the given arrangement of symbols in the expression in the OP.
You say that / is the same as ÷ (formally it is) and that the OP was "all on one line", which it literally was. BUT, without typesetting tools and using only normal keyboard characters, all such expressions typically are literally "all on one line" even though, in some cases, mathematically they are not. Of course, it that case, the onus is on the poster to form the expression in an unambiguous manner. Which, in the expression in the OP it was easily possible to do in more than one way.
In this case the OP was itself something of a 'translation' from the original source, which muddied the water even more.
The point is not that the answer 9 is wrong and the answer 1 is correct, rather that as it isn't possible (for an average human) to be sure what was meant. So either you have to go with how YOU interpret it, based on experience and judgement, or ask what actually WAS intended, which rather defeats the point of the question...
modellingman wrote:Spaces have no significance in mathematical expressions.
No. But they have significance in how humans interpret the meaning of text put in front of them.
modellingman wrote:In terms of the OP, how would this "implied brackets" convention resolve an expression such as
6 / 2 × (2+1)
Unambiguously. Because it is unambiguous.
modellingman wrote:Some posters have gone further than this. For at least one, spaces around a "/" mean the expression is split vertically with a horizontal divisor operation line inserted at the split point (so no longer "an expression all on one line").
That's right - along with the absence of spaces between the 2 and the (2+1), implicitly tying them together.
modellingman wrote:To such posters there are similar questions. Under such a splitting convention, how much of the expression before and after the split is placed in the upper and lower components respectively? Do other spaces in the expression limit the scope of the split?
The question should be: "Why such a misleadingly formed expression (bearing in mind the limitation of the character set being used)? It is simplicity itself to format it unambiguously."
modellingman wrote:I look forward to seeing from posters the many references from mathematical texts that fully define these conventions, which somehow escaped my mathematical education. Until then, I shall maintain (along with my fellow graduates of mathematics) the much simpler convention that spaces have no significance, that "/" is simply another character for "÷" and that an expression where all the characters are physically on the same line is precisely that, ie "an expression all on one line".
Again, you may be formally correct but, for most it was firstly a question of interpreting what the poster
intended. (Which could, of course, be something else.)
I don't think we can learn anything very significant from the OP, other than to understand the importance of clarity when writing mathematical expressions using normal keyboard characters.