It depends on which interpretation of multiplication by juxtaposition you use. Academically, it implies grouping and multiplication which gives 1. The more literal interpretation implies multiplication only which gives 9. Both are valid.

It doesn't take implicit notation into account.

@@GanonTEK If the writer uses left to right PEMDAS, then no ambiguity. Things are ambbiguous because you don't know if the writer is following convention.

@@davidwebster9788 L to R gives both answers. The ambiguity occurs with the implicit notation used. This must be interpreted before you can use any rules or laws or properties of maths. Academically, juxtaposition implies grouping and multiplication so 6/2(1+2) explicitly means 6/(2*(1+2)) Using PEMDAS then gives 1 Literally/programming-wise, juxtaposition implies only multiplication so 6/2(1+2) explicitly means 6/2*(1+2). Using the same PEMDAS gives 9. Both are common interpretations so both are valid. Both use the exact same order of operations. Notation conventions are language conventions and supercede anything else.

@@GanonTEK ok, what is implicit notation?

@@davidwebster9788 It's notation (language) where something is implied (not explicity written down). For example, Sin²x is implicit notation for (SIn x)² It implies brackets and the exponent in a different place. Sin3y implies Sin(3*y) It implies brackets and multiplication around the 3 and y. Sin²3y does both (Sin(3*y))².

You have to combine factors before doing that, otherwise you are flipping one factor and not the other. 2x, where x is any function in parentheses is a monomial. You can't just flip one factor in a monomial and not the others. 1÷2wxyz becomes 1 * (1/a), where a = 2wxyz (all the factors combined)

No, both 6÷2(1+2) and 6•1/2(1+2) are ambiguous for the same reason: they both have implicit multiplication after division on one line. That's the main problem. You need to write 6•(1/2)•(1+2) or (6/2)•(1+2), for example, to be unambiguous there and something like 6•(1/2)•1/(1+2) for 1

Depends on which interpretation of multiplication by juxtaposition you use. 1 is just as valid. You can see it on many scientific calculators even. It's bad writing. The question should never be written that way. It's very poor writing.

It gives both answers since the ambiguity is with the implicit notation which occurs before you use the order of operations at all.

what about it

The two answers come from the two different interpretations of juxtaposition in use. Academically, juxtaposition implies grouping and multiplication (1). Literal/programming-wise, juxtaposition implies multiplication only (9). The O in BOMDAS is better represented by Order. It means exponents/indices. Nothing to do with multiplication itself as M is the multiplication step which comes after it. So, academically 6÷2(1+2) explicitly means 6÷(2×(1+2)) Using BODMAS gives 6÷(2×(3)) = 6÷(6) = 1 Literally/programming-wise, 6÷2(1+2) explicitly means 6÷2×(1+2) Using BODMAS gives 6÷2×(3) = 3×3 = 9 This is the interpretation you chose. So, the same order of operations gives both answers. So, your explanation isn't wrong. It's just one of the two possible answers.

@@GanonTEK Alright, but what's Juxtaposition ?

@@yor_201 It's when you have them side by side with no operator between them. It's also called implicit multiplication or multiplication with the operator omitted.

6:2(1+2)=1 you always start with (x)

It depends on which interpretation of juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally/programming-wise, juxtaposition implies multiplication only (9). So the same order of operations gives both answers then since they are used after you interpret any implicit notation.

. * / \ 2 f f is any function in parentheses. It's a separate tree. f = ? / \ ? ? Figuring out f=3 doesn't perform an operation on the main tree. The node directly above f on the main tree is a multiplication by a factor that is outside the function. Algebra says this... all variables (in this case f, an unknown function) have coefficients. You can't use the 2 in some other operation just because you figure out what f equals. You haven't even performed an operation on the main tree. You only figured out one leaf.

@@v-chris156 It all comes down to which implicit notation convention you use. This is done *before* you make your tree so any tree argument is irrelevant.

​@@GanonTEK You suggests that: Academically, juxtaposition implies grouping and multiplication, which would lead to an answer of 1. Literally/programming-wise, juxtaposition implies multiplication only, which leads to an answer of 9. *However, this interpretation is not supported by standard mathematical conventions.* In standard mathematics, the order of operations (PEMDAS/BODMAS) is clear. Juxtaposition in Mathematics: Juxtaposition (placing two things side by side) in mathematical expressions typically implies multiplication. However, *this does not change the standard order of operations.* Standard Interpretation: 6÷2(3)→6÷2×3→3×3=9 *Misinterpretation and Clarification:* The confusion arises when one mistakenly assumes that 2(3) should be treated as a single grouped entity without proper application of the division. This is incorrect under standard mathematical rules. Incorrect Grouping Interpretation: 6÷[2(3)]→6÷6=1 This interpretation does not follow the correct order of operations. The multiplication and division must be handled left to right as they appear in the expression. *Conclusion:* Standard mathematical rules do not support the academic versus programming context distinction stated by you. The expression 6÷2(1+2) evaluates to 9 following the correct application of PEMDAS/BODMAS rules. Any deviation from this implies a misunderstanding or misapplication of these rules. Therefore, the only correct answer, according to mathematical conventions, is 9.

@@happygood18 Common mistake there. The order of operations has nothing to do with notation conventions. Notation conventions are language conventions. These must be interpreted before any and all rules/proerties/laws/axioms of maths can be applied. You're putting the cart before the horse so your argument is moot. PEMDAS etc. does not have implicit notation as a part of it. It's completely separate. The academic interpretation is widely in use (e.g. the American Physical Society style guide and Bracketing Mathematical Expressions H9) and in plenty of other academic writing. WolframAlpha's Solidus article mentions the ambiguity even. Scientific calculators also disagree on the answer here as some use one notation convention and some use the other. Conclusion: It's ambiguous notation as already confirmed by the American Mathematical Society, ISO, and multiple professors and mathematicians.

That's not true. Also, the academic interpretation of juxtaposition implies grouping so the 2 and (1+2) are grouped before you go near the ÷ symbol at all. Academically, it's explicitly 6÷(2×(1+2)) which is 1. Literally/programming-wise, it's explicitly 6÷2×(1+2) which is 9. Even over in America where 9 is more popular, the American Mathematical Society stated it was ambiguous notation, as have multiple professors and mathematicians. Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. (6/2)(1+2) and 6/(2(1+2)) are the correct forms.

That's one of the valid answers, yes.

@@GanonTEK its the only valid answer

@@More.A-gf5rd That's incorrect. Since academically juxtaposition implies grouping 1 is valid when using that notation convention. It's why many scientific calculators give 1. It's perfect valid and very common. The expression itself is not acceptable according to modern international standards like ISO-80000-1.

@@GanonTEK ooohh ,thanks for sharing, I thought according to BODMAS , Division has higher priority that multiply,and my phone calaculor also giving 9 .anyway thanks for clearing misconception.

@@More.A-gf5rd You're very welcome