Node:Reals and Rationals, Next:, Previous:Integers, Up:Numbers

#### 21.2.3 Real and Rational Numbers

Mathematically, the real numbers are the set of numbers that describe all possible points along a continuous, infinite, one-dimensional line. The rational numbers are the set of all numbers that can be written as fractions P/Q, where P and Q are integers. All rational numbers are also real, but there are real numbers that are not rational, for example the square root of 2, and pi.

Guile represents both real and rational numbers approximately using a floating point encoding with limited precision. Even though the actual encoding is in binary, it may be helpful to think of it as a decimal number with a limited number of significant figures and a decimal point somewhere, since this corresponds to the standard notation for non-whole numbers. For example:

```0.34
-0.00000142857931198
-5648394822220000000000.0
4.0
```

The limited precision of Guile's encoding means that any "real" number in Guile can be written in a rational form, by multiplying and then dividing by sufficient powers of 10 (or in fact, 2). For example, `-0.00000142857931198` is the same as `142857931198` divided by `100000000000000000`. In Guile's current incarnation, therefore, the `rational?` and `real?` predicates are equivalent.

Another aspect of this equivalence is that Guile currently does not preserve the exactness that is possible with rational arithmetic. If such exactness is needed, it is of course possible to implement exact rational arithmetic at the Scheme level using Guile's arbitrary size integers.

A planned future revision of Guile's numerical tower will make it possible to implement exact representations and arithmetic for both rational numbers and real irrational numbers such as square roots, and in such a way that the new kinds of number integrate seamlessly with those that are already implemented.

 real? obj Scheme Procedure scm_real_p (obj) C Function
 Return `#t` if obj is a real number, else `#f`. Note that the sets of integer and rational values form subsets of the set of real numbers, so the predicate will also be fulfilled if obj is an integer number or a rational number.

 rational? x Scheme Procedure scm_real_p (x) C Function
 Return `#t` if x is a rational number, `#f` otherwise. Note that the set of integer values forms a subset of the set of rational numbers, i. e. the predicate will also be fulfilled if x is an integer number. Real numbers will also satisfy this predicate, because of their limited precision.