extended real number system
Very lowAcademic, Technical
Definition
Meaning
The set of real numbers (ℝ) augmented with two distinct elements, positive infinity (+∞) and negative infinity (−∞).
In mathematical analysis, this is the topological closure of the real number line, providing a framework for handling limits and operations involving unbounded quantities, with defined arithmetic and order relations for the infinite elements.
Linguistics
Semantic Notes
A technical mathematical term. It is a proper noun referring to a specific, formally defined set. Not to be confused with non-standard analysis, hyperreal numbers, or projective geometry (which uses a single point at infinity). The primary conceptual move is to treat infinity as formal symbols subject to specific rules, not as numbers in the conventional sense.
Dialectal Variation
British vs American Usage
Differences
No significant differences in definition or usage between British and American academic mathematics. The notation [−∞, +∞] or ℝ ∪ {−∞, +∞} is universal.
Connotations
None; purely technical.
Frequency
Exclusively used in university-level mathematics, measure theory, and analysis textbooks and research. Equally rare in both dialects outside these contexts.
Vocabulary
Collocations
Grammar
Valency Patterns
The extended real number system is denoted by ℝ̄.We consider the function f taking values in the extended real number system.This property holds in the extended real number system.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None”
Usage
Context Usage
Business
Never used.
Academic
Used in advanced mathematics, particularly real analysis, measure theory, and integration theory, to rigorously handle limits and sums that may be infinite.
Everyday
Never used.
Technical
Used in formal mathematical writing and proofs where unbounded limits or measures are discussed.
Examples
By Part of Speech
verb
British English
- We can extend the real number system to include infinities.
- The system was extended to simplify limit theorems.
American English
- We need to extend the real number system for this proof.
- He extended the system by formally adjoinging ±∞.
adverb
British English
- The limit is defined extended-really, allowing for infinite values.
- (This adverbial form is extremely rare and non-standard.)
American English
- The variable is considered extended-really, meaning it can be ∞. (This adverbial form is extremely rare and non-standard.)
adjective
British English
- An extended-real-valued function can take infinite values.
- The extended real number system is a convenient framework.
American English
- We use an extended-real-valued measure.
- This is an extended real number system property.
Examples
By CEFR Level
- (Not applicable for A2 level.)
- (Not applicable for B1 level. Concept requires C1+ mathematical literacy.)
- In some maths, you can add infinity to the number line.
- The 'extended real number system' is a topic for university maths.
- The Lebesgue integral is defined for functions taking values in the extended real number system.
- Working in the extended reals, the supremum of an unbounded set is simply +∞.
Learning
Memory Aids
Mnemonic
Imagine the real number line as a long road. The 'extended' system adds two 'end-of-the-road' signs at the extremes: one for +∞ (going forever to the right) and one for −∞ (going forever to the left).
Conceptual Metaphor
COMPLETION AS CLOSURE: The system is conceptually 'closed' by adding endpoints to an unbounded line, similar to adding caps to an infinite tube.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating as 'расширенная система реальных чисел' where 'реальных' implies 'real-world'. Use 'расширенная система вещественных чисел' or 'расширенная числовая прямая' which are standard mathematical terms.
- The concept is precise; do not confuse with 'комплексные числа' (complex numbers) or 'гипервещественные числа' (hyperreal numbers).
Common Mistakes
- Using 'extended real number system' to mean 'complex numbers'.
- Treating ∞ as a number that can be freely used in all arithmetic operations (e.g., ∞ − ∞ is undefined).
- Capitalising it as a general term (it is not a proper noun like 'Riemann Hypothesis').
Practice
Quiz
What is the primary purpose of the extended real number system?
FAQ
Frequently Asked Questions
It is a formal symbol, not a number in the standard algebraic sense. It is an element of the set ℝ̄ with specific, defined order relations and restricted arithmetic operations (e.g., ∞ + a = ∞ for a ≠ −∞, but ∞ − ∞ is undefined).
The (affinely) extended real number system adds two distinct infinities, +∞ and −∞, preserving the concept of direction. The projectively extended real line adds only one point at infinity, turning the line into a circle, which is useful in complex analysis and geometry.
No. In the standard extended real number system, 0 ⋅ ∞ is left undefined because no consistent definition can preserve all the usual rules of arithmetic. This is crucial in measure and integration theory, where defining it depends on context (e.g., in Lebesgue integration, 0 ⋅ ∞ is defined as 0 by convention).
It allows mathematicians to state theorems about limits, suprema, infima, and integrals in a clean, unified way without constantly making special cases for unbounded sequences or functions. For example, every set of extended real numbers has a supremum.