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This is true for all values of , so the solution set is all real numbers. But clearly not all real numbers are solutions to the original equation. But clearly not all real numbers are solutions to the original equation.
The set of all real numbers is uncountable, ... (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, ...
More generally, the solution set to an arbitrary collection E of relations (E i) (i varying in some index set I) for a collection of unknowns (), supposed to take values in respective spaces (), is the set S of all solutions to the relations E, where a solution () is a family of values (()) such that substituting () by () in the collection E makes all relations "true".
The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions
[24] [25] Cantor described in terms of ternary expansions, as "the set of all real numbers given by the formula: = + + + + where the coefficients arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."
Taking it down all the way to 2 will be the solution to the Twin Prime Conjecture. ... The major example Cantor proved is that the set of real numbers is bigger, written |ℝ|>ℵ₀. But the ...
A natural example of such a question concerning positive-dimensional systems is the following: decide if a polynomial system over the rational numbers has a finite number of real solutions and compute them. A generalization of this question is find at least one solution in each connected component of the set of real solutions of a polynomial ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...