winding number
C2 (Very Low Frequency, Technical)Formal, Academic, Technical
Definition
Meaning
An integer that counts the total number of times a curve winds around a given point in the plane in a counterclockwise direction.
In complex analysis, the winding number of a closed curve around a point is a topological invariant. In electrical engineering and physics, the concept can describe the number of turns in a coil or the phase accumulation in a system.
Linguistics
Semantic Notes
Always used as a singular countable noun ('the winding number is...'). The concept is abstract and mathematical, relating to topology, complex analysis, and vector fields.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage. The term is identically technical in both dialects.
Connotations
Purely technical; no colloquial or figurative connotations.
Frequency
Extremely low frequency in general language, confined almost exclusively to university-level mathematics, physics, and engineering texts.
Vocabulary
Collocations
Grammar
Valency Patterns
The winding number of [curve] around [point] is [integer].[Curve] has a winding number of [integer] about [point].Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “N/A”
Usage
Context Usage
Business
Virtually never used.
Academic
Core term in advanced mathematics (complex analysis, topology), physics (phase defects, vortices), and some engineering fields (control theory).
Everyday
Never used.
Technical
The primary domain. Used to quantify topological properties, stability of dynamical systems, and phase singularities.
Examples
By Part of Speech
verb
British English
- N/A
American English
- N/A
adverb
British English
- N/A
American English
- N/A
adjective
British English
- N/A
American English
- N/A
Examples
By CEFR Level
- N/A
- N/A
- The engineer explained that the winding number of the coil affected its inductance.
- In complex analysis, a contour's winding number about a singularity determines the value of the related integral via the residue theorem.
- The topological stability of the defect was guaranteed by its non-zero winding number.
Learning
Memory Aids
Mnemonic
Imagine a piece of string winding around a pencil. The number of complete loops it makes is the winding number. If it goes counterclockwise, the number is positive; if clockwise, negative.
Conceptual Metaphor
COUNTING ENCIRCLEMENTS (The abstract mathematical concept is understood via the physical action of winding a rope or path around a pole.)
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate as 'виндинг намбер' or 'извилистое число'. The correct translation is '**индекс**' or, more descriptively, '**число оборотов**'.
Common Mistakes
- Using 'winding number' to describe a number that is not an integer (it must be an integer).
- Confusing it with a 'winding count' in a physical coil (related but not identical in abstract rigor).
- Pronouncing 'winding' as in 'windy weather' (/ˈwɪndɪŋ/) instead of the verb 'to wind' (/ˈwaɪndɪŋ/).
Practice
Quiz
What must the winding number of a simple closed curve around a point be if the point is outside the curve?
FAQ
Frequently Asked Questions
Yes, by definition, the winding number is an integer. It counts complete revolutions.
Yes. A positive winding number indicates counterclockwise encirclement, while a negative winding number indicates clockwise encirclement.
In many mathematical contexts, especially in complex analysis and vector calculus, they are synonyms. 'Index' is a slightly more general term that can be applied in broader topological settings.
Virtually never in everyday conversation. It is a specialist concept. The closest analogy might be counting how many times a rope is wound around a post.