Real and Imaginary Numbers: Set of All?

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Discussion Overview

The discussion revolves around the question of whether the set of complex numbers can be considered the set of all numbers. Participants explore various definitions and interpretations of what constitutes a "number," including real numbers, imaginary numbers, and other mathematical constructs such as quaternions and transfinite numbers.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the set of complex numbers, defined as a + bi, encompasses all real and imaginary numbers, thus fitting the description of all numbers.
  • Others argue that defining "all numbers" solely through complex numbers is insufficient, as it excludes other mathematical constructs like quaternions, octonions, and vectors.
  • A participant introduces the idea that if transfinite numbers are included, the concept of a set of all numbers becomes problematic, suggesting it would be a proper class instead.
  • Another viewpoint emphasizes that the term "all numbers" lacks a universal meaning, with different interpretations being equally valid or invalid depending on context.
  • Some participants discuss the necessity of including irrational and transcendental numbers to fully represent the reals, while others challenge the completeness of this approach.
  • There is a contention regarding the mapping of numbers to points on a line, with some asserting that this property is crucial for defining numbers, while others question the significance of algebraic structure in this context.
  • Disagreements arise over the inclusion of n-tuples and whether they can be considered numbers, with some participants asserting that they do not fit satisfactory definitions of "number."

Areas of Agreement / Disagreement

Participants generally do not reach a consensus on what constitutes the set of all numbers, with multiple competing views and interpretations remaining throughout the discussion.

Contextual Notes

Participants express various assumptions about the definitions of numbers, the relevance of algebraic structure, and the implications of including different types of numbers, leading to unresolved mathematical and conceptual complexities.

  • #31
micromass said:
I don't really see why this must be true.

really? I'm surprised at you.

how does one define "an infinitely repeating decimal"? that is, how can a person be sure that the infinite sum:

\sum_{k=1}^\infty \frac{3}{10^k}

actually "converges"?

ok, the usual proof that 0.\overline{333} = 1/3 goes like this:

x = 0.33333...
10x = 3.33333... <--here. this step. how does one justify what "multiplying an infinite decimal MEANS?"

9x = 3
x = 3/9 = 1/3.

it all looks very convincing, but the step i am calling into question is a serious issue. looks like "hand-waving" to me. why? because even a little thought tells us "infinite things" DON'T behave like finite things. so if you have an infinite sum, you need to back up any claims about working with that sum.

and that notion of "convergence" is what we use to justify that "infinite decimals" mean something. so my point is, UNTIL you have a solid notion of what a "convergent series" IS, talking about "infinite decimals" doesn't make any SENSE.

so if you are going to use infinite decimals, you are already pre-supposing some facts about a completion of "some number system" (i don't care if the numerical (digital) representation is base 10, or base 2, or base p).

so yes, one of the drawbacks of "decimals" (or binary representations, doesn't matter), is that many rational numbers don't have a terminating decimal form. and the "proofs" that high-school kids are taught (how to find/covert fractions to decimals) are true, but not VALID. I'm not saying this is bad pedagogical practice, we're taught what the areas of several regions are, without having a good definition of "area", either.

******

let me back up a bit, and address the main topic. the real question that we are dancing around, is: what is a number? this is a very good question, and there are several good candidates:

a) real numbers
b) complex numbers
c) rational numbers
d) computable numbers
e) constructible numbers
f) algebraic numbers
g) matrices
h) elements of a division algebra (perhaps commutative, preferably)
i) vectors
j) p-adic numbers

for various reasons, people believe that one or more of these are "unsatisfactory", because they do not capture some "intuitive" idea of what a number should BE.

MY feeling about the answer to this question, is another question: what is a number supposed to DO?
 

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