Editing 2028: Complex Numbers

As such they are two-dimensional {{w|Euclidean vector|vectors}}, with an interesting rule for multiplication. The justification for this rule is to consider a complex number as an expression of the form <math>a+bi</math>, where <math>i^2 = -1</math>, i.e. ''i'' is the square root of negative 1. Applying the common rules of algebra and the definition of ''i'' yields rules for addition and multiplication above.

Regular two-dimensional vectors are pairs of values, with the same rule for addition, and no rule for multiplication.

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 complex numbers are really just vectors, and can be defined without consideration of the square root of a negative number.

The teacher, [[Miss Lenhart]], counters that to ignore the natural construction of the negative 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}} (the application of calculus to complex-valued functions), but she also agrees that mathematicians are too cool for "regular vectors."

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, geometric algebra aka {{w|Clifford algebra| Clifford algebra}} is a construct allowing one to do arithmetic operations on vectors of a higher dimensional space. The algebra of quaternions, often used to handle rotations in 3D computer graphics, is an example of a Clifford algebra. {{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.  

Randall's joke in the mouseover text is a wordplay combining the concepts of (meta-)abelian groups and change in the order of word orders with the general idea of "meta".