Explain how and why the cancellation of 6’s in 16/64 to get 1/4 is a fallacious statement. Based on what we know from elementary and middle school teachers, most of us know that 16/64 correctly equals 1/4 because 16/64 is simplified with a common divisibility of 16. However, there is another way to prove that 16/64 equals 1/4 without dividing the numerator and denominator by 16. Who can explain how and why that method leads to a fallacious statement?

Baladdaa9 2022-07-16 Answered
Explain how and why the cancellation of 6’s in 16 64 to get 1 4 is a fallacious statement.
Based on what we know from elementary and middle school teachers, most of us know that 16/64 correctly equals 1/4 because 16/64 is simplified with a common divisibility of 16. However, there is another way to prove that 16/64 equals 1/4 without dividing the numerator and denominator by 16. Who can explain how and why that method leads to a fallacious statement?
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Answers (2)

escobamesmo
Answered 2022-07-17 Author has 18 answers
The way someone might have justified that:
"Just remember that
10 20 = 1 0 2 0 = 1 2
Therefore
16 64 = 1 6 6 4 = 1 4
There is actually a problem on project euler regarding this type of fractions. Those that can be fallaciously simplified to something that holds as true.
Why it does not work:
There is a widely-used simplification that is
a b a d = a b a d
That works because we have a product. The above fraction is just syntatic sugar for
a b 1 a 1 d
But the product is commutative and therefore we have
a b 1 a 1 d = a b 1 a 1 d = b d
The problem with the digits is that 16 is not 1 6 just as 64 6 4. That means 1 64 is not syntatic sugar for 1 6 1 4 and the 6s won't cancel. It only works when the numbers end in 0 because if k and j end in 0, then k is the product of k with 10 and j is the product of j with 10. Then we have:
k j = k 10 j 10 = k 10 j 10 = k j

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anudoneddbv
Answered 2022-07-18 Author has 2 answers
The wrong proof is more of a joke than a serious fallacy:
16 64 = 16 / / 64 = 1 4
This joke exploits the notational ambiguity that writing two symbols next to each other can either mean multiplication or -- if the symbols happen to be digits -- be part of the usual decimal notation for numbers, in which case it means something quite different from multiplying the digits together.
In the joke proof we pretend that 16 and 64 mean 1 6 and 6 4 (which of course they don't) and then "cancel the common factor" of 6
This doesn't really work because the 6 is not a factor.

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