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The problem is that we divided both sides by , which involves the indeterminate operation of dividing by zero when = It is generally possible (and advisable) to avoid dividing by any expression that can be zero; however, where this is necessary, it is sufficient to ensure that any values of the variables that make it zero also fail to satisfy ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP ...
Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them. Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work. [126]
In some interpretations of the Marxian transformation problem, total "(production) prices" for output must equal total "values" by definition, and total profits must by definition equal total surplus value. However, Marx himself explicitly denied in chapter 49 of the third volume of Das Kapital that such an exact mathematical identity actually ...
The problem is that what is "socially necessary" depends entirely on whether or not there is demand for the finished product, i.e., the knotted cord. In this way, introducing the "socially necessary" qualifier into the labor theory of value simply converts the theory into a roundabout and imprecise description of supply and demand.
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.
In Theories of Surplus Value (1862–1863), he discusses the problem very clearly. [58] His first attempt at a solution occurs in a letter to Engels, dated 2 August 1862. [59] In Capital, Volume I (1867) [60] he noted that "many intermediate terms" were still needed in his progressing narrative, to arrive at the answer.