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Euler's identity is also a special case of the more general identity that the n th roots of unity, for n > 1, add up to 0: = = Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,
Specializing to the case = =, =, one immediately recovers a restatement of the fundamental theorem of algebra: a polynomial P in [] has a root in if and only if deg P ≠ 0. For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials. [ 2 ]
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers , the set of rational numbers and the set of algebraic numbers are all countably infinite and therefore are null sets when considered as subsets of the real numbers.
For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x. If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring. If a ring R contains the zero ring as a subring, then R itself is the zero ring. [6]
The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law. Let {} be a sequence of independent random variables, then the event {=} is a tail event. Thus by Kolmogorov 0-1 law, it has either probability 0 or 1 to happen.
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Let (H, , ) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator.Then there is a sequence of non-zero real eigenvalues λ i, i = 1, …, N, with N equal to the rank of A, such that |λ i | is monotonically non-increasing and, if N = +∞, + =