NumeratorDenominator

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`NumeratorDenominator`

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NumeratorDenominator[expr]

gives the list {Numerator[expr],Denominator[expr]} of expr.

Details and Options

Numerator picks out terms that do not have superficially negative exponents. Denominator picks out the remaining terms.
An exponent is "superficially negative" if it has a negative number as a factor.
The standard representation of rational expressions as products of powers means that you cannot simply use Part to extract numerators.
NumeratorDenominator can be used on rational numbers.
NumeratorDenominator automatically threads over lists.

Examples

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Basic Examples (3)Summary of the most common use cases

Extract the numerator and denominator of a rational number:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-i9875r

Out[1]=1

Extract the numerator and denominator of a rational expression:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-fp31w

Out[1]=1

Extract the numerator and denominator of a symbolic expression:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-xu5bwl

Out[1]=1

Scope (7)Survey of the scope of standard use cases

Rational numbers:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-9bhlk3

Out[1]=1

Gaussian rationals:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-jf4nld

Out[1]=1

Rational expressions:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-qieutj

Out[1]=1

Select terms with syntactically positive and negative exponents:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-dxef23

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlfmdeubbtq-esqxkf

Out[2]=2

Reconstruct the original expression:

In[3]:=3

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https://wolfram.com/xid/0mlfmdeubbtq-8dpag7

Out[3]=3

NumeratorDenominator automatically threads over lists:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-tm4

Out[1]=1

Compute the numerator and denominator over the integers modulo 5:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-9n7tba

Out[1]=1

Compute the numerator and denominator while incorporating common trigonometric identities:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-t43vxw

Out[1]=1

Options (2)Common values & functionality for each option

Modulus (1)

Find numerators and denominators over integers modulo :

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-x8541

Out[1]=1

Trig (1)

Numerators and denominators of trigonometric functions:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-ceg9xk

Out[1]=1

Applications (2)Sample problems that can be solved with this function

Compute the numerator and denominator with respect to a specified modulus:

In[7]:=7

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https://wolfram.com/xid/0mlfmdeubbtq-9fk5ir

Out[7]=7

Compute the limit at infinity of a rational function:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-q9u57j

Out[1]=1

Extract the numerator and denominator:

In[2]:=2

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https://wolfram.com/xid/0mlfmdeubbtq-400s8p

Out[2]=2

Apply Exponent to find the degree of the numerator and denominator:

In[3]:=3

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https://wolfram.com/xid/0mlfmdeubbtq-euku12

Out[3]=3

We can see that the denominator has a larger degree. Thus, the function approaches zero as grows large:

In[4]:=4

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https://wolfram.com/xid/0mlfmdeubbtq-yarit3

Out[4]=4

This can be confirmed using Limit:

In[5]:=5

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https://wolfram.com/xid/0mlfmdeubbtq-dp46bm

Out[5]=5

Properties & Relations (1)Properties of the function, and connections to other functions

An expression is a quotient of its numerator and denominator:

In[1]:=1

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https://wolfram.com/xid/0mlfmdeubbtq-3rgdv

In[2]:=2

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https://wolfram.com/xid/0mlfmdeubbtq-e4mi0f

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0mlfmdeubbtq-cprc6f

Out[3]=3

Neat Examples (1)Surprising or curious use cases

Plot semitransparent blue squares at positions {i, j} for which i/j is an irreducible fraction:

In[9]:=9

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https://wolfram.com/xid/0mlfmdeubbtq-f7byz5

Out[9]=9

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