40 + 40 x 0 + 1= ?

Short answer: 41

Long answer: is this to be solved by parsing or by algebra? If it's to be solved by parsing, we need a set of parsing rules, in other words a convention. Grade school teaches things like BEDMAS/PEMDAS, but that's a fairly complex rule operating on groups. Instead let's go with one particular way of computer program parsing.

Read the first number until the operator is found. Create a tree leaf containing the operator, with a left branch containing the first number read, and a right branch being empty. Read the next symbol: if it's a parenthesis, start over but with the right branch becoming a new "leaf" to hold the next operator. If it's another number, put it into the right branch. Now simplify by applying the leaf operator to both its branches, and storing the result inside the leaf and clipping the branches. Read the next symbol, if it's an operator create a leaf with a left branch containing the resulting value previously computed and a right branch containing nothing... repeat.

So for the above expression: we read 40, plus, 40, we stop and add those to get 80, we read times, we read 0, we stop and multiply those to get 0, we read plus, 1, stop and add to get 1, and stop with the answer 1.

Since according to rules like PEMDAS the answer should be 41, we say the above expression was "malformed". You can get the right answer by adding parentheses in the right places, or you can use a more complicated parser that conforms to PEMDAS.

Why do we want PEMDAS's answer anyway? The answer comes from algebra. $$40 + 40*0 + 1 = x$$ is the algebraic question, which we must apply transformations to using algebraic rules. The particular rules are chosen in the same way that the particular parsing method is chosen--because we want the answer to be a particular something. In this case, we can use real-valued definitions and axioms of a vector space. The definitions describe addition and multiplication as operations on two numbers. The associativity axiom lets us rewrite as $$40 + 1 + 40*0 = x$$. Applying the definition of times, we rewrite as $$40 + 1 + 0 = x$$. Applying the definition of plus, we reduce to $$41 + 0 = x$$. Applying the definition of plus again, we reduce to $$41 = x$$.

Note that the definition of scalar multiplication implicitly takes precedence over addition. From $$40 + 1 + 40*0 = x$$, we might have said $$41 + 40*0 = 81*0 = 0 = x$$. But from the definitions, "40+1" constitute two distinct elements "40" and "1", while "40*0" does not--it constitutes one distinct element "x" in X, which in this case is "0". So the road to $$x=0$$ would be erroneous within our vector space axioms. The associative transformation above was also unnecessary since that multiplicative substitution takes precedence regardless (just like PEMDAS says), it's there just for fun.

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Posted on 2011-10-15 by Jach

Tags: fodder, math

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