ordinary point
C2 (Technical/Professional)Formal, Technical, Academic
Definition
Meaning
In mathematics, a point on a curve at which the curve is smooth and well-behaved, as opposed to a singular point.
In a more general context, can refer to a standard, unremarkable location or stage in a process, or in complex systems analysis, a point that does not exhibit unusual properties.
Linguistics
Semantic Notes
The primary meaning is domain-specific to mathematics, particularly complex analysis, differential geometry, and the study of differential equations. Its use in everyday contexts is rare and likely metaphorical.
Dialectal Variation
British vs American Usage
Differences
No difference in the mathematical or technical meaning. The compound may be slightly more likely to be hyphenated (ordinary-point) in older British mathematical texts, but the open form is standard in both regions.
Connotations
Identical; purely technical with no regional connotative differences.
Frequency
Frequency is equally low and confined to technical fields in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The function f(z) is analytic at the ordinary point z₀.We must determine whether x=0 is an ordinary point or a singular point of the differential equation.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “[No common idioms. The term itself is technical.]”
Usage
Context Usage
Business
Virtually never used.
Academic
Common in mathematics, physics, and engineering publications dealing with differential equations, complex analysis, or power series solutions.
Everyday
Extremely rare. If used, it would be in a metaphorical sense (e.g., 'This is no ordinary point in our company's history').
Technical
The primary context. Precisely defined in mathematical literature.
Examples
By Part of Speech
verb
British English
- The solution can be series-expanded about any point that is ordinary.
- We first need to ordinary-point the equation, a process of checking for singularities.
American English
- The method requires that we identify which points are ordinary.
- To solve via power series, you must ordinary-point the differential equation.
adverb
British English
- [No standard adverbial form. Very rare/improvised.]
- The function behaved ordinarily point-wise across the interval.
American English
- [No standard adverbial form. Very rare/improvised.]
- The curve proceeded ordinarily, point by point, until the singularity.
adjective
British English
- The ordinary-point analysis is crucial for the power series method.
- He studied the ordinary-point behaviour of the system.
American English
- The ordinary-point criterion is satisfied.
- We assume ordinary-point conditions for the theorem to hold.
Examples
By CEFR Level
- This dot on the line is an ordinary point. (Simplified, non-technical)
- On a smooth graph, every point is an ordinary point. (Basic mathematical)
- To apply the power series method, we must first locate the ordinary points of the differential equation.
- A function is said to be analytic at an ordinary point.
- The distinction between an ordinary point and a regular singular point is fundamental to the Frobenius method for solving linear ODEs.
- While x=0 is a singular point of Bessel's equation, every other finite x is an ordinary point.
Learning
Memory Aids
Mnemonic
Imagine a smooth, ORDINARY road (the curve). Any point on that smooth stretch is an ORDINARY POINT. A pothole or a fork in the road would be a SINGULAR POINT.
Conceptual Metaphor
ANALYSIS IS NAVIGATION. An ordinary point is a smooth, navigable part of the path (the function/curve). A singular point is an obstacle, a fork, or a dead-end.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate as 'ординарная точка' (this is incorrect). The standard Russian mathematical term is 'обыкновенная точка'.
- Avoid the direct calque 'обычная точка' in technical contexts, as it is less precise.
Common Mistakes
- Using 'ordinary point' in everyday language where 'normal point' or 'regular point' is intended.
- Confusing 'ordinary point' with 'critical point' (a point where the derivative is zero).
- Incorrectly hyphenating as 'ordinary-point' in modern technical writing.
Practice
Quiz
Which of the following is the direct antonym of 'ordinary point' in complex analysis?
FAQ
Frequently Asked Questions
In most mathematical contexts, yes. 'Ordinary point' and 'regular point' are synonyms, though 'ordinary point' is often used specifically in the context of differential equations and power series.
It is highly specialised. Any non-mathematical use would be metaphorical and very rare, likely to be misunderstood without clear context.
For an ODE in standard form y'' + P(x)y' + Q(x)y = 0, a point x = x₀ is an ordinary point if both P(x) and Q(x) are analytic (i.e., have convergent power series expansions) at x₀.
At an ordinary point, P(x) and Q(x) are analytic. At a regular singular point, they are not analytic, but (x-x₀)P(x) and (x-x₀)²Q(x) are analytic. At an irregular singular point, even these multiplied functions are not analytic.