| Difference Of Two Squares |
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| elementary algebra | |
| mathematical identities | |
a from elementary to get a sum of four terms, and set ba as an application of the Commutative Law . The resulting identity is one of the most commonly used in all of mathematics. The proof just given indicates the scope of the identity in ''R''. Also, conversely, if this identity holds in a Ring ''R'' for all pairs of elements ''a'' and ''b'' of the ring, then ''R'' is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get a and if this is equal to ''a''2-''b''2, then we have a and by Associativity and the rule that ''r'' − ''r'' = 0, we can rewrite this as ba If the original identity holds, then, we have ''ba'' − ''ab'' = 0 for all pairs ''a'', ''b'' of elements of ''R'', so the ring ''R'' is commutative. |
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