Associative property of addition
Basic addition is a pairwise operation. To add more than two numbers, you must combine them in pairs successively until all the numbers have been used. For example, to add Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = 3 + 4 + 5} you first pick two consecutive numbers, say 3 and 4, and add them: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = 7 + 5} . Now there are just two numbers remaining, which you can add to get the final answer: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = 12} .
The associative property of addition is the fact that the answer does not depend on how the pairings are done. For example, we could have started with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 + 5 = 9} , and then done Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = 3 + 9 = 12} . We use parentheses to indicate the order of addition used. For example, writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3 + 4)+ 5} means that you first add Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 + 4} then add the result by 5. The associative property is expressed by the formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3+ 4) + 5 = 3 +(4 + 5)} .
With more than three numbers, there can be many ways to do the addition. For example, with 4 numbers one of the ways is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 + 3 + 4 + 5 = ((2 + 3) + 4 )+ 5 = (5 + 4 )+ 5 = 9+ 5 = 14} .
The associative property of addition is different from the commutative property.