6/2(1 + 2)


  • What is 6/2(1 + 2)?

    I had a guy on Discord ask me this question, so I decided to post this answer because it was an interesting question:

    6/2(1 + 2)
    = 6/2*(1 + 2)
    = 3*(1 + 2)
    = 3*(3)
    = 3*3
    = 9? Or is it?

    The issue with this problem is that there's convention, but context often defines the convention you use. PEMDAS (in my opinion) is not a good gauge to go by. It can be interpreted that multiplication goes before division, but that's not true (or maybe it can be?). You almost need another acronym to understand what the acronym means. A better way to approach this problem is to understand the underlying structures:

    6/2*(1 + 2)
    

    can be referring to

    (6/2)*(1 + 2)
    

    or

    6/(2*(1 + 2))
    

    or something else entirely.

    On paper and written (because you can clearly write what the denominator and numerator are), it can be better understood (and yet, even that is sometimes not enough). As text and in this form, it is even less clear.

    In the end, the best way to go about this is to define your terms or use more brackets/parentheses. Different answers can be justified depending on how you define operation order. Operation order is convention, and thus it is malleable. You can even define how parentheses work.

    As long as your answer is logical and proceeds from your axioms, you cannot be wrong (thus the beauty of mathematics). Mathematics is not as boring as a set of rules. Rather, it is a universe which in which mathematicians have created guidelines and shortcuts (i.e. symbols to represent operations). You are only advised to stick to them. Conventions have changed in the past hundreds of years and they will change in the future. Thus, if you're ever confused about how a problem "should" be done, just define your terms and you can do it however you like (as long as your answer follows logically from your starting points). Your mathematics teacher may not appreciate this approach, but it's the more mathematically correct one. It is my belief that teachers have been making mathematics unappealing and more boring than it ever was meant to be. Mathematics is more of an art form than a collection of boring rules.

    If you'd like to read more on this, take a look at the book Burn Math Class.

    A quote from that page:
    "Focusing on how mathematics is created rather than on mathematical facts, Wilkes teaches the subject in a way that requires no memorization and no prior knowledge beyond addition and multiplication. From these simple foundations, Burn Math Class shows how mathematics can be (re)invented from scratch without preexisting textbooks and courses. We can discover math on our own through experimentation and failure, without appealing to any outside authority. When math is created free from arcane notations and pretentious jargon that hide the simplicity of mathematical concepts, it can be understood organically--and it becomes fun!"

    If you don't like mathematics, maybe it's because of the way you were taught.

    Teachers be like "No imagination, ingenuity, or exploration allowed."
    Students be like "I hate math."


  • Another thing to note:

    6/2 = 6*(1/2)
    

    So... maybe division is the same as multiplication, but it's the type of number you're multiplying by that's different.

    a - b = a + -b
    a/b = a*(1/b)
    

    Maybe subtraction and division are commutative, but not in the way we think...

    a + -b = -b + a
    

    and

    a*(1/b) = (1/b)*a
    
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