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In mathematics, a quaternion algebra over a field F is a central simple algebra A over F [1] [2] that has dimension 4 over F.Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
The term antonym (and the related antonymy) is commonly taken to be synonymous with opposite, but antonym also has other more restricted meanings. Graded (or gradable) antonyms are word pairs whose meanings are opposite and which lie on a continuous spectrum (hot, cold).
In Spanish dar (basic meaning "to give"), when applied to lessons or subjects, can mean "to teach", "to take classes" or "to recite", depending on the context. [22] Similarly with the French verb apprendre , which usually means "to learn" but may refer to the action of teaching someone. [ 23 ]
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy.There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or ...
A distinction between an exercise and a mathematical problem was made by Alan H. Schoenfeld: [2] Students must master the relevant subject matter, and exercises are appropriate for that. But if rote exercises are the only kinds of problems that students see in their classes, we are doing the students a grave disservice. He advocated setting ...
g 1, g 2 denotes the ordered pair of the two group elements. *' can be viewed as the naturally induced addition of +. In group theory , a branch of mathematics , an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action .
Let K be an algebraic number field, and let O K be its ring of integers.Let b 1, ..., b n be an integral basis of O K (i.e. a basis as a Z-module), and let {σ 1, ..., σ n} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C).
These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration. [1] Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although ...