6÷2(1+2) = ?
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yes and no
the fault lies somewhat with the question
while writing on paper we skip the multiplication sign or the division sign by writing the numbers as numerators or denominators
in text you are supposed to write the signs.
regardless, (x)(y) will be read as x multiplied by y
and (1/x)(y) will be read as y/x, not 1/xy
which is what's happening here
I mean if you just write 6/(3(1+2)) or (6/3)(1+2) (or even better, use fractions, which makes it much easier to read), there is simply no doubt about what is meant. No mathematician would write an expression like this, everyone would use fractions. If you have to rely on arbitrary conventions like pemdas or whatever to get your point across, your point just wasn't clear enough
1
edit : To be more clear so as to not create more confusion. The answer is that the expression by itself is simply an incorrect expression since it is ambiguous in interpretation. So for a computer programmer it is ok to say 9 bcoz in programmatic math, division and multiplication are evaluated serially
The expression is 6÷2(1+2) and not 6÷2×(1+2)
Yes that × operation makes a huge difference. If the expression were written as the latter one, it would evaluate to 9. But since there is no × between them, it makes the 2 a coefficient of the bracket, and not simply a multiplier. And coefficients are evaluated first.
For example consider the expression "10a ÷ 5b" with a and b being unspecified constants. In such a case you will perform the 5×b operation before the division because it is a coefficient. This expression is (10×a)÷(5×b) and CANNOT be evaluated as (10×a÷5)×b
first off, "10a ÷ 5b" is different from "10a ÷ (5b)"
Because we are writing divisions and fractions in a straight line like this, you need to be clear on which numbers are the denominators and which aren't. If there are more than 1 element, then group 'em together in a bracket. Else, they are considered to be separated and only the first one is the denominator.
ex:
"10a ÷ 5b" legit means "10 × a ÷ 5 × b" so only 5 is dividing 10a and b is later multiplied.
"10a ÷ (5b)" means 5 and b are together and both are the denominator in the fraction with numerator 10a.
if 2 is a coefficient of (1+2), then (1+2) has to go together with 2 under the line, like this
"6 ÷ (2(1+2))"
no parenthesis means only 2 is dividing 6 which is then multiplied with (1+2) after.
If we're to translate "6 ÷ 2(1+2)" into the bigger format it would look like this:
6
— (1 + 2) = 9
2
And then "6 ÷ (2(1+2))" (which is what you mean), would look like this:
6
———— = 1
2 (1+2)
Unfortunately here’s why your wrong, when something is divided by something, it is considered it’s own part, therefore it would be the expression 6/2 , making it 9, IF IT WAS TRULY 1, then it should be 6 ÷[2(2+1)], which means it would be 1, but unfortunately that is not the equation as it’s 6 ÷ 2(2+1), which means it’s(6/2)*(2+1), that 2 is not part of the (2+1) unfortunately, and this is why every math calculator thinks your right even a scientific one.
Math in computer programming has different conventions than academic math.
I do see your point and if you're speaking in terms of computer programming then yes multiplication and division operations are evaluated serially.
But every math textbook I have used would treat a coefficient different than a multiplier.
It's a popular argument people get into all the time about how you treat the expression a÷bc. Is it (a÷b)×c or is it a÷(bc)
And the answer is that the expression is just poorly written since it is not presented in an unambiguous way. So if you're a computer programmer I can understand why you'd think 9 but from a strictly mathematical point of view it's 1
An actual mathematician would take a look at this expression and before giving you the answer would point out that you should use proper brackets. And then he'd tell you that 1 is the mathematically correct answer whereas 9 is the programmatically correct answer
If a source has concretely stated that "the correct answer is 9" then that is a bad source since the first thing a good source in this case would point out that the expression is written incorrectly.
Any real mathematician wouldn't write a stupid expression like this anyways
That is so true and is literally the main reason for the entire argument. It's not a mathematical expression if it's ambiguous and there's multiple ways to interpret it.
However the fact that people now consider 9 as the accepted answer is surprising to me but it's probably because people are more concerned with the programming aspect of mathematics than the scientific.
You are correct. A fraction should be used here.
Otherwise you run into this issue:
https://imgur.com/a/c78Wm77
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