Ad
related to: how to prove something algebraically related
Search results
Results From The WOW.Com Content Network
The legal term probity means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status. [6] Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. [7]
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
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. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...
The condition J F ≠ 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to F exists at every point where J F is non-zero. For example, the map x → x + x 3 has a smooth global inverse, but the inverse is not polynomial.
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem.Charles Hermite first proved the simpler theorem where the α i exponents are required to be rational integers and linear independence is only assured over the rational integers, [4] [5] a result sometimes referred to as Hermite's theorem. [6]
There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian geometric arguments. The main idea here is to prove that the existence of a non-constant polynomial p(z) without zeros implies the existence of a flat Riemannian metric over the sphere S 2. This leads to a ...