What is 6÷2(1+2)?

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#1 30 December, 2016 - 13:06
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What is 6÷2(1+2)?

What is the answer to the following math problem?

$$6\div2(1+2)=$$

Give it a try before scrolling down for the discussion. :)

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This video claims to have the correct answer

The video concludes that the correct answer is 9, but many comments on the video still argue that the answer is 1,

What do you think? 9 or 1? :)

30 December, 2016 - 13:14
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The problem is that you're not supposed to use the ÷ sign in maths, and that "2(1+2)" is also ambiguous and not the right way to put it. If it was written properly, like for example
___6___
2⋅(1+2)

... then there wouldn't be any confusion :)

30 December, 2016 - 13:59
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Most calculators will still give 9 if written that way:

I can only get Wolfram Alpha to give me what would appear to be the correct answer if I enter $$\frac{6}{2 \cdot (1+2)}$$ in LaTeX format:
https://www.wolframalpha.com/input/?i=%5Cfrac%7B6%7D%7B2+%5Ccdot+(1%2B2)%7D

Is $$6/2 \cdot (1+2)$$ different from $$\frac{6}{2 \cdot (1+2)}$$? A calculator will answer 9 for the former and 1 for the latter.

30 December, 2016 - 14:21
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Bear with me if I fail to make any sense here - generally speaking, I don't express myself very clearly when I'm complaining about other people not expressing themselves clearly...

What I was trying to say is that you're supposed to write things like that in such a way that it's clear which parts of the equation you're dividing the 6 by. If everything's under the line, then the answer's 1; if only the first 2 is under the line, the answer's 9. But if you write it like it's written up there, you can basically read it however you want. And maths is meant to be always written in a way that doesn't leave any room for ambiguity...

I blame computers, incidentally. Because it's hard to write mathematical formulas with a standard keyboard, this whole problem has sprung into existence. I think we need to scrap the whole thing and build a new internet that works with moveable type.

30 December, 2016 - 15:25
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I guess that the solution would be to write things with extra brackets like $$6/(2(1+2))$$. That seems to work correctly.

30 December, 2016 - 15:43
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A bit more reading suggests that some calculators interpret a slash as a division sign and others as a fraction line.

Wolfram Alpha says:

If left unevaluated, x/y is called a fraction, with x known as the numerator and y known as the denominator.

Does that mean that they are using the pre 1917 rule that is mentioned in the video above? Shouldn't everything to the right of the slash be treated as a group?

Wolfram Alpha also suggests removing the ambiguity:

Special care is needed when interpreting the meaning of a solidus in in-line math because of the notational ambiguity in expressions such as a/bc. Whereas in many textbooks, "a/bc" is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages such as the Wolfram Language, it means (a/b)×c. For clarity, parentheses should therefore always be used when delineating compound denominators.

Common examples of failure to parenthesize include $$E/kT$$ (where E is energy, k is Boltzmann's constant, and T is temperature; Arfken 1985, p. 950), its variant $$c/ \lambda T$$ (where c is a constant and $$\lambda$$ is wavelength; Weast 1981, pp. F-109 and F-111), the exponent $$-(x-mu)^2/2sigma^2$$ in the normal distribution (where sigma is the standard deviation; Hastings 2000, p. 217). Other miscellaneous examples occur even in the standard references for computer math systems (e.g., Wolfram 2003, pp. 776, 779, and 787).

30 December, 2016 - 21:29
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This video suggests at around 2:29 that it's just bad notation.

… Speaking of notation, this infuriating bit of nonsense has been circulating around recently, and that there has been so much discussion of it is a sign that we’ve been trained to care about notation way too much. Do you multiply here first, or divide here first?

The answer is that this is a badly formed sentence. It’s like saying, “I would like some juice or water with ice”. Do you mean you’d like either juice with no ice or water with ice, or do you mean you’d like either juice with ice or water with ice? You can make claims about conventions and what’s right and wrong, but really the burden is on the author of the sentence to put in some commas and make things clear. Mathematicians do this by adding parentheses, and avoiding this ÷ sign.

Math is not marks on a page. The mathematics is in what those marks represent. You can make any rules you want about stuff, as long as you’re consistent with them. The end.

There are also a few answers on Quora:
https://www.quora.com/What-does-6%C3%B72-1+2-equal

And another opinion that it is ambiguous:

And another video that has a few more ambiguities, like 8-2+1 and 6/3/3.

3 January, 2017 - 18:01
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