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rational, rationalize

rational, rationalize Function

Syntax:

rational number → rational

rationalize number → rational

Arguments and Values:

number—a real.

rational—a rational.

Description:

rational and rationalize convert reals to rationals.

If number is already rational, it is returned.

If number is a float, rational returns a rational that is mathematically equal in value to the float. rationalize returns a rational that approximates the float to the accuracy of the underlying floating-point representation.

rational assumes that the float is completely accurate.

rationalize assumes that the float is accurate only to the precision of the floating-point representation.

Examples:

(rational 0) → 0 
(rationalize -11/100) → -11/100 
(rational .1) → 13421773/134217728 ;implementation-dependent 
(rationalize .1) → 1/10

Affected By:

The implementation.

Exceptional Situations:

Should signal an error of type type-error if number is not a real. Might signal arithmetic-error.

Notes:

It is always the case that

(float (rational x) x) ≡ x

and

(float (rationalize x) x) ≡ x

That is, rationalizing a float by either method and then converting it back to a float of the same format produces the original number.

Expanded Reference: rational, rationalize​

Converting floats with rational​

rational converts a float to a rational that is mathematically exactly equal to the float's value. Since floats are stored in binary, the result can have a large denominator.

(rational 0)
=> 0
(rational 0.5)
=> 1/2
(rational 0.25)
=> 1/4
(rational .1)
=> 13421773/134217728

Converting floats with rationalize​

rationalize finds a simpler rational approximation that is within the floating-point precision of the input.

(rationalize 0.5)
=> 1/2
(rationalize 0.1)
=> 1/10
(rationalize 0.33)
=> 33/100

Difference between rational and rationalize​

The key distinction is that rational treats the float as exact, while rationalize accounts for float imprecision.

;; 0.1 cannot be exactly represented in binary floating point
(rational 0.1)
=> 13421773/134217728
(rationalize 0.1)
=> 1/10

;; Both give the same result when the float is exact
(rational 0.5)
=> 1/2
(rationalize 0.5)
=> 1/2

Identity for rational inputs​

If the argument is already rational (integer or ratio), both functions return it unchanged.

(rational 5)
=> 5
(rationalize 5)
=> 5
(rational 3/7)
=> 3/7
(rationalize 3/7)
=> 3/7
(rationalize -11/100)
=> -11/100

Round-trip guarantee​

Converting to rational and back to float always recovers the original float.

(float (rational 3.14) 1.0)
=> 3.14
(float (rationalize 3.14) 1.0)
=> 3.14