#### 4.3.4 Fixnum arithmetic ¶

The usual arithmetic operations like `+` and `<` are called generic arithmetic operations because they work for all (appropriate) kinds of number.

A fixnum is an exact integer that is small enough to fit in a machine word. In MIT/GNU Scheme, fixnums are 26 bits on 32-bit machines, and 58 bits on 64-bit machines; it is reasonable to assume that fixnums are at least 24 bits. Fixnums are signed; they are encoded using 2’s complement.

All exact integers that are small enough to be encoded as fixnums are always encoded as fixnums—in other words, any exact integer that is not a fixnum is too big to be encoded as such. For this reason, small constants such as `0` or `1` are guaranteed to be fixnums. In addition, the lengths of and valid indexes into strings and vectors are also always fixnums.

If you know that a value is always a small fixnum, you can substitute the equivalent fixnum operation for the generic operation. However, care should be exercised: if used improperly, these operations can return incorrect answers, or even malformed objects that confuse the garbage collector. For a listing of all fixnum operations, see Fixnum Operations in MIT/GNU Scheme Reference Manual.

A fruitful area for inserting fixnum operations is in the index operations in tight loops.