2028: Complex Numbers

2018-08-06T10:55:23Z

Heng: /* Explanation */ Improved the description of geometric algebra and its relationship to Clifford algebras.

{{comic
| number = 2028
| date = August 3, 2018
| title = Complex Numbers
| image = complex_numbers.png
| titletext = I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that algebreic geometry and geometric algebra are the same thing.
}}

==Explanation==
{{incomplete|Created by a MATHEMATICIAN - Do NOT delete this tag too soon.}}

The {{w|complex number}}s can be thought of as pairs <math>(a,\ b)\in\mathbb{R}\times\mathbb{R}</math> of real numbers with rules for addition and multiplication.

: <math>(a,\ b) + (c,\ d) = (a+c,\ b+d)</math>

: <math>(a,\ b) \cdot (c,\ d) = (ac - bd,\ ad + bc)</math>

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, 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, often used to handle rotations in 3D computer graphics, is an example of a 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.

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".

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[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!

==Trivia==
This comic is similar to [[1724: Proofs]].
{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Miss Lenhart]]
[[Category:Math]]

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2028: Complex Numbers