That is a classic. The problem is that there is no universal agreement on how to lớn parse expressions with a set of multiplications and divisions, or with more than one division. Does 1/2a mean 1/(2b) or (1/2)a? It is a matter of convention. Mathematica treats it as (1/2)a, but you may find books where it is meant as 1/(2a), perhaps because this way you save sầu typing parentheses. Usually the ambiguity can be decided from the context, as only one interpretation makes sense. I teach my students to avoid expressions such as a/b/c and always write parentheses to lớn make sure they are not misunderstood. There is a Wikipedia article on the "Order of operations".The calculation 6 ÷ 2(1+2) = 6 ÷ 2 + 4 = 7 is a gross mistake in my view.
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Thank you for your response Gianluca,
For my own edification, what context clues would there be to lớn solve sầu the equation any other way? Which interpretation makes the most sense in this context? 9 or 1? Would a word problem make this easier to understvà as well?
Also I very much appreciate any responses in this thread--All of us have sầu access khổng lồ the same Wikipedia & websites that have greater detail...however for my end users who are obstinate about unreliable sources I wanted lớn come khổng lồ the most credible place.Answer
Yes it comes down to lớn whether a space in front of parenthesis in an equation has any significance when parsing a formula. Mathematica doesn"t care and even inserts a space lớn make the formula appear nice. Of course in Mathematice as in many other notations, the multiplication sign can be replaced by a space lượt thích 4 5=45 obviously leaving out the space there would be the number 45. with parenthesis the space might not be necessary 2(1+2) = 2 (1+2)=2(1+2), I have not seen the interpretation 2(1+2)=(2*(1+2)), but it is all a matter of agreed convention.
In:= 6 / 2 (1 + 2)Out= 9When in doubt, the best way to insure clarity one can write out the formula lượt thích this:
In:= 6 / 2*(1 + 2)Out= 9In:= 6 / (2*(1 + 2))Out= 1AnswerIt all depends on what they call the precedence of the operators. For Mathematica (mctq.vn Language) the Divide has a higher precedence than Times. So it will first vì divisions then multiplications:
PrecedencePrecedence470400But in other cases you also have to lớn look at how it groups:
6/2/3could be 6/(2/3) or (6/2)/3, giving different answers.
have sầu a look here:
To sum up: the author of that equation is just sloppy; you have to assume something in order to solve it, và depending on the conventions... so therefore always add extra parenthesis lớn rules out those cases...
& similarly things lượt thích a^b^c could be (a^b)^c or a^(b^c), so always add parenthesis when there is no clear answer. You can avoid part of it by always doing multiplication first & then division:
Thank you all!
All things considered would an appropriate answer be "Need more information to solve" since the author gave sầu no context clues to lớn assist?
Yes, without further hints it is not good notation, you need to know the rules of the game. It is like using parenthesis like
*5)we don"t know how lớn deal with that without some extra rules...
Some clayên the equation is ambiguous, but the notation is comtháng. It is understood that we apply the Distributive Law in the Parentheses step of PEMDAS. 2(1+2) = (2x1) + (2x2) = 6.
The only number you can divide 6 by khổng lồ arrive sầu at 9 is 2/3.
2(1+2) != 2/3
2(1+2) == 6
The left to lớn right solvers, by replacing implied multiplication with implicit multiplication, are flipping the parenthetical expression khổng lồ the inverse.
2÷(1+2) == 2/3
6÷2(1+2) != 6÷2*(1+2)
The Desmos online calculator will return the correct answer when using the keypad.
The Casio fx9860GII SD returns the correct answer to the equation as given 6÷2(1+2) và provides the alternate notation if you were expecting a result of 9.
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6÷2(2+1), the answer is actually 9.For the people who used this principle x/2y, 2y is obviously together in a group. Polynomials is a simpliest size of an equation the doesn"t need parentheses. for example, if the problem is written as 6÷, in algebraic expression it can easily written as 6÷2y or 6/2y with y=2+1.For the people who said that we should use the distribution, using distribution in 2(2+1) is actually a multiplication method which means you break the rule of PEMDAS that Multiplication and Division are tied in priority which who comes first from left khổng lồ right should be solve sầu first.1st step: 6÷2(2+1), Inside the parentheses, 6÷2(3), the parentheses sign is indicate a multiplication, eq. of multiplication signs 2nd Step: 6÷2(3), since () is indicate a multiplication, you can rewrite the problem as 6÷2x3, since Division comes first, used Left khổng lồ Right rule, solve sầu 6÷2=3 3(3) or 3x3 or 3*33rd Step 3(3)=3x3= ****9****.