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  2. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form . In general, the sine and cosine of most angles of the form β / 2 n {\displaystyle \beta /2^{n}} can be expressed using nested square roots of 2 in terms of β {\displaystyle \beta } .

  3. Methods of computing square roots - Wikipedia

    en.wikipedia.org/wiki/Methods_of_computing...

    A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...

  4. Conjugate (square roots) - Wikipedia

    en.wikipedia.org/wiki/Conjugate_(square_roots)

    In particular, the two solutions of a quadratic equation are conjugate, as per the in the quadratic formula =. Complex conjugation is the special case where the square root is i = − 1 , {\displaystyle i={\sqrt {-1}},} the imaginary unit .

  5. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

  6. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    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 a + c = x ± y {\displaystyle {\sqrt {a+{\sqrt {c}}}}={\sqrt {x}}\pm {\sqrt {y}}} if and only if a 2 − c ...

  7. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    For example, the golden ratio, (+) /, is an algebraic number, because it is a root of the polynomial x 2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number + is algebraic because it is a root of x 4 + 4.

  8. Integer square root - Wikipedia

    en.wikipedia.org/wiki/Integer_square_root

    One can prove [citation needed] that = is the largest possible number for which the stopping criterion | + | < ensures ⌊ + ⌋ = ⌊ ⌋ in the algorithm above.. In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than 1 should be used to protect against round-off errors.

  9. Equating coefficients - Wikipedia

    en.wikipedia.org/wiki/Equating_coefficients

    A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals + to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that