Complex Numbers
Complex Numbers
[Cueball (the student) is raising his hand and writing with his other hand. He is sitting down at a desk, which has a piece of paper on it.] Cueball: Does any of this really have to do with the square root of -1? Or do mathematicians just think they're too cool for regular vectors?
[Miss Lenhart (the teacher) is standing in front of a whiteboard.] Miss Lenhart: Complex numbers aren't just vectors. They're a profound extension of real numbers, laying the foundation for the fundamental theorem of algebra and the entire field of complex analysis.
[Miss Lenhart is standing slightly to the right in a blank frame.] Miss Lenhart: And we're too cool for regular vectors. Cueball (off-screen): I knew it!

The complex numbers can be thought of as pairs of real numbers with rules for addition and multiplication.


As such, they can be modeled as two-dimensional vectors , with standard vector addition and an interesting rule for multiplication. The justification for this rule is to consider a complex number as an expression of the form , where , 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 some uses of complex numbers could be modeled with vectors alone, without consideration of the square root of a negative number.
The teacher, Miss Lenhart , counters that to ignore the natural construction of the complex numbers would hide the relevance of the fundamental theorem of algebra (Every polynomial of degree n has exactly n roots, when counted according to multiplicity) and much of 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, however, that 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 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 Abelian group is one where the operation is commutative, that is, where the terms of the operation can be exchanged: 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. 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 geometric algebra (a 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. 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 title 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".
This comic is similar to the earlier Miss Lenhart comic 1724: Proofs .
[Cueball (the student) is raising his hand and writing with his other hand. He is sitting down at a desk, which has a piece of paper on it.] Cueball: Does any of this really have to do with the square root of -1? Or do mathematicians just think they're too cool for regular vectors?
[Miss Lenhart (the teacher) is standing in front of a whiteboard.] Miss Lenhart: Complex numbers aren't just vectors. They're a profound extension of real numbers, laying the foundation for the fundamental theorem of algebra and the entire field of complex analysis.
[Miss Lenhart is standing slightly to the right in a blank frame.] Miss Lenhart: And we're too cool for regular vectors. Cueball (off-screen): I knew it!
