The syntax for integer values is an optional sign (+ or -) followed by one or more digits. Ratios are written as an optional sign and a sequence of digits, representing the numerator, a slash (/), and another sequence of digits representing the denominator. All rational numbers are “canonicalized” as they’re read—that’s why 10 and 20/2 are both read as the same number, as are 3/4 and 6/8. Rationals are printed in “reduced” form—integer values are printed in integer syntax and ratios with the numerator and denominator reduced to lowest terms.

It’s also possible to write rationals in bases other than 10. If preceded by #B or , a rational literal is read as a binary number with 0 and 1 as the only legal digits. An #O or #o indicates an octal number (legal digits 0-7), and #X or #x indicates hexadecimal (legal digits 0-F or -f). You can write rationals in other bases from 2 to 36 with #nR where n is the base (always written in decimal). Additional “digits” beyond 9 are taken from the letters A-Z or a-z. Note that these radix indicators apply to the whole rational—it’s not possible to write a ratio with the numerator in one base and denominator in another. Also, you can write integer values, but not ratios, as decimal digits terminated with a decimal point.6 Some examples of rationals, with their canonical, decimal representation are as follows:

The basic format for floating-point numbers is an optional sign followed by a nonempty sequence of decimal digits possibly with an embedded decimal point. This sequence can be followed by an exponent marker for “computerized scientific notation.”8 The exponent marker consists of a single letter followed by an optional sign and a sequence of digits, which are interpreted as the power of ten by which the number before the exponent marker should be multiplied. The letter does double duty: it marks the beginning of the exponent and indicates what floating- point representation should be used for the number. The exponent markers s, f, d, l (and their uppercase equivalents) indicate short, single, double, and long floats, respectively. The letter e indicates that the default representation (initially single-float) should be used.

Numbers with no exponent marker are read in the default representation and must contain a decimal point followed by at least one digit to distinguish them from integers. The digits in a floating-point number are always treated as base 10 digits—the #B, #X, #O, and #R syntaxes work only with rationals. The following are some example floating-point numbers along with their canonical representation:

But you can write them however you want—if a complex is written with one rational and one floating-point part, the rational is converted to a float of the appropriate representation. Similarly, if the real and imaginary parts are both floats of different representations, the one in the smaller representation will be upgraded.

However, no complex numbers have a rational real component and a zero imaginary part—since such values are, mathematically speaking, rational, they’re represented by the appropriate rational value. The same mathematical argument could be made for complex numbers with floating-point components, but for those complex types a number with a zero imaginary part is always a different object than the floating-point number representing the real component. Here are some examples of numbers written the complex number syntax: