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Math Help - Why PEMDAS?

  1. #1
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    Why PEMDAS?

    I understand that the order of operations is done as PEMDAS because exponentation is just repeated multiplication and multiplication is just repeated addition and you could even make division and subtraction into multiplication and addition respectively and consequently you could really write everything in terms of addition WHEN YOU'RE DEALING WITH NATURAL NUMBERS.

    e.g. 2+3*4^2
    = 2+3*(4*4)
    = 2 + 3*(4+4+4+4)
    = 2 + (4+4+4+4) + (4+4+4+4)+ (4+4+4+4)

    but what's the explanation for why we still use PEMDAS for rational numbers?
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    Quote Originally Posted by lamp23 View Post
    I understand that the order of operations is done as PEMDAS because exponentation is just repeated multiplication and multiplication is just repeated addition and you could even make division and subtraction into multiplication and addition respectively and consequently you could really write everything in terms of addition WHEN YOU'RE DEALING WITH NATURAL NUMBERS.

    e.g. 2+3*4^2
    = 2+3*(4*4)
    = 2 + 3*(4+4+4+4)
    = 2 + (4+4+4+4) + (4+4+4+4)+ (4+4+4+4)

    but what's the explanation for why we still use PEMDAS for rational numbers?
    It is a convention that reduces the number of sets of brackets needed to make an expression unambiguous. Like all conventions it works best when everyone uses it.

    CB
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    Hmmm, I think there's a mistake in my reasoning actually. By going from line 1 to line 2 in my example I am assuming that the exponent applies to the 4 which means I'm assuming that the exponent should be performed first.
    Nevertheless, is the idea that exponents and multiplication can be broken down into addition (when dealing with naturals) related to why we choose PEMDAS as a convention?
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    Quote Originally Posted by lamp23 View Post
    Hmmm, I think there's a mistake in my reasoning actually. By going from line 1 to line 2 in my example I am assuming that the exponent applies to the 4 which means I'm assuming that the exponent should be performed first.
    Nevertheless, is the idea that exponents and multiplication can be broken down into addition (when dealing with naturals) related to why we choose PEMDAS as a convention?
    Try for example \displaystyle 4 + 3 \times 7.

    When you use the definition of multiplication as repeated addition, \displaystyle 4 + 7 + 7 + 7 = 25.

    So \displaystyle 4 + 3 \times 7 = 25.


    What would have happened if you had evaluated this left to right?

    \displaystyle 4 + 3 \times 7 = 7 \times 7 = 49.

    Does that make sense, when you consider that the multiplication was supposed to represent a repated addition of 7 three times?


    The reason we use PEDMAS (or BODMAS if you're in Australia) is because it is the only reason that makes sense.
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    Well actually by writing it as 4+7+7+7 , we're assuming a particular order of operations. Multiplication could still be thought of as repeated addition if you made it 7x7 and then wrote it as 7+7+7+7+7+7+7
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    The point is that everything boils down to addition - if you expect to be able to perform multiplication, then you expect that you would need to be able to evaluate it using addition. Therefore, it makes sense to get everything to addition BEFORE trying to do any evaluating.
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    Right, but to know what to boil down to addition requires a convention, right?

    P.S. To the mod, I posted this in Analysis because I would like to know the answer an Analysis book would give rather than Pre-Algebra textbooks which often just state the rules without explaining why.
    Last edited by lamp23; February 13th 2011 at 10:26 PM.
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    Quote Originally Posted by lamp23 View Post
    Right, but to know what to boil down to addition requires a convention, right?

    P.S. To the mod, I posted this in Analysis because I would like to know the answer an Analysis book would give rather than Pre-Algebra textbooks which often just state the rules without explaining why.
    The answer is in post #2. It is nothing to do with being able to break stuff down to addition but is everything to do with disambiguation.

    (it is also nothing to do with analysis)

    CB
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    Quote Originally Posted by CaptainBlack View Post
    The answer is in post #2. It is nothing to do with being able to break stuff down to addition but is everything to do with disambiguation.

    (it is also nothing to do with analysis)

    CB
    I agree now but now I'd like to know why choose this particular convention.
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    Quote Originally Posted by lamp23 View Post
    I agree now but now I'd like to know why choose this particular convention.
    And the answers to that question are in posts 4 and 6...
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    Quote Originally Posted by Prove It View Post
    And the answers to that question are in posts 4 and 6...
    Well no, right? Because I don't see any reason why another culture couldn't have decided to use the notation of 4+3*7 to represent 7 boxes of 7 marbles in each box and then still write 7*7 as repeated addition: 7+7+7+7+7+7+7
    Therefore, it would still be broken down to addition.
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    I don't see any reason why another culture couldn't have decided to not use the Arabic numeral system. There's no reason why we must express numbers in base 10. We just do. It is how math has evolved in the Middle East, then the West, and eventually, everywhere. PEMDAS is a convention, although there is no reason why one has to do it that way, except that it is how it is.
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    Quote Originally Posted by Rabolisk13 View Post
    I don't see any reason why another culture couldn't have decided to not use the Arabic numeral system. There's no reason why we must express numbers in base 10. We just do. It is how math has evolved in the Middle East, then the West, and eventually, everywhere. PEMDAS is a convention, although there is no reason why one has to do it that way, except that it is how it is.
    I understand that our convention is not the only possible convention but nevertheless why was/is it the convention chosen?
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    Quote Originally Posted by lamp23 View Post
    I agree now but now I'd like to know why choose this particular convention.
    Other than it is adequate?

    Why did the world settle on VHS rather than BetaMax for a domestic video tape standard (since you probably barely remember video tape I will tell you that in this case the technically inferior solution dominated)?

    There is no objective reason other than one gains an initial advantage in market penetration, then because doing what the majority do has an advantage an initial small random greater usage is amplified to near 100% usage.

    Not every thing has a reason, or at least what we would recognise as one. Nor does the best solution invariably dominate - I presume you are typing on a qwerty keyboard, which for computer use is ludicrously non-optimal.

    Another example of a convention where different culture have settled on different solutions is the side of the road that we drive on. I drive on the left, because everyone else here does so. Others drive on the right for much the same reason. The difference is due to an accident of history.

    CB
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    Quote Originally Posted by lamp23 View Post
    I understand that our convention is not the only possible convention but nevertheless why was/is it the convention chosen?
    Would you like someone to tell you that Newton flipped a coin to decide?

    There is no reason for a convention- that's essentially what convention means. It's the same reason some nations drive cars on the right side of the street and others on the left. The only difference is that since different nations and different cultures have to be able to talk "mathematics" to each other, we all agree on the same convention rather than having separate conventions.
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