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Difference Of Squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity :a^2-b^2 = (a+b)(a-b) in elementary algebra. Proof The proof of the factorization identity is straightforward. Starting from the left-hand side, apply the distributive law to get :(a+b)(a-b) = a^2+ba-ab-b^2 By the commutative law, the middle two terms cancel: :ba - ab = 0 leaving :(a+b)(a-b) = a^2-b^2 The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring. Conversely, if this identity holds in a ring ''R'' for all pairs of elements ''a'' and ''b'', then ''R'' is commutative. To see this, apply the distributive law to the right-hand side of the equation and get :a^2 + ba - ab - b^2. For this to be equal to a^2 - b^2, we must have :ba - ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
