Formatting a fraction as a mixed number

Question

Is there a command that will take a rational number and rewrite it in a mixed-number-like form? That is, I'd like to apply a command to something like 10/7 and get the result 1 + 3/7 (or 3/7 + 1 would be fine, too). With polynomial division, the Apart[] command does the trick pretty well, but I haven't been able to find anything comparable for numbers.

Answer 1

Here is a definition for mixedForm that works for all cases, i.e. proper and improper fractions and integers.

Clear[mixedForm]
mixedForm[Rational[x_, y_]] := 
    If[Abs@x > y, HoldForm[#1 + #2/y], x/y] & @@ (Sign@x QuotientRemainder[Abs@x, y])
mixedForm[x_Integer] := x

Some examples:

mixedForm /@ {2, 4/5, 10/3, -3/4, -5/2}
Out[1]= {2, 4/5, 3 + 1/3, -3/4, -2 - 1/2}

Compare with Eli's, which produces 0s if the number is an integer or a proper fraction

ImproperForm /@ {2, 4/5, 10/3, -3/4, -5/2}
Out[2]= {2 + 0, 0 + 4/5, 3 + 1/3, -1 + 1/4, -3 + 1/2}

Answer 2

Another solution based on FractionalPart and IntegerPart would be :

Fraction[x_Rational]:= 
    Function[{z, y}, If[z!=0, HoldForm[z + y], HoldForm[y]],    
                     {HoldAll}] @@ {IntegerPart[x], FractionalPart[x]}

Fraction[x_Integer] := x

This approach produces slightly different results than R.M.'s solution :

Fraction /@ {2, 4/5, 10/3, -3/4, -5/2}

{2, 4/5, 3 + 1/3, -3/4, -2 -1/2 }

Answer 3

Using the function from the notebook here:

ImproperForm[x_] := 
 Function[{z, y}, HoldForm[z + y], {HoldAll}] @@ {Floor[x],x - Floor[x]}

Usage:

In[12]:= ImproperForm[10/7]
Out[12]= 1+3/7

Answer 4

This should work with any real, rationalizing it (so, it becomes an approximate result)... The integer part of a negative number is considered the floor of the number. I'm not sure what's desired in those cases...

ClearAll[MixedForm];

Format[MixedForm[r_?NumericQ]] := Module[{rat = 
  If[Element[r, Rationals], r,Rationalize[N@r, 0]], x, y},
   {x, y} = Through@{Numerator, Denominator}[rat];
   With[{num = x, den = y},
    With[{quot = Quotient[num, den], rem = Mod[num, den]},
     Defer[quot] + Defer[rem/den] /. Defer[_?PossibleZeroQ] -> 0]]];

MixedForm /: (h_Symbol /; MemberQ[Attributes[h], NumericFunction])[
   bef___, MixedForm[stuff_], aft___] := MixedForm@h[bef, stuff, aft];

SetAttributes[MixedForm, {Listable, Flat}];

Unprotect[$OutputForms];
AppendTo[$OutputForms, MixedForm];
Protect[$OutputForms];

Ok, first, we test it

MixedForm /@ {2, 4/5, 10/3, -3/4, -5/2}

{2, 4/5, 3 + 1/3, -1 + 1/4, -3 + 1/2}

Works as expected, if you expected the integer part to be the floor (which I don't think it's the most natural thing but I'm not sure)

We see that the function can also take machine precision numbers

MixedForm /@ {2.`, 0.8`, 3.333333333333333`, -0.75`, -2.5`}

gives the same result.

For consistency with built-ins, the retuned cell has the label with //MixedForm prepended. Unlike some of the other options based on HoldForm, you can copy the result and do arithmetics with it... You can also do some arithmetics with the returned values

{2, 4/5, 10/3, -3/4, -5/2} // MixedForm

{2, 4/5, 3 + 1/3, -1 + 1/4, -3 + 1/2}



% + 9

{11, 9 + 4/5, 12 + 1/3, 8 + 1/4, 6 + 1/2}

EDIT

Added the Listable and Flat attribute and changed the previous definition to only match numeric values. Now we don't need to map it

{2, 4/5, 10/3, -3/4, -5/2} // MixedForm

returns what's expected... But we lose the label in these cases

Answer 5

Here is an attempt to implement both formatting for entire expressions and basic arithmetic.

EDIT: fixed according to Artes' solution.

ClearAll[mixedForm]    
Format[mixedForm[x : Except[_Rational]]] := x /. q_Rational :> mixedForm[q]    
Format[mixedForm[q_Rational]] ^:= Interpretation[
   If[Abs[q] < 1, q, HoldForm[# + #2] & @@ {IntegerPart@q, FractionalPart@q}],
   q
 ]

mixedForm[q_] + (mixedForm[x_] | x_) ^:= mixedForm[q + x]
mixedForm[q_] * (mixedForm[x_] | x_) ^:= mixedForm[q * x]
mixedForm[q_] ^ (mixedForm[x_] | x_) ^:= mixedForm[q ^ x]

Now:

ff[2, 4/5, 10/3, gg[-3/4, -5/2]] // mixedForm
(* Out[1]= ff[2, 4/5, 3 + 1/3, gg[-(3/4), -2 - 1/2]] *)

x = mixedForm[8/3];
y = mixedForm[9/5];    
x + y
(* Out[2]= 4 + 7/15 *)

x + 1/2
(* Out[3]= 3 + 1/6 *)