{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.square-root","code":"thing-q134237","url":"https://ver.cy/models/thing/q134237/","name":"square root","alternateNames":["square root of a matrix","square root of a 2 by 2 matrix","theta characteristic"],"kind":"thing","status":"published","version":"0.2.0-wave.4","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A number that, multiplied by itself, gives a given number, so that every positive real number has a positive principal square root and a negative square root, with the radical sign denoting the principal square root only, with extensions to complex numbers, to matrices where a square root of a matrix is a matrix whose square is the original, including explicit formulas for 2 by 2 matrices, and to abstract settings such as theta characteristics in algebraic geometry; square roots appear throughout mathematics, science and computation.","purpose":"Let an agent explain square roots and their properties, relay computation methods, extensions to complex numbers and matrices, and history from mathematical references, describe applications, and distinguish the principal square root from other roots, from squaring and from higher nth roots.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Inverse of squaring","Two roots for positive numbers","Irrational values","Matrix and abstract extensions"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a square root is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a square root, what is the principal root, and how does it differ from other roots and from squaring?","kind":"definition"},{"text":"Is the question about real numbers, complex numbers, matrices or algebraic geometry?","kind":"boundary"}]}]},{"id":"properties","name":"Properties","description":"Properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What are the algebraic properties of square roots, and why is the square root of 2 irrational?","kind":"definition"},{"text":"Which entry fits irrational number?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"methods","name":"Methods","description":"Methods.","findings":[{"id":"methods-finding","name":"Methods","description":"Methods.","questions":[{"text":"How are square roots computed by hand, by Newton s method and in hardware?","kind":"action"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"numeric","name":"Numeric","description":"Numerical issues.","findings":[{"id":"numeric-finding","name":"Numeric","description":"Numeric.","questions":[{"text":"How do floating-point systems compute square roots, and what precision issues arise?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"extend","name":"Extend","description":"Extensions.","layers":[{"id":"complex","name":"Complex","description":"Complex numbers.","findings":[{"id":"complex-finding","name":"Complex","description":"Complex.","questions":[{"text":"How are square roots of negative and complex numbers defined, and what is the branch cut?","kind":"provenance"},{"text":"Which entry fits complex number?","kind":"action"}]}]},{"id":"matrices","name":"Matrices","description":"Matrices and geometry.","findings":[{"id":"matrices-finding","name":"Matrices","description":"Matrices.","questions":[{"text":"What is a square root of a matrix, how is the 2 by 2 case solved, and what is a theta characteristic?","kind":"provenance"},{"text":"Which entry fits square root of a matrix?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"Applications and history.","layers":[{"id":"applications","name":"Applications","description":"Applications.","findings":[{"id":"applications-finding","name":"Applications","description":"Applications.","questions":[{"text":"Where do square roots appear in geometry, statistics, physics and computing?","kind":"provenance"},{"text":"Which entry fits Pythagorean theorem?","kind":"action"}]}]},{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Babylonian, Greek, Indian and Renaissance mathematicians treat square roots?","kind":"provenance"},{"text":"Which entry fits the history of mathematics?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute square roots and state the precision used.","Explain principal roots, negative roots and extensions to complex numbers and matrices.","Check the domain of a square root before using it in a calculation."],"mustNot":["Present a rounded root as exact without saying so.","Silently drop a negative or complex root that a problem needs.","Give answers to graded work as a student's own without the teacher allowing it."],"requiresHuman":["Using a computed root in a safety-critical design calculation."]},"ethics":{"considerations":["Rounding errors in roots can spread through engineering and financial calculations.","Learners need understanding, not only answers."],"affectedParties":["Students and teachers","Engineers and analysts"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics as common knowledge.","roles":[],"masterSystems":[]},"relations":[{"target":"Q601053","type":"parent","note":"registry parent class"},{"target":"nth root","type":"related","note":"in registry terms"},{"target":"square","type":"related","note":"as an operation"},{"target":"matrix","type":"related","note":"as matrix square roots"},{"target":"Newton s method","type":"related","note":"among other algorithms"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.square-root","Wikidata: Q134237 (https://www.wikidata.org/wiki/Q134237)"]},"properties":{"applicability":"not-applicable","items":["square root of 2: 1.41421356 value - irrational","radical sign introduced: 1525 year - Christoph Rudolff","Newton s method: converges quadratically note"]},"recognition":{"applicability":"optional","items":["Number whose square is a given number","Principal square root, square root of a matrix, 2 by 2 matrix formula, theta characteristic","Cube and nth roots are other roots; squaring is the inverse operation","Not a visible object; the radical symbol."]},"capabilities":{"applicability":"required","items":["explain the concept","relay computation and extensions","describe applications","distinguish related operations"]},"hazards":{"applicability":"required","items":["Forgetting the negative root or the principal root convention","Errors with square roots of negatives","Non-unique matrix square roots"]},"interfaces":{"applicability":"required","items":["IEEE 754 specifies correctly rounded square root","No regulation"]},"context":{"applicability":"required","items":["Not applicable; a mathematical operation.","square roots of positive reals","square roots of negative numbers and complex numbers","square roots of matrices including the 2 by 2 case","square roots in modular arithmetic","theta characteristics as square roots of the canonical bundle","algorithms for computing square roots"]}},"sources":[{"title":"Wikidata item Q134237: square root","url":"https://www.wikidata.org/wiki/Q134237","note":"identity and sense of the item"},{"title":"Wikipedia: Square root","url":"https://en.wikipedia.org/wiki/Square_root","note":"general description of the item"}],"openQuestions":["Should square root of a matrix and theta characteristic be separate entries?","How should mathematical references be linked?","The registry entry has merged aliases naming an advanced concept in algebraic geometry; should it be split off?"],"resources":{"spec":"/models/things/publications/thing-q134237/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q134237/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T13:11:03Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:26+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}