transcendental number: meaning, definition, pronunciation and examples
LowTechnical / Academic (mathematics, number theory)
Quick answer
What does “transcendental number” mean?
A real or complex number that is not a root of any non-zero polynomial equation with integer coefficients.
Audio
Pronunciation
Definition
Meaning and Definition
A real or complex number that is not a root of any non-zero polynomial equation with integer coefficients.
A number that 'transcends' algebraic operations, existing beyond the reach of standard algebraic equations; most real numbers are transcendental, with well-known examples including π (pi) and e (Euler's number).
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Terminology is identical in both mathematical traditions.
Connotations
Identical—purely technical mathematical term.
Frequency
Very low frequency in general discourse; used only in specialised mathematical contexts. Slightly more likely to appear in UK contexts like A-Level Further Maths or university courses.
Grammar
How to Use “transcendental number” in a Sentence
The [Noun, e.g., constant e] is a transcendental number.To prove/discover that [Noun Phrase] is transcendental.Vocabulary
Collocations
Examples
Examples of “transcendental number” in a Sentence
adjective
British English
- Liouville's work on transcendental constants was groundbreaking.
- The proof established the number's transcendental nature.
American English
- The search for transcendental constants is a key area.
- He published a paper on transcendental function values.
Usage
Meaning in Context
Business
Not used.
Academic
Used in mathematics, number theory, and theoretical computer science.
Everyday
Almost never used; would be mentioned only in advanced educational contexts or popular science.
Technical
Core term in mathematical analysis and number theory.
Vocabulary
Synonyms of “transcendental number”
Strong
Neutral
Weak
Vocabulary
Antonyms of “transcendental number”
Watch out
Common Mistakes When Using “transcendental number”
- Confusing 'transcendental' with 'transcendent'.
- Thinking all irrational numbers (like √2) are transcendental (they are not; √2 is algebraic).
- Incorrect: 'transcendental number theory' used to refer to any advanced number theory.
FAQ
Frequently Asked Questions
Yes. All transcendental numbers are irrational numbers, but not all irrational numbers are transcendental (e.g., √2 is irrational but algebraic).
In mathematics, 'transcendental' is the correct term for these numbers. 'Transcendent' is more commonly used in philosophy and theology.
No, they are extremely common—almost all real numbers are transcendental. However, proving a specific number is transcendental is often very difficult.
Yes, any rational number (like 3 or ½) or the square root of a non-perfect square (like √5) is an algebraic number, not transcendental.
A real or complex number that is not a root of any non-zero polynomial equation with integer coefficients.
Transcendental number is usually technical / academic (mathematics, number theory) in register.
Transcendental number: in British English it is pronounced /ˌtræn.senˈden.təl ˈnʌm.bər/, and in American English it is pronounced /ˌtræn.senˈden.t̬əl ˈnʌm.bɚ/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: π (pi) and e are 'too good' for algebra—they **transcend** it.
Conceptual Metaphor
BEYOND CAPTURE (A transcendental number is a quarry that cannot be caught in the net of algebraic equations).
Practice
Quiz
Which of the following is a transcendental number?