BUILT-IN MATHEMATICA SYMBOL

Rationalize

Rationalize[x]
converts an approximate number x to a nearby rational with small denominator.

yields the rational number with smallest denominator that lies within dx of x.

Rationalize[x] yields x unchanged if there no rational number close enough to x to satisfy the condition , with chosen to be .

Convert to a rational number:

Out[1]=

Find rational approximations to within a given tolerance:

Rationalize works with exact numbers:

Rationalize all numbers in an expression:

No rational number is by default considered "close enough" to N[Pi]:

Force a rational approximation to be found:

Successive rational approximations to

:

Plot the error in progressively better rational approximations to

:

Plot the error in progressively better approximations to

:

Create a polynomial with approximate numerical coefficients:

Find a rough approximation involving only rational numbers:

Factor the result:

If Rationalize returns a rational number

, then

:

When Rationalize[x] returns x unchanged, there is no rational number satisfying this:

Get the rational approximations with smallest denominator error dx through machine precision:

The residual of the inequality is positive for all of these rational approximations:

both give rational approximations for real x:

gives a rational that is equivalent to x up to the precision of x:

gets a rational directly from the bitwise representation of x:

Rationalize and RootApproximant both give exact quantities approximating real x:

RootApproximant[x] gives an algebraic number equivalent to x up to the precision of x:

gives a rational number equivalent to x up to the precision of x: