* 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 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 Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Every diagonal matrix is symmetric, since all off-diagonal entries are zero.

Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Every maximally symmetric space has constant curvature.

Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular.

Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix.

Every symmetric operator is closable.

Every self-adjoint operator is maximal symmetric.

Every self-adjoint operator is densely defined, closed and symmetric.

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

Every symmetric association scheme is commutative.

Every convex centrally symmetric polyhedron P in R < sup > 3 </ sup > admits a pair of opposite ( antipodal ) points and a path of length L joining them and lying on the boundary ∂ P of P, satisfying

< li > Every positive definite matrix is invertible and its inverse is also positive definite.

Every aspect of the line matrix printer is designed to deliver higher reliability, fast throughput, and greater resistance to rough handling and hazardous environmental conditions.

Every orthogonal matrix has determinant either 1 or − 1.

Every symplectic matrix is invertible with the inverse matrix given by

Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.

Every constraint is in turn a pair ( usually represented as a matrix ), where is an-tuple of variables and is an-ary relation on.

Every hypergraph has an incidence matrix where

Every real m-by-n matrix yields a linear map from R < sup > n </ sup > to R < sup > m </ sup >.

* Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.

* Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

We call a field E a splitting field for A if A ⊗ E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed, in the case when A is a division algebra, then a maximal subfield of A is a splitting field.

* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.

Every object in the drawing can be subjected to arbitrary affine transformations: moving, rotating, scaling, skewing and a configurable matrix.

Every pixel from the secret image is encoded into multiple subpixels in each share image using a matrix to determine the color of the pixels.