+ right-expr
- right-expr
left-expr + right-expr
left-expr - right-expr
left-expr * right-expr
left-expr / right-expr
left-expr mod right-expr
left-expr ^ right-expr
The arithmetic operators are +, -, *, /, mod, and ^.
The meanings (semantic) of those operators generally depend on the types
of the operands involved, and they are defined in the various chapters
describing the types. However basically the meanings are as follows.
+ denotes the addition, and - the subtraction of ring and field
elements. * is the multiplication of group elements, / is the
multiplication of the left operand with the inverse of the right operand.
mod is only defined for integers and rationals and denotes the modulo
operation. + and - can also be used as unary operations. The unary
+ is ignored and unary - is equivalent to multiplication by -1. ^
denotes powering of a group element if the right operand is an integer,
and is also used to denote operation if the right operand is a group
element.
The precedence of those operators is as follows. The powering operator
^ has the highest precedence, followed by the unary operators + and
-, which are followed by the multiplicative operators *, /, and
mod, and the additive binary operators + and - have the lowest
precedence. That means that the expression -2 ^ -2 * 3 + 1 is
interpreted as (-(2 ^ (-2)) * 3) + 1. If in doubt use parentheses
to clarify your intention.
The associativity of the arithmetic operators is as follows.^ is not
associative, i.e., it is illegal to write 2^3^4, use parentheses to
clarify whether you mean (2^3) ^ 4 or 2 ^ (3^4). The unary
operators + and - are right associative, because they are written to
the left of their operands. *, /, mod, +, and - are all left
associative, i.e., 1-2-3 is interpreted as (1-2)-3 not as 1-(2-3).
Again, if in doubt use parentheses to clarify your intentions.
The arithmetic operators have higher precedence than the comparison
operators (see Comparisons and In) and the logical operators (see
Operations for Booleans). Thus, for example, a * b = c and
d is interpreted, ((a * b) = c) and d.
gap> 2 * 2 + 9; # a very simple arithmetic expression
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GAP 3.4.4