Editing Operator associativity (section)

== Examples ==

Associativity is only needed when the operators in an expression have the same precedence. Usually <code>+</code> and <code>-</code> have the same precedence. Consider the expression <code>7 - 4 + 2</code>. The result could be either <code>(7 - 4) + 2 = 5</code> or <code>7 - (4 + 2) = 1</code>. The former result corresponds to the case when <code>+</code> and <code>-</code> are left-associative, the latter to when <code>+</code> and <code>-</code> are right-associative.

In order to reflect normal usage, [[addition]], [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]] operators are usually left-associative,<ref>Education Place: [http://eduplace.com/math/mathsteps/4/a/index.html The Order of Operations]</ref><ref>[[Khan Academy]]: [https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations The Order of Operations], timestamp [https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s 5m40s]</ref><ref>Virginia Department of Education: [http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 Using Order of Operations and Exploring Properties], section 9</ref> while for an [[exponentiation]] operator (if present)<ref name="Codeplea_2016">[https://codeplea.com/exponentiation-associativity-options Exponentiation Associativity and Standard Math Notation] Codeplea. 23 Aug 2016. Retrieved 20 Sep 2016.</ref>{{Better source needed|reason=A [[WP:RSSELF|blog isn't a reliable source]].|date=May 2024}} there is no general agreement. Any [[assignment (computer science)|assignment]] operators are typically right-associative. To prevent cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.

=== A detailed example ===

Consider the expression <code>5^4^3^2</code>, in which <code>^</code> is taken to be a right-associative exponentiation operator. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the right-associativity of <code>^</code>, in the following way:
# Term <code>5</code> is read.
# Nonterminal  <code>^</code> is read. Node: "<code>5^</code>".
# Term <code>4</code> is read. Node: "<code>5^4</code>".
# Nonterminal <code>^</code> is read, triggering the right-associativity rule. Associativity decides node: "<code>5^(4^</code>".
# Term <code>3</code> is read. Node: "<code>5^(4^3</code>".
# Nonterminal <code>^</code> is read, triggering the re-application of the right-associativity rule. Node "<code>5^(4^(3^</code>".
# Term <code>2</code> is read. Node "<code>5^(4^(3^2</code>". 
# No tokens to read.  Apply associativity to produce parse tree "<code>5^(4^(3^2))</code>".
This can then be evaluated depth-first, starting at the top node (the first <code>^</code>):
# The evaluator walks down the tree, from the first, over the second, to the third <code>^</code> expression.
# It evaluates as: 3{{sup|2}} = 9. The result replaces the expression branch as the second operand of the second <code>^</code>.
# Evaluation continues one level up the [[parse tree]] as: 4{{sup|9}} = {{formatnum:262144}}. Again, the result replaces the expression branch as the second operand of the first <code>^</code>. 
# Again, the evaluator steps up the tree to the root expression and evaluates as: 5{{sup|262144}} ≈ {{val|6.2060699|e=183230}}. The last remaining branch collapses and the result becomes the overall result, therefore completing overall evaluation.
A left-associative evaluation would have resulted in the parse tree  <code>((5^4)^3)^2</code> and the completely different result (625{{sup|3}}){{sup|2}} = {{formatnum:244140625}}{{sup|2}} ≈ {{val|5.9604645|e=16}}.