Extract Numerators and Denominators of Rational Expressions

To extract the numerator and denominator of a rational symbolic expression, use the numden function. The first output argument of numden is a numerator, the second output argument is a denominator. Use numden to find numerators and denominators of symbolic rational numbers.

[n,d] = numden(1/sym(3))

n =
1
 
d =
3

Use numden to find numerators and denominators of a symbolic expressions.

syms x y
[n,d] = numden((x^2 - y^2)/(x^2 + y^2))

n =
x^2 - y^2
 
d =
x^2 + y^2

Use numden to find numerators and denominators of symbolic functions. If the input is a symbolic function, numden returns the numerator and denominator as symbolic functions.

syms f(x) g(x)
f(x) = sin(x)/x^2;
g(x) = cos(x)/x;
[n,d] = numden(f)

n(x) =
sin(x)
 
d(x) =
x^2

[n,d] = numden(f/g)

n(x) =
sin(x)
 
d(x) =
x*cos(x)

numden converts the input to its one-term rational form, such that the greatest common divisor of the numerator and denominator is 1. Then it returns the numerator and denominator of that form of the expression.

[n,d] = numden(x/y + y/x)

n =
x^2 + y^2
 
d =
x*y

numden works on vectors and matrices. If an input is a vector or matrix, numden returns two vectors or two matrices of the same size as the input. The first vector or matrix contains numerators of each element. The second vector or matrix contains denominators of each element. For example, find numerators and denominators of each element of the 3-by-3 Hilbert matrix.

H = sym(hilb(3))

H =
[   1, 1/2, 1/3]
[ 1/2, 1/3, 1/4]
[ 1/3, 1/4, 1/5]

[n,d] = numden(H)

n =
[ 1, 1, 1]
[ 1, 1, 1]
[ 1, 1, 1]
 
d =
[ 1, 2, 3]
[ 2, 3, 4]
[ 3, 4, 5]