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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo p; the essential point is that a single-term polynomial over the integral domain D/p cannot decompose as a product in which at least one of the factors has more than one term (because in such a product there ...
The rectangle that bounds an equilateral triangle of side 2, or a regular hexagon of side 1, has size square root of 3 by square root of 4, with a diagonal of square root of 7. A Logarex system Darmstadt slide rule with 7 and 6 on A and B scales, and square roots of 6 and of 7 on C and D scales, which can be read as slightly less than 2.45 and ...
The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation in the form of square roots and fourth roots. [21]
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the mth root of an integer N as the ratio a ⁄ b of two integers a and b, that share no common prime factor, except in cases in which b = 1.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 2 − 2 = 0. The golden ratio (denoted φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − ...