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Record W2739535508 · doi:10.5948/9781614441199.009

Rational and Irrational Numbers

2015· book-chapter· en· W2739535508 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueAmerican Mathematical Society eBooks · 2015
Typebook-chapter
Languageen
FieldSocial Sciences
TopicGlobal History, Politics, and Ideology
Canadian institutionsUniversity of Alberta
Fundersnot available
KeywordsIrrational numberMathematicsGeometry

Abstract

fetched live from OpenAlex

Tweedledum and Tweedledee invited Alice for a treat. They put eight tarts on the table. They looked so deliciously inviting that Alice picked one up and started munching right away. “Where did you boys get these tarts?” asked Alice, suddenly suspicious. “You don't need to know,” said Tweedledum. “We bought them from the Duchess's Cook,” said Tweedledee. “You haven't got the money. You boys stole them from the Queen of Hearts.” “She has got plenty, and won't miss a few,” said Tweedledum. “Put them back,” ordered Alice, “or you boys will be in big trouble.” “We can't,” said Tweedledee. “It wasn't easy getting into the Queen's pantry, and if we go back, we are sure to be caught this time. However, nobody knows anything so far, except you. You are not going to tell on us, are you?” “Oh dear,” said Alice. “I don't suppose I can now, seeing that I have finished eating one of them already. So I won't tell, but promise me that there won't be a next time.” “We promise,” the twins said together. “Let us share the tarts among us.” “Fairly, I suppose,” said Alice. “Of course,” said Tweedledum. “If I remember correctly,” said Tweedledee, “we each get two, with two left over.” “Not this time,” said Alice. “We better not leave any remainder for the Queen to find.” “What is a fair share then?” asked the twins. Fractions If we share eight tarts fairly among three people and leave no remainders, each fair share is more than two tarts but less than three. Let us focus on just one tart at a time. To share it fairly among three people, it must be cut up into three equal pieces, and each person would get one of them. The amount of each share is to be represented by a new number such that the sum of three copies of it is equal to 1. We call this number one-third and write it as ⅓. With eight tarts, each will get 8/3 tarts. In general, if the divisor is a positive integer n , each fair share amounts to 1/ n . This number is called the reciprocal of n . We could also have called it the multiplicative inverse of n , because the product of n and 1/ n is equal to the multiplicative identity 1.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMeta-epidemiology (narrow), Science and technology studies, Insufficient payload (model declined to judge)
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: none
GenreCandidate signal: Other · Consensus signal: Other
Teacher disagreement score0.443
Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.006
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0010.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.039
GPT teacher head0.305
Teacher spread0.266 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it