Difference between revisions of "Associative property of multiplication"
| Line 5: | Line 5: | ||
With more than three numbers, there can be many ways to do the multiplication. For example, with 4 numbers one of the ways is <math>2 \times 3 \times 4 \times 5 = ((2 \times 3) \times 4 )\times 5 = (6 \times 4 )\times 5 = 24\times 5 = 120</math>. | With more than three numbers, there can be many ways to do the multiplication. For example, with 4 numbers one of the ways is <math>2 \times 3 \times 4 \times 5 = ((2 \times 3) \times 4 )\times 5 = (6 \times 4 )\times 5 = 24\times 5 = 120</math>. | ||
| − | The associative property of multiplication is different from the commutitative property. For example, multiplication of matrices has the associative property but is not commutative. If <math>A</math>, <math>B</math>, and <math>C</math> are three matrices, then <math>A\times(B \times C)=(A\times B) \times C </math> but <math>A\times \neq B \times A </math>. | + | The associative property of multiplication is different from the commutitative property. For example, multiplication of matrices has the associative property but is not commutative. If <math>A</math>, <math>B</math>, and <math>C</math> are three matrices, then <math>A\times(B \times C)=(A\times B) \times C </math> but <math>A\times B \neq B \times A </math>. |
Revision as of 13:24, April 3, 2007
Basic multiplication is a pairwise operation. To multiply more than two numbers, you must combine them in pairs successively until all the numbers have been used. For example, to multiply 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 \times 4 \times 5} you first pick two consecutive numbers, say 3 and 4, and multiply 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 = 12 \times 5} . Now there are just two numbers remaining, which you can multiply to get the final answer: .
The associative property of multiplication is the fact that the answer does not depend on how the pairings are done. For example, we could have started with , and then done . We use parentheses to indicate the order of multiplication used. For example, writing means that you first multiply 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 \times 4} then multiply 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\times 4) \times 5 = 3 \times(4 \times 5)} .
With more than three numbers, there can be many ways to do the multiplication. 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 \times 3 \times 4 \times 5 = ((2 \times 3) \times 4 )\times 5 = (6 \times 4 )\times 5 = 24\times 5 = 120} .
The associative property of multiplication is different from the commutitative property. For example, multiplication of matrices has the associative property but is not commutative. If 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 A} , 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 B} , and 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 C} are three matrices, then 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 A\times(B \times C)=(A\times B) \times C } but 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 A\times B \neq B \times A } .