링크를 타고 가 보니... (예전에도 있었는지 모르겠는데)
페이지 말미에 이에 대한 설명이 나와 있습니다.
3줄 요약하면
1. 수학자 사이의 일반적!인 합의는 "곱하기 기호(×)를 생략한 곱은 (다른작업?)보다 우선순위가 있다"는 것
2. 그래서 내 답은 5 이다. (본문에 적용하면 48÷2(9+3) = 48÷24 = 2)
3. 메일 보내지 마라. 최종 결론은 없으니까.
Sideways Formatting and Multiplication-by-Juxtaposition
This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing. (It's become annoying popular to post these to Facebook.)
Simplify 16 ÷ 2[8 − 3(4 − 2)] + 1.
I simplify in the usual way:
16 ÷ 2[8 − 3(4 − 2)] + 1
16 ÷ 2[8 − 3(2)] + 1
16 ÷ 2[8 − 6] + 1
16 ÷ 2[2] + 1 (**)
16 ÷ 4 + 1
4 + 1
5
The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1".
That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses seem somehow to outrank division, so the first 2 in the starred line is often regarded as going with the [2] that follows it, rather than with the "16 divided by" that precedes it. That is, the multiplication that is indicated by placement against parentheses (or brackets, etc) is often regarded (by science-y folks) as being "stronger" somehow than "regular" multiplication which is indicated by a symbol of some sort, such as "×".
Typesetting the entire problem in a graphing calculator verifies the existence of this hierarchy, at least in some software:
calculator screen-shot: 16 / 2(8) = 1, but 16 / 2 * 8 = 64
Note that different software packages will process this expression differently; even different models of Texas Instruments graphing calculators will process this expression differently. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask! And, when typing things out sideways, be very careful of your parentheses, and make your meaning clear, so as to avoid precisely this ambiguity.
(Please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there is no such final verdict. Telling me to do things your way will not solve the issue!) (For an example of the sort of e-mails I get on this, continue to the next page, which also contains more fractional-form examples.)
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링크를 타고 가 보니... (예전에도 있었는지 모르겠는데)
페이지 말미에 이에 대한 설명이 나와 있습니다.
3줄 요약하면
1. 수학자 사이의 일반적!인 합의는 "곱하기 기호(×)를 생략한 곱은 (다른작업?)보다 우선순위가 있다"는 것
2. 그래서 내 답은 5 이다. (본문에 적용하면 48÷2(9+3) = 48÷24 = 2)
3. 메일 보내지 마라. 최종 결론은 없으니까.
Sideways Formatting and Multiplication-by-Juxtaposition
This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing. (It's become annoying popular to post these to Facebook.)
Simplify 16 ÷ 2[8 − 3(4 − 2)] + 1.
I simplify in the usual way:
16 ÷ 2[8 − 3(4 − 2)] + 1
16 ÷ 2[8 − 3(2)] + 1
16 ÷ 2[8 − 6] + 1
16 ÷ 2[2] + 1 (**)
16 ÷ 4 + 1
4 + 1
5
The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1".
That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses seem somehow to outrank division, so the first 2 in the starred line is often regarded as going with the [2] that follows it, rather than with the "16 divided by" that precedes it. That is, the multiplication that is indicated by placement against parentheses (or brackets, etc) is often regarded (by science-y folks) as being "stronger" somehow than "regular" multiplication which is indicated by a symbol of some sort, such as "×".
Typesetting the entire problem in a graphing calculator verifies the existence of this hierarchy, at least in some software:
calculator screen-shot: 16 / 2(8) = 1, but 16 / 2 * 8 = 64
Note that different software packages will process this expression differently; even different models of Texas Instruments graphing calculators will process this expression differently. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask! And, when typing things out sideways, be very careful of your parentheses, and make your meaning clear, so as to avoid precisely this ambiguity.
(Please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there is no such final verdict. Telling me to do things your way will not solve the issue!) (For an example of the sort of e-mails I get on this, continue to the next page, which also contains more fractional-form examples.)