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  2. Rationalisation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Rationalisation_(mathematics)

    In elementary algebra, root rationalisation (or rationalization) is a process by which radicals in the denominator of an algebraic fraction are eliminated.. If the denominator is a monomial in some radical, say , with k < n, rationalisation consists of multiplying the numerator and the denominator by , and replacing by x (this is allowed, as, by definition, a n th root of x is a number that ...

  3. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_decomposition

    In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

  4. Conjugate (square roots) - Wikipedia

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

    As (+) = and (+) + =, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator , by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation ).

  5. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, tan ⁠ 3 π / 16 ⁠, tan ⁠ 7 π / 16 ⁠, tan ⁠ 11 π / 16 ⁠, and tan ⁠ 15 π / 16 ⁠ satisfy the irreducible polynomial x 4 − 4x 3 − 6x 2 + 4x + 1 = 0, and so are conjugate algebraic integers. This is the ...

  6. Periodic continued fraction - Wikipedia

    en.wikipedia.org/wiki/Periodic_continued_fraction

    He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other.

  7. Complex conjugate - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate

    In polar form, if and are real numbers then the conjugate of is . This can be shown using Euler's formula . The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates ).

  8. Rational function - Wikipedia

    en.wikipedia.org/wiki/Rational_function

    In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K .

  9. Fractional ideal - Wikipedia

    en.wikipedia.org/wiki/Fractional_ideal

    The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal = itself. This group is called the group of fractional ideals of . The principal fractional ideals form a subgroup.

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