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In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
The four 4th roots of −1, none of which are real The three 3rd roots of −1, one of which is a negative real. An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:
If the discriminant of such a polynomial is negative, then both roots of the quadratic equation have imaginary parts. In particular, if b and c are real numbers and b 2 − 4 c < 0, all the convergents of this continued fraction "solution" will be real numbers, and they cannot possibly converge to a root of the form u + iv (where v ≠ 0 ...
has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as: (+) = For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
In elementary mathematics, the additive inverse is often referred to as the opposite number, [3] [4] or its negative. [5] The unary operation of arithmetic negation [6] is closely related to subtraction [7] and is important in solving algebraic equations. [8] Not all sets where addition is defined have an additive inverse, such as the natural ...
In mathematics, a negative number is the opposite of a positive real number. [1] Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset.
The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root § Principal square root of a complex number.
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