2028: Complex Numbers

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Complex Numbers

Title text: I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that algebraic geometry and geometric algebra are the same thing.

Explanation

The complex numbers can be thought of as pairs $(a,\ b)\in\mathbb{R}\times\mathbb{R}$ of real numbers with rules for addition and multiplication.

$(a,\ b) + (c,\ d) = (a+c,\ b+d)$

$(a,\ b) \cdot (c,\ d) = (ac - bd,\ ad + bc)$

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 $a+bi$, where $i^2 = -1$, 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: $a \cdot b = b \cdot a$. The title text argues that the "link" between algebra and geometry in "algebraic 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.

Transcript

[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

The word "algebraic" in the title text was initially spelt "algebreic". This was later fixed.

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Discussion

I assume this is strictly a coincidence, but in reference to the title-text, I'll just mention that Caucher Birkar [the mathematician whose Fields Medal was stolen minutes after he received it in Rio de Janeiro on Weds (1Aug2018)] received the award for work in algebraic geometry. Arcanechili (talk) 16:34, 3 August 2018 (UTC)

• Perhaps it's causal not coincidental. Medal theives and perhaps Randall might read the news also. [[1]] 162.158.79.209 00:34, 4 August 2018 (UTC)

I've added a basic description of Abelian groups in the title text, and that's about as much as I know about such topics. I'm not sure what a "meta-Abelian group" is, is that an Abelian group of other groups? Also, could someone add basic descriptions of algebreic geometry and geometrical algebra? 172.68.94.40 18:42, 3 August 2018 (UTC)

In the title text, since groups are a concept within mathematics, it seems odd to consider mathematics as a whole forming any sort of group within itself, which I suspect is the first part of the pun. Secondly, since groups involve the commutative property, I think the last part is a pun about the order of the words algebra and geometry, as if they're commutative themselves! Ianrbibtitlht (talk) 19:19, 3 August 2018 (UTC)

I meant to say 'abelian' groups involve the commutative property, and the meta prefix is referring to the fact that it's about the names rather than the mathematical details - i.e. commutative in metadata only. Ianrbibtitlht (talk) 19:24, 3 August 2018 (UTC)

I guess the joke is that informally mathematicians form a group (a number of people classed together), what would strictly be a set in mathematics. While in mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies specific conditions. --JakubNarebski (talk) 21:18, 3 August 2018 (UTC)

It's a false dilemma. Complex numbers are vectors ($\mathbb{C}$ is a two-dimensional $\mathbb{R}$-vector space, and more generally every field is a vector space over any subfield), but that doesn't change anything about the fact that $i$ is by definition a square root of -1. Zmatt (talk) 20:38, 3 August 2018 (UTC)

Important concepts in math usually show up naturally in many apparently unrelated areas. Each area will name and define a concept that makes sense for the problems being considered. One of the joys of math is proving that multiple, apparently unrelated, definitions are equivalent. When definitions are equivalent you cannot pick "the one true definition" -- any of them will do. However the principle of maximum laziness leads to the one with the easiest notion being used as a canonical definition.162.158.75.130 18:17, 7 August 2018 (UTC)

Fun factoid: not only is $\mathbb{C}$ the unique proper field extension of finite degree over $\mathbb{R}$ (since $\mathbb{C}$ is algebraically closed), but the converse is true as well: $\mathbb{R}$ is the only proper subfield of finite index in $\mathbb{C}$. They're like a weird married couple. Zmatt (talk) 20:53, 3 August 2018 (UTC)

Altho there are no "meta-abelian" groups there are metabelian groups. If xy=yx then the commutator [x,y]=xyx^{-1}y^{-1}=1. The group generated by the commutators -- the commutator subgroup -- is thus a measure of how far a group is from being abelian. A metabelian group is a nonabelian group whose commutator subgroup is abelian. Thus a metabelian group is one made of a stack of two abelian groups. It is "meta-abelian" in that sense. A standard example is the group of invertible upper-trianglular matrices. The commutators all have 1s on the diagonals.

One should note that the concept of complex numbers actually is older than vector spaces. So while it is true that complex numbers are a cool variant of vectors, historically that's not true, because vectors were more or less unknown when complex numbers were used for the first time. --162.158.90.6 09:59, 4 August 2018 (UTC)

Shouldn't the description of a group involve two operations? There is a binary operation that gloms two things together to make a new thing, but there's also a unary operation that takes only one thing and makes a new thing -- the inverse. Without the unary operation, you only have a semigroup.108.162.215.160 09:40, 5 August 2018 (UTC)

No. The inverse operation arises as a consequence of the fact that it's a group. A group satisfies four conditions: 1. it is closed under the operation, 2. the operation is associative 3. there is an identity e such that a op e = e op a = a. 4. For every element a, there is a unique element b such that a op b = b op a = e. The inverse function falls out as a result of conditions 3 and 4 Jeremyp (talk) 10:26, 6 August 2018 (UTC)

Real numbers with regular multiplication is not a group, as zero does not have an inverse element. The example would work with addition --162.158.134.112 13:48, 9 August 2018 (UTC)Random guy who read the article

"Does any of this really have to do with vectors? Or do mathematicians just think they're too cool for regular floating point arrays? SteveBaker (talk) 16:00, 6 July 2020 (UTC)