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evaluation - How to NOT simplify fraction


How can I prevent Mathematica from simplifying fractions? What I would like to do is run a test on a list of fractions to see how many are in their reduced form.


For example, I have a list:


1/10, 2/10, 5/10

I do not want Mathematica to treat it as:


1/10, 1/5, 1/2


I would like it left exactly as it was originally entered for comparison, i.e. if one list contains 2/10 and the other list contains 1/5 they should be treated as different elements.


I tried converting the elements to strings but Mathematica still simplified the fractions...


What is the best way to maintain the entire list in the original form?


Something like:


a = {1/10, 2/10, 3/10, 4/10, 5/10};
Internal`RationalNoReduce[Numerator[a], Denominator[a]]

still gives


Internal`RationalNoReduce[{1, 1, 3, 2, 1}, {10, 5, 10, 5, 2}]


Answer



Your expression will be evaluated unless it is entered in a held form, or a form that natively does not evaluate. For example:


a = Hold[1/10, 2/10, 3/10, 4/10, 5/10];

b = {{1,10}, {2,10}, {3,10}, {4,10}, {5,10}};

The second representation is a bit easier to work with as the first, which while not evaluating is nevertheless parsed differently from what you might expect:


a // FullForm



Hold[Times[1, Power[10, -1]], Times[2, Power[10, -1]], Times[3, Power[10, -1]], 
 Times[4, Power[10, -1]], Times[5, Power[10, -1]]]

Note that every fraction is not parsed as Rational but as Times and Power.


You can work with either format however. First the easy example (ref: Apply):


GCD @@@ b


{1, 2, 1, 2, 5}


A GCD of one means that the fraction is in reduced form, while any other value means it is not. You could for example Select the fractions that are or are not in reduced form:


Select[b, 1 == GCD @@ # &]


{{1, 10}, {3, 10}}

To work with the first format you could convert it to the second format with:


List @@ (a /. n_*(d_^-1) :> {n, d})



{{1, 10}, {2, 10}, {3, 10}, {4, 10}, {5, 10}}

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