# Tag Archives: algebra

Published on Feb 18, 2016
This problem has over 2 million comments on Facebook and is getting coverage in mainstream media and blogs. Can you figure out this Facebook fruit algebra puzzle involving apples, bananas, and coconuts? The video presents what many people consider to be the correct answer.

Published on Jun 28, 2015
What is the car’s parking spot number? Hong Kong elementary students were asked to solve this problem in 20 seconds. The problem stumped a lot of adults and went viral in 2014.

# the Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.[1]

The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert‘s axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.

An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group.

This can be made precise in various contexts with slightly different ways of formulation. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.

# Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can bediagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decompositioneigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

# Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices
(under matrix multiplication) that is defined by polynomial equations.

An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of
M.

# magmas

## Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

From magma to group, via two alternative paths. Key:
M = magma, d = divisibility, a = associativity,
Q = quasigroup, S = semigroup, e = identity.
L = loop, i = invertibility, N = monoid, G = group
Note that both divisibility and invertibility imply
the existence of the cancellation property.

# Fields

In abstract algebra, a field is an algebraic structure with notions of additionsubtractionmultiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fieldsp-adic fields, and so forth.
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomialswith coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.
As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possessmultiplicative inverses. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.)
As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions: