#### 4.3` `Numbers

Numbers in The Racket Guide introduces numbers.

All numbers are complex numbers. Some of them are real numbers, and all of the real numbers that can be represented are also rational numbers, except for +inf.0 (positive infinity), +inf.f (single-precision variant, when enabled via read-single-flonum), -inf.0 (negative infinity), -inf.f (single-precision variant, when enabled), +nan.0 (not-a-number), and +nan.f (single-precision variant, when enabled). Among the rational numbers, some are integers, because round applied to the number produces the same number.

See Reading Numbers for information on the syntax of number literals.

Orthogonal to those categories, each number is also either an exact number or an inexact number. Unless otherwise specified, computations that involve an inexact number produce inexact results. Certain operations on inexact numbers, however, produce an exact number, such as multiplying an inexact number with an exact 0. Operations that mathematically produce irrational numbers for some rational arguments (e.g., sqrt) may produce inexact results even for exact arguments.

In the case of complex numbers, either the real and imaginary parts are both exact or inexact with the same precision, or the number has an exact zero real part and an inexact imaginary part; a complex number with an exact zero imaginary part is a real number.

Inexact real numbers are implemented as double-precision IEEE floating-point numbers, also known as flonums, or as single-precision IEEE floating-point numbers, also known as single-flonums. Single-flonums are supported only when (single-flonum-available?) reports #t. Although we write +inf.f, -inf.f, and +nan.f to mean single-flonums, those forms read as double-precision flonums by default, since read-single-flonum is #f by default. When single-flonums are supported, inexact numbers are still represented as flonums by default, and single precision is used only when a computation starts with single-flonums.

Inexact numbers can be coerced to exact form, except for the inexact numbers +inf.0, +inf.f, -inf.0, -inf.f, +nan.0, and +nan.f, which have no exact form. Dividing a number by exact zero raises an exception; dividing a non-zero number other than +nan.0 or +nan.f by an inexact zero returns +inf.0, +inf.f, -inf.0 or -inf.f, depending on the sign and precision of the dividend. The +nan.0 value is not = to itself, but +nan.0 is eqv? to itself, and +nan.f is similarly eqv? but not = to itself. Conversely, (= 0.0 -0.0) is #t, but (eqv? 0.0 -0.0) is #f, and the same for 0.0f0 and -0.0f0 (which are single-precision variants). The datum -nan.0 refers to the same constant as +nan.0, and -nan.f is the same as +nan.f.

Calculations with infinities produce results consistent with IEEE double- or single-precision floating point where IEEE specifies the result; in cases where IEEE provides no specification, the result corresponds to the limit approaching infinity, or +nan.0 or +nan.f if no such limit exists.

The precision and size of exact numbers is limited only by available memory (and the precision of operations that can produce irrational numbers). In particular, adding, multiplying, subtracting, and dividing exact numbers always produces an exact result.

A fixnum is an exact integer whose two’s complement representation fits into 30 or 31 bits (depending on the Racket variant) on a 32-bit platform or 61 or 63 bits (depending on the Racket variant) on a 64-bit platform. No allocation is required when computing with fixnums. See also the racket/fixnum module, below.

Two fixnums that are = are also the same according to eq?. Otherwise, the result of eq? applied to two numbers is undefined, except that numbers produced by the default reader in read-syntax mode are interned and therefore eq? when they are eqv?.

Two real numbers are eqv? when they are both inexact with the same precision or both exact, and when they are = (except for +nan.0, +nan.f, 0.0, +0.0f0, -0.0, and -0.0f0, as noted above). Two complex numbers are eqv? when their real and imaginary parts are eqv?. Two numbers are equal? when they are eqv?.

See Reading Numbers for information on reading numbers and Printing Numbers for information on printing numbers.