* 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 field has an algebraic closure.

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

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.

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

Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation ( s ) defining the structure.

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.

* Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties ( where none of the sets in the decomposition are subsets of each other ).

Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.

* Every ( biregular ) algebraic automorphism of a projective space is projective linear.

* Every algebraic extension of k is separable.

* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.

Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub ( A ) for some algebra A.

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

* Every finite poset is directed complete and algebraic.

* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.

Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line.

Every planar graph has an algebraic dual, which is in general not unique ( any dual defined by a plane embedding will do ).