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# Rationalize

 Rationalize[x]converts an approximate number x to a nearby rational with small denominator. Rationalizeyields 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:
Convert to a rational number:
 Out[1]=
 Scope   (5)
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:
 Applications   (3)
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:
SetPrecision and Rationalize both give rational approximations for real x:
Rationalize gives a rational that is equivalent to x up to the precision of x:
SetPrecision 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:
Rationalize gives a rational number equivalent to x up to the precision of x:
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