Rational fraction approximation
[N,D] = rat(X)
[N,D] = rat(X,tol)
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. The rat function attempts to do this. Rational approximations are generated by truncating continued fraction expansions. The rats function calls rat, and returns strings.
Ordinarily, the statement
s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7
s = 0.7595
the printed result is
s = 319/420
This is a simple rational number. Its denominator is 420, the least common multiple of the denominators of the terms involved in the original expression. Even though the quantity s is stored internally as a binary floating-point number, the desired rational form can be reconstructed.
To see how the rational approximation is generated, the statement rat(s) produces
1 + 1/(-4 + 1/(-6 + 1/(-3 + 1/(-5))))
And the statement
[n,d] = rat(s)
n = 319, d = 420
The mathematical quantity π is not a rational number, but the MATLAB® quantity pi that approximates it is a rational number. pi is the ratio of a large integer and 252:
However, this is not a simple rational number. The value printed for pi with format rat, or with rats(pi), is
This approximation was known in Euclid's time. Its decimal representation is
and so it agrees with pi to seven significant figures. The statement
3 + 1/(7 + 1/(16))
This shows how the 355/113 was obtained. The less accurate, but more familiar approximation 22/7 is obtained from the first two terms of this continued fraction.
The rat(X) function approximates each element of 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). For x = sqrt(2) , the error with k terms is about 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.