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  2. Zero-product property - Wikipedia

    en.wikipedia.org/wiki/Zero-product_property

    If is a prime number, then the ring of integers modulo has the zero-product property (in fact, it is a field). The Gaussian integers are an integral domain because they are a subring of the complex numbers. In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication ...

  3. Persistence of a number - Wikipedia

    en.wikipedia.org/wiki/Persistence_of_a_number

    The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest number of multiplicative persistence 3.

  4. Multiplicity (mathematics) - Wikipedia

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

    The graph crosses the x-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity. A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an x 0 {\displaystyle x_{0}} such that f ( x 0 ) > 0 {\displaystyle f(x_{0})>0} .

  5. Fundamental theorem of algebra - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

    To establish that every complex polynomial of degree n > 0 has a zero, it suffices to show that every complex square matrix of size n > 0 has a (complex) eigenvalue. [13] The proof of the latter statement is by contradiction. Let A be a complex square matrix of size n > 0 and let I n be the unit matrix of the same size. Assume A has no eigenvalues.

  6. Multiplication sign - Wikipedia

    en.wikipedia.org/wiki/Multiplication_sign

    The multiplication sign (×), also known as the times sign or the dimension sign, is a mathematical symbol used to denote the operation of multiplication, which results in a product. [ 1 ] The symbol is also used in botany , in botanical hybrid names .

  7. Bézout's theorem - Wikipedia

    en.wikipedia.org/wiki/Bézout's_theorem

    The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality. Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed.

  8. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...

  9. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    The roots of the corresponding scalar polynomial equation, λ 2 = λ, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the nullity of P, while the multiplicity of 1 is the rank of P. Another example is a matrix A that satisfies A 2 = α 2 I for some scalar α. The eigenvalues must be ± ...