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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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?

escobamesmo · Accepted Answer

The way someone might have justified that:&quot;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&#039;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      ′

anudoneddbv · Accepted Answer

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&#039;t) and then &quot;cancel the common factor&quot; of   6This doesn&#039;t really work because the   6 is not a factor.

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