This question was initially asked in the Yahoo website, and spread rapidly through Facebook recently, especially in Taiwan, it even debated by parliamentary members. As reported by Sin Chew Daily, a popular Malaysian Chinese Newspaper, there are more than 3 million people responded to it. About 2 million voted for the answer 9, and about 1.5 million voted for the answer 1. Actually, similar questions have been asked long time ago, but human tend to be forgetful or ignorant of history.
Some mathematics experts said the answer is 9, because they opined that calculation should follow the order of operations called BODMAS, which stands for Bracket first, O for orders (powers and square roots, etc.), Divide and Multiply (from left to right), Add and Subtract (from left to right), or PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Since “x” and “¸” are of the same level of operation order, they must be carried out from left to right as or as if they appear in the equation, so, the answer obtained is 9. But, this is wrong!
There are various suggestions that the question should be presented in more clearly defined ways such as:
By adding an “x” just before the bracket: 6¸2x(1+2) = 6¸2x3 = 3x3 = 9
By adding a bracket just before the other: (6¸2)(1+2) = 3x3 = 9
By adding a square bracket after “¸”: 6¸[2(1+2)] = 6¸[2x3] = 6¸6 = 1
By doing so, the answer will be definite. However, there is nothing wrong with the original question as it is written 6¸2(1+2). The above additions although could remove doubts and useful for primary students, but are sometimes redundant. There should be no issue about the correctness of the question if there is only one answer and basic principles are followed.
If the operation is based on BODMAS rules and the first “2” in front of the bracket is merely treated as a number:
(1) However, if you type on a calculator with the equation’s numbers and symbols exactly as written, it should give the correct answer as 1. The calculator must be correct although it does not provide explanation. If the calculator created and programmed by human is wrong, what else can we trust?
Unfortunately, it is really found that: If use Casio fx-570MS and fx-350MS calculators the answer is 1. If use Casio FC-100V and fx570ES calculators the answer is 9, this is wrong. All teachers and people must be informed or warned that something wrong with calculators. The MS series and ES series do not agree each other probably due to certain mechanism, programming error or conceptual mistake. The mathematician, the software engineer, programmer, school teachers, education ministry, global system, or who to be blamed? But not the designer of “¸” and ( ).
(2) In fact, the first “2” is not just a number, but a factor for “(1+2)”, so 2(1+2) must be wholly performed first as follows:
The first “2” can be a factor because it is true that when 6¸(2+4) is factorized will becomes 6¸2(1+2) or 6¸[2(1+2)], but usually the [ ] is omitted and treated as redundant.
(3) We know that 6¸2(1+2) can be written as follows using an “over” sign, or fraction bar, i.e. the horizontal line representing division and separate the numerator and denominator. This is the important concept learned in primary and secondary schools.
It is not correct to write and calculate as follows.
(4) If represent (1+2) with y, i.e. y = (1+2), and the first “2” is a factor of y, therefore 2y is bound together as one term, then,
Substitute y = (1+2) back to it:
(5) Similarly, equalize a bracket to a multiplication is not always true, as can be seen in the following examples.
(6) Let consider a scientific example, the ideal gas law, PV = nRT can be rearranged to n = PV¸RT or n = VP¸TR, etc.
So, those who have never properly studied the calculation of ideal gas law will probably give the wrong answer 9!
(7) One obscure rule which may override BODMAS is the factorization and defactorization or expansion, which can always be a priority to be performed first when necessary or sometimes done later or just ignored. Under the situation of the question, defactorization will be of priority.
If we defactorize/expand the equation,
The other way round, if we factorize the following equation, it should give the same answer of 1.
This means the first “2” is a factor. It can be bracketed, although always is omitted, and the above equation becomes,
If we substitute a = 1, b = 2, c = 3 in the question and carry out factorization, and substitute back the values of a, b and c:
If we try to factorize the above equation the following way,
The trick here is, we cannot factorize two terms separated by a “¸” sign. Otherwise any number can be the answer.
e.g.
(8) The main issue now is the meaning of “¸” and its substitute/alternative namely the fraction bar “¾” as well. “¸” is the division symbol used to separate between two immediate terms side-by-side. The term before ¸ is the numerator, in this case is 6. The term after ¸ is the denominator, in this case is 2(1+2). In the presence of “¸”, we cannot split the second term 2(1+2) into two separate terms/parts as 2 and (1+2) and use the former part for division but leave the second part for multiplication. Therefore, 2(1+2) is not exactly same meaning as 2x(1+2), but more than that.
(9) One example of equivalence for further understanding the meaning of “¸” and ( ).
The following answer is incorrect:
This proves that we cannot simply drop the bracket and change it to multiplication, and “force” it to follow the BODMAS.
(10) One money-counting matter about money mongers. A billionaire plans to donate 6 million ringgits to two families, each family has a single parent and two children, the money is to be distributed evenly per person.
If you calculate as follows, each person will get 1 million ringgits, total 6 million ringgits.
If you calculate as follows, each person will get 9 million ringgits, total 54 million ringgits!
Where is the money come from? It is not a magic, it is just a mental trick! That’s only sort of one of many ways money mongers become rich.
Some conclusions
The back and forth of mathematical operations must always give consistent results. Otherwise, the Laws of Conservation do not apply, or, creation and destroy of energy and mass, and even natural counting will be a magic or just miracle.
It is always a common perception that a scientific theory or equation needs mathematics to backup. Now, we can see that even a simple arithmetic mathematics need scientific concepts and measurements to support the validity.
A mathematic operation is not just following an incomplete set of rules like BODMAS and ignoring certain hidden logics. A question or scientific equation must have its meaning and purpose, so the answer must also be meaningful and purposeful, and fulfill what is being asked and sought for, otherwise rules are not just rules but also for abuse, facts will be misinterpreted, ethics will twist around, etc. This is what happened to the mass-energy equivalence equation: E = mc2.
Hew Nam Fong, Senior Lecturer, HELP CAT, 12-May-2011. http://hewnf.blogspot.com/
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