Editing 2028: Complex Numbers
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The usual way to introduce complex numbers is by starting with ''i'' and deducing the rules for addition and multiplication, but Cueball is correct to say that some uses of complex numbers could be modeled with vectors alone, without consideration of the square root of a negative number. | The usual way to introduce complex numbers is by starting with ''i'' and deducing the rules for addition and multiplication, but Cueball is correct to say that some uses of complex numbers could be modeled with vectors alone, without consideration of the square root of a negative number. | ||
| β | The teacher, [[Miss Lenhart]], counters that to ignore the natural construction of the complex numbers would hide the relevance of the {{w|fundamental theorem of algebra}} (Every polynomial of degree ''n'' has exactly ''n'' roots, when counted according to multiplicity) and much of {{w|complex analysis}} (calculus with complex numbers; the study of analytic and meromorphic functions), but she also agrees that mathematicians are too cool for "regular vectors." Just because the complex numbers can be interpreted through vector space, | + | The teacher, [[Miss Lenhart]], counters that to ignore the natural construction of the complex numbers would hide the relevance of the {{w|fundamental theorem of algebra}} (Every polynomial of degree ''n'' has exactly ''n'' roots, when counted according to multiplicity) and much of {{w|complex analysis}} (calculus with complex numbers; the study of analytic and meromorphic functions), but she also agrees that mathematicians are too cool for "regular vectors." Just because the complex numbers can be interpreted through vector space, ohwever, doesn't mean that they ''are'' just vectors, any more than being able to construct the natural numbers from set logic mean that natural numbers are ''really'' just sets. |
In mathematics, a {{w|group (mathematics)|group}} is the pairing of a binary operation (say, multiplication) with the set of numbers that operation can be used on (say, the real numbers), such that you can describe the properties of the operation by its corresponding group. An {{w|Abelian group}} is one where the operation is commutative, that is, where the terms of the operation can be exchanged: <math> a \cdot b = b \cdot a</math> The title text argues that the "link" between algebra and geometry in "algebreic [sic] geometry" and "geometric algebra" is the operation in an Abelian group, such that both of those fields are equivalent. Algebraic geometry and geometric algebra are mostly unrelated areas of study in mathematics. {{w|Algebraic geometry}} studies the properties of sets of zeros of polynomials. It runs relatively deep. Its tools were used for example in Andrew Wiles' celebrated proof of Fermat's Last Theorem. For its part, a {{w|geometric algebra| geometric algebra}} (a {{w|Clifford algebra| Clifford algebra}} with some specific properties) is a construct allowing one to do algebraic manipulation of geometric objects (e.g., vectors, planes, spheres, etc.) in an arbitrary space that has a resultant geometric interpretation (e.g., rotation, displacement, etc.). The algebra of quaternions, which is often used to handle rotations in 3D computer graphics, is an example of geometric algebra, as is the algebra of complex numbers. {{w|Metabelian group|Meta-Abelian groups}} (often contracted to metabelian groups) is a class of groups that are not quite abelian, but close to being so. | In mathematics, a {{w|group (mathematics)|group}} is the pairing of a binary operation (say, multiplication) with the set of numbers that operation can be used on (say, the real numbers), such that you can describe the properties of the operation by its corresponding group. An {{w|Abelian group}} is one where the operation is commutative, that is, where the terms of the operation can be exchanged: <math> a \cdot b = b \cdot a</math> The title text argues that the "link" between algebra and geometry in "algebreic [sic] geometry" and "geometric algebra" is the operation in an Abelian group, such that both of those fields are equivalent. Algebraic geometry and geometric algebra are mostly unrelated areas of study in mathematics. {{w|Algebraic geometry}} studies the properties of sets of zeros of polynomials. It runs relatively deep. Its tools were used for example in Andrew Wiles' celebrated proof of Fermat's Last Theorem. For its part, a {{w|geometric algebra| geometric algebra}} (a {{w|Clifford algebra| Clifford algebra}} with some specific properties) is a construct allowing one to do algebraic manipulation of geometric objects (e.g., vectors, planes, spheres, etc.) in an arbitrary space that has a resultant geometric interpretation (e.g., rotation, displacement, etc.). The algebra of quaternions, which is often used to handle rotations in 3D computer graphics, is an example of geometric algebra, as is the algebra of complex numbers. {{w|Metabelian group|Meta-Abelian groups}} (often contracted to metabelian groups) is a class of groups that are not quite abelian, but close to being so. | ||
