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Page "Nth root" ¶ 18
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Every and positive Every positive integer n > 1 can be represented in exactly one way as a product of prime powers: Every positive number a has two square roots:, which is positive, and, which is negative. < li > Every positive definite matrix is invertible and its inverse is also positive definite. Every positive integer appears exactly once somewhere on this list. Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negative roots, see complex roots below. Every line of a GEDCOM file begins with a level number where all top-level records ( HEAD, TRLR, SUBN, and each INDI, FAM, OBJE, NOTE, REPO, SOUR, and SUBM ) begin with a line with level 0, while other level numbers are positive integers. Every positive rational number can be represented by an Egyptian fraction. * Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to * Every positive integer can be written as the sum of 73 or fewer sixth powers ( see Waring's problem ). Every positive integer is the sum of at most 37 fifth powers ( see Waring's problem ). * Every positive integer except 1 is a PV number. Every positive integer can be expressed as the sum of at most 19 fourth powers ; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers ( see Waring's problem ). A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups. Every C *- algebra has an approximate identity of positive elements of norm ≤ 1 ; indeed, the net of all positive elements of norm ≤ 1 ; in A with its natural order always suffices. ' Every positive law, or every law simply and strictly so called, is set, directly or circuitously, by a sovereign person or body, to a member or members of the independent political society wherein that person or body is supreme. Every positive rational number can be expanded as an Egyptian fraction. Every positive integer is the sum of at most 143 seventh powers ( see Waring's problem ). Every residue class in this group contains exactly one square free integer, and it is common, therefore, only to consider square free positive integers, when speaking about congruent numbers. Every positive rational number q may be expressed as a continued fraction of the form

Every and real * Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. * Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions. * Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. * Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional. Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification. Every real number, whether integer, rational, or irrational, has a unique location on the line. Every real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >. Every ordered field is a formally real field. Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares. * Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding. Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by, where √ is called the radical sign or radix. Every dual number has the form z = a + bε with a and b uniquely determined real numbers. Every real number has an additive inverse ( i. e. an inverse with respect to addition ) given by. Every nonzero real number has a multiplicative inverse ( i. e. an inverse with respect to multiplication ) given by ( or ). Every real number, rational or not, is equated to one and only one cut of rationals. Every sedenion is a real linear combination of the unit sedenions 1, < var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >, Every octonion is a real linear combination of the unit octonions: Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions. In his book Nirvana: The Stories Behind Every Song, Chuck Crisafulli writes that the song " stands out in the Cobain canon as a song with a very specific genesis and a very real subject ". * Every real number greater than zero or every complex number except 0 has two square roots. Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.

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