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Numerator

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Numerator[expr]
    gives the numerator of expr regarded as a rational expression.
    Picks out factors of expr that do not carry a superficially negative
    exponent; constants and symbols pass through as-is.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Numerator[(x-1)(x-2)/(x-3)^2]
Out[1]= (-2 + x) (-1 + x)

In[2]:= Numerator[3/7 + I/11]
Out[2]= 33 + 7*I

Implementation notes

builtin_numerator calls the shared extract_num_den splitter and returns the numerator (freeing the denominator); Denominator is the mirror. extract_num_den handles literal rationals (n/d), complex numbers (clearing the common denominator of the real/imaginary parts), Power[b, e]/Exp[e] (a negative integer or rational exponent — or a Plus exponent with superficially-negative terms — moves the factor into the denominator), and Times[...] (recurse on each factor, partition the results into numerator and denominator products). Anything else is its own numerator over denominator 1. It does not combine a Plus over a common denominator — that is Together's job — so Numerator[a/b + c/d] returns the input's surface numerator, not the combined one. Numerator carries ATTR_LISTABLE | ATTR_PROTECTED.

  • Protected, Listable.
  • Picks out terms which do not have superficially negative exponents.
  • Can be used on rational and complex numbers.

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Numerator[6/8]
Out[1]= 3

In[1]:= Numerator[(x+1)/(x-1)]
Out[1]= 1 + x

In[1]:= Numerator[x^(-2)]
Out[1]= 1

In[1]:= Numerator[a/b + c/d]
Out[1]= a/b + c/d

Combine the sum into a single fraction first, then Numerator returns the genuine combined top:

In[1]:= Numerator[Together[a/b + c/d]]
Out[1]= b c + a d

Several factors are sorted into numerator and denominator by the sign of their exponents — only z^(-1) is pushed below the bar:

In[1]:= Numerator[2 x/(3 y) * z^(-1)]
Out[1]= 2 x

Notes

Numerator extracts the numerator of the structural rational form of its argument. A rational constant is first reduced to lowest terms, so Numerator[6/8] = 3. For symbolic quotients it returns the literal top of the expr/expr form, giving 1 + x for (x+1)/(x-1). Factors with negative exponents are treated as denominators, so Numerator[x^(-2)] = 1. Note that Mathilda does not auto-combine a sum into a single fraction first: Numerator[a/b + c/d] returns the unevaluated sum, since the expression is a Plus, not a single quotient. Apply Together first if you want the combined numerator.