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In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
In elementary arithmetic, a standard algorithm or method is a specific method of computation which is conventionally taught for solving particular mathematical problems. . These methods vary somewhat by nation and time, but generally include exchanging, regrouping, long division, and long multiplication using a standard notation, and standard formulas for average, area, and vol
The third power of the trinomial a + b + c is given by (+ +) = + + + + + + + + +. This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c.. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0)Each equation follows by definition [A1]; the first with a + b, the second with b.
Let R be a commutative ring and M an R-module.Let E = R ⊕ M be the direct sum of abelian groups. Define the multiplication on E by (,) (,) = (, +).Note that identifying (a, x) with a + εx where ε squares to zero and expanding out (a + εx)(b + εy) yields the above formula; in particular we see that E is a ring.
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
Police in south Texas are investigating after three police officers were stabbed Tuesday evening while responding to a domestic violence call, officials said.
The complex numbers form a 2-dimensional commutative algebra over the real numbers. The quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions). Every polynomial ring R[x 1, ..., x n] is a commutative R-algebra.