Search results
Results from the WOW.Com Content Network
The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0, a mathematical truth. But the same substitution applied to the original equation results in x/6 + 0/0 = 1, which is mathematically meaningless.
(The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ...
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have the same denominator then they can be added by adding their numerators and keeping the common denominator.
Let = + + +be a polynomial, and , …, be its complex roots (not necessarily distinct). For any constant c, the polynomial whose roots are +, …, + is = = + + +.If the coefficients of P are integers and the constant = is a rational number, the coefficients of Q may be not integers, but the polynomial c n Q has integer coefficients and has the same roots as Q.
In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L .
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.