square root
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.
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 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.
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 one, denoted by the radical sign, 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.
What it is for: Not applicable; a mathematical operation.
It can be explain the concept; relay computation and extensions; describe applications; distinguish related operations.
Distinguishing features
Inverse of squaring
Two roots for positive numbers
Irrational values
Matrix and abstract extensions
What it looks like
Not a visible object; the radical symbol.
Physical character
square root of 2: 1.41421356 value - irrational
radical sign introduced: 1525 year - Christoph Rudolff
Newton s method: converges quadratically note
How it is recognised
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
Related models
is a kind of - in registry terms
is the inverse of - as an operation
is extended to - as matrix square roots
is computed by - among other algorithms
In practice
Families and kinds
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
Standards and regulation
IEEE 754 specifies correctly rounded square root
No regulation
Failure modes and hazards
Forgetting the negative root or the principal root convention
Errors with square roots of negatives
Non-unique matrix square roots
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What a square root is.
Science.
Definition
Definition.
Definition
Definition.
- What is a square root, what is the principal root, and how does it differ from other roots and from squaring? definition
- Is the question about real numbers, complex numbers, matrices or algebraic geometry? boundary
Properties
Properties.
Properties
Properties.
- What are the algebraic properties of square roots, and why is the square root of 2 irrational? definition
- Which entry fits irrational number? action
Compute Computation.
Practice.
Methods
Methods.
Methods
Methods.
- How are square roots computed by hand, by Newton s method and in hardware? action
- Which references are standard? provenance
Numeric
Numerical issues.
Numeric
Numeric.
- How do floating-point systems compute square roots, and what precision issues arise? provenance
- Which sources are cited? provenance
Extend Extensions.
Science.
Complex
Complex numbers.
Complex
Complex.
- How are square roots of negative and complex numbers defined, and what is the branch cut? provenance
- Which entry fits complex number? action
Matrices
Matrices and geometry.
Matrices
Matrices.
- What is a square root of a matrix, how is the 2 by 2 case solved, and what is a theta characteristic? provenance
- Which entry fits square root of a matrix? action
Context Applications and history.
Context.
Applications
Applications.
Applications
Applications.
- Where do square roots appear in geometry, statistics, physics and computing? provenance
- Which entry fits Pythagorean theorem? action
History
History.
History
History.
- How did Babylonian, Greek, Indian and Renaissance mathematicians treat square roots? provenance
- Which entry fits the history of mathematics? action
What the second pass must settle
- 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?