什么是运算符的结合性? 为什么它很重要？

It's the order of binding of operands to an operator. Basically:

a - b + c

might be evaluated as (assuming - and + have the same precedence):

((a - b) + c) or,
(a - (b + c))

If operators are left associative (bind immediately to the left operand), it'll be evaluated as the first. If they are right associative, it'll be evaluated as the second.

There are three kinds of associativity:

The Associative property in mathematics

Order of Operations in programming languages

Associativity in CPU caches.

The Associative property in mathematics is a property of operators such as addition (+). This property allows you to rearrange parentheses without changing the value of a statement, i.e.:

(a + b) + c = a + (b + c)

In programming languages, the associativity (or fixity) of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses; i.e. in what order each operator is evaluated. This can differ between programming languages.

In CPU caches, associativity is a method of optimizing performance.

If you mean operator associativity:

It defines the way expressions are parsed. It gives a standard, so every expression is parsed the same way.

It's mostly important for operations which have the same precedense, when there could be side effects.

For operators, associativity means that when the same operator appears in a row, then which operator occurence we apply first. In the following, let Q be the operator

a Q b Q c

If Q is left associative, then it evaluates as

(a Q b) Q c

And if it is right associative, then it evaluates as

a Q (b Q c)

It's important, since it changes the meaning of an expression. Consider the division operator with integer arithmetic, which is left associative

4 / 2 / 3    <=>    (4 / 2) / 3    <=> 2 / 3     = 0

If it were right associative, it would evaluate to an undefined expression, since you would divide by zero

4 / 2 / 3    <=>    4 / (2 / 3)    <=> 4 / 0     = undefined

it is the order of evaluate for operators of the same precedence. The LEFT TO RIGHT or RIGHT TO LEFT order matters. For

3 - 2 - 1

if it is LEFT to RIGHT, then it is

(3 - 2) - 1

and is 0. If it is RIGHT to LEFT, then it is

3 - (2 - 1)

and it is 2. In most languages, we say that the minus operator has a LEFT TO RIGHT associativity.

Update 2020:

The situation about 3 - 2 - 1 might seem trivial, if the claim is, "of course we do it from left to right". But in other cases, such as if done in Ruby or in NodeJS:

$ irb
2.6.3 :001 > 2 ** 3 ** 2
=> 512

The ** is "to the power of" operator. The associativity is from right to left. And it is

 2 ** (3 ** 2)

which is 2 ** 9, i.e., 512, instead of

(2 ** 3) ** 2

which is 8 ** 2, i.e., 64.

If you are referring to "operator associativity" - it is how a language determines how operators of the same precedence are grouped in the absence of parentheses.

For example, the + and - operators in C-based languages have the same precedence. When you write an expression that uses both of them (without parentheses) the compiler must determine what order to evaluate them in.

If you write 12 - 5 + 3, the possible evaluations include:

1. (12 - 5) + 3 = 10
2. 12 - (5 + 3) = 4

Depending on the order you evaluate the expression in, you can get different results. In C-based languages, + and - have left associativity, which means that the expression above would evaluate as the first case.

All language have strongly-defined rules for both precedence and associativity. You can learn more about the rules for C# here. The general concepts of operator associativity and precedence are well covered on wikipedia.

Most of the previous examples have used constants. If the arguments happen to be function calls, the order that the calls are made in may be determined by the association rules, depending of course on your compiler. And if those functions have side effects ..

We all know that precedence is important but so is associativity in interpreting the meaning of an expression. For a really simple intro try Power of Operators.

Simple!!

Left Associative means we evaluate our expression from left to right


Right Associative means we evaluate our expression from right to left

We know *, /, and % have same precedence, but as per associativity, answer may change:

For eg: We have expression: 4 * 8 / 2 % 5

Left associative:   (4 * 8) / 2 % 5 ==> (32 / 2) % 5 ==> 16 % 5 ==> 1


Right associative:  4 * 8 /(2 % 5) ==>  4 * ( 8 / 2) ==> 4 * 4 ==> 16

Associativity comes under the order of computation in the programming language concepts. The order of computation determines the meaning of the expression. It has two main rules,

1. Precedence rules
2. Associativity rules

precedence rules define the order in which "adjacent" operators of different types are evaluated. Every programming language has its own operator precedence table regarding its operators.

Coming back to the associativity,

It defines the order of execution of adjacent operations with the same precedence. It has 3 flavors,

left-associativity
right-associativity
non-associativity

If an operator is left-associative it evaluates from left to right likewise if it is right-associative it evaluates from right to left.