Rational fraction approximation
Approximate the value of
using a rational representation of the quantity
The mathematical quantity
is not a rational number, but the quantity
pi that approximates it is a rational number since all floating-point numbers are rational.
Find the rational representation of
format rat pi
ans = 355/113
The resulting expression is a character vector. You also can use
rats(pi) to get the same answer.
rat to see the continued fractional expansion of
R = rat(pi)
R = 3 + 1/(7 + 1/(16))
The result is an approximation by continued fractional expansion. If you consider the first two terms of the expansion, you get the approximation
, which only agrees with
pi to 2 decimals.
However, if you consider all three terms printed by
rat, you can recover the value
355/113, which agrees with
pi to 6 decimals.
Specify a tolerance for additional accuracy in the approximation.
R = rat(pi,1e-7)
R = 3 + 1/(7 + 1/(16 + 1/(-294)))
The resulting approximation,
104348/33215, agrees with
pi to 9 decimals.
Create a 4-by-4 matrix.
format short; X = hilb(4)
X = 1.0000 0.5000 0.3333 0.2500 0.5000 0.3333 0.2500 0.2000 0.3333 0.2500 0.2000 0.1667 0.2500 0.2000 0.1667 0.1429
Express the elements of
X as ratios of small integers using
[N,D] = rat(X)
N = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 D = 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7
The two matrices,
View the elements of
X as ratios using
format rat X
X = 1 1/2 1/3 1/4 1/2 1/3 1/4 1/5 1/3 1/4 1/5 1/6 1/4 1/5 1/6 1/7
In this form, it is clear that
N contains the numerators of each fraction and
D contains the denominators.
X— Input array
Input array, specified as a numeric array of class
Complex Number Support: Yes
Tolerance, specified as a scalar.
N./D - X < tol. The default tolerance
R— Continued fraction
Continued fraction, returned as a character array with
m is the number of elements in
The accuracy of the rational approximation via continued fractions
increases with the number of terms.
Numerator, returned as a numeric array.
Denominator, returned as a numeric array.
Even though all floating-point numbers are rational numbers, it is sometimes desirable to approximate them by simple rational numbers, which are fractions whose numerator and denominator are small integers. Rational approximations are generated by truncating continued fraction expansions.
rat function approximates each element
X by a continued fraction of the form
The Ds are obtained by repeatedly picking
off the integer part and then taking the reciprocal of the fractional
part. The accuracy of the approximation increases exponentially with
the number of terms and is worst when
X = sqrt(2).
X = sqrt(2) , the error with
2.68*(.173)^k, so each additional term
increases the accuracy by less than one decimal digit. It takes 21
terms to get full floating-point accuracy.