* 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.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

Every polynomial P in x corresponds to a function, ƒ ( x )

Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial.

* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every delta operator ' has a unique sequence of " basic polynomials ", a polynomial sequence defined by three conditions:

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

* Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

* Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q =

* Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

Every polynomial function is a rational function with.

* Every irreducible polynomial over k has distinct roots.

* Every polynomial over k is separable.

Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring.

In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

* Every irreducible polynomial in K which has a root in L factors into linear factors in L.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.

* Hardy and Littlewood listed as their Conjecture I: " Every large odd number ( n > 5 ) is the sum of a prime and the double of a prime.

Every odd number q satisfies for.

: Every odd number greater than 7 can be expressed as the sum of three odd primes.

Every second number ( all even numbers ) is eliminated, leaving only the odd integers:

Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number.

Every 2 years ( on odd years ) the Glandore Classic Boat Regatta is held during the second week of July.

Every odd year Glandore hosts its " Classic Boat Regatta " which takes place over the space of a week.

Every odd year ( i. e., 2011 ), the race travels the south route from Ophir to Kaltag through the ghost town of Iditarod.

Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree.

Every year, conferences with topics pertaining to choral conductors are held-in even numbered years, a division conference is held in each division, and in odd numbered years, a national conference takes place in a major U. S. city.

Every other year, in odd calendar years, three trustees face re-election.

Every supermatrix can be written uniquely as the sum of an even supermatrix and an odd one.

Every other year, in odd years, the UUBF holds a Convocation.