transcendental number
Let an agent handle transcendental numbers by definition, examples, proofs and open problems, stating proof status accurately.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle transcendental numbers by definition, examples, proofs and open problems, stating proof status accurately.
A real or complex number that is not algebraic, meaning it is not a root of any non-zero polynomial with rational coefficients, such as pi, e and Liouville numbers; almost all real numbers are transcendental, yet proving a specific number transcendental is often hard.
What it is for: Number theory and understanding the structure of the real numbers.
It can be explain transcendence versus irrationality; list proven transcendental numbers; state open problems accurately; relate to classical construction problems.
Distinguishing features
Non-algebraic
Uncountably many
Hard to prove for specific numbers
Linked to impossibility of squaring the circle
What it looks like
Numbers such as pi and e, with non-repeating, non-algebraic expansions.
How it is recognised
Not a root of any integer polynomial
Examples pi, e, Liouville constant
Irrational algebraic numbers like the square root of 2 are not transcendental
Related models
is a kind of - category
is contrasted with - complement
includes - examples
is studied in - field
In practice
Families and kinds
Liouville numbers
classical constants such as pi and e
numbers from the Gelfond-Schneider theorem
uncomputable numbers (related)
Failure modes and hazards
Claiming unproven numbers are transcendental
Confusing irrational and transcendental
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.transcendental-number
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Definition What it means.
Precise definitions matter.
Meaning
Non-algebraic.
Meaning
Definition.
- What does it mean for a number to be transcendental? definition
- How does it differ from irrational? definition
Status
Proven or open.
Status
Proof status.
- Is this number proven transcendental, and by whom? provenance
- Or is its status open? boundary
Examples Known numbers.
Examples need proofs.
Classic
Pi and e.
Classic
Classic examples.
- When were e and pi proven transcendental? provenance
- Which methods were used? provenance
Constructed
Liouville.
Constructed
Constructed examples.
- How did Liouville construct the first transcendental numbers? definition
- Why do they work? definition
Theorems Results.
Theorems extend knowledge.
Gelfond-Schneider
a^b.
Gelfond-Schneider
Gelfond-Schneider theorem.
- What does the Gelfond-Schneider theorem say? provenance
- Which numbers does it cover? definition
Open
Unknown cases.
Open
Open problems.
- Which well-known constants have unknown transcendence status? provenance
- Is that stated clearly? boundary
Applications Consequences.
Transcendence settles old problems.
Squaring the circle
Impossibility.
Squaring the circle
Squaring the circle.
- Why does the transcendence of pi make squaring the circle impossible? definition
- Which proof established it? provenance
Measure
Almost all.
Measure
Almost all reals.
- In what sense are almost all real numbers transcendental? definition
- Why is it hard to name examples? definition
What the second pass must settle
- Should key theorems be separate entries?
- How should proof status be tracked?
- How should open constants be listed?