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  2. Complex conjugate root theorem - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate_root_theorem

    It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots;

  3. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.

  4. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...

  5. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example, the equation: = has a unique real root, ≈ 1.1673, that cannot be expressed in terms of only radicals and arithmetic operations.

  6. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. Galois theory allows proving that when the three roots are real, and none is rational ( casus irreducibilis ), one cannot express the roots in terms of real radicals.

  7. Square root - Wikipedia

    en.wikipedia.org/wiki/Square_root

    Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 5 2 (5 squared). In mathematics, a square root of a number x is a number y such that =; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1]

  8. nth root - Wikipedia

    en.wikipedia.org/wiki/Nth_root

    For a positive real number x, denotes the positive square root of x and denotes the positive real n th root. A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, ⁠ + ⁠ and ⁠ ⁠, where i is the imaginary unit.

  9. Cube root - Wikipedia

    en.wikipedia.org/wiki/Cube_root

    If this definition is used, the cube root of a negative number is a negative number. The three cube roots of 1. If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair.

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