mathematical induction: meaning, definition, pronunciation and examples
Low (C2)Academic, Technical
Quick answer
What does “mathematical induction” mean?
A rigorous proof technique in mathematics used to prove statements that are claimed to be true for all natural numbers. It consists of two steps: proving the base case (usually for n=1) and proving that if the statement holds for an arbitrary integer k (the inductive hypothesis), then it also holds for k+1.
Audio
Pronunciation
Definition
Meaning and Definition
A rigorous proof technique in mathematics used to prove statements that are claimed to be true for all natural numbers. It consists of two steps: proving the base case (usually for n=1) and proving that if the statement holds for an arbitrary integer k (the inductive hypothesis), then it also holds for k+1.
While primarily used for natural numbers, the concept can be applied to well-ordered sets. The term is sometimes used metaphorically in philosophical and logical discourse to describe reasoning from specific instances to a general conclusion, though this is distinct from its formal mathematical meaning.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or application. The phrase is identical and functions as a compound noun in both dialects.
Connotations
Identical technical connotations in both dialects.
Frequency
Identical low frequency, exclusive to academic and technical contexts.
Grammar
How to Use “mathematical induction” in a Sentence
PROVE [statement] BY mathematical inductionAPPLY mathematical induction TO [problem][Statement] HOLDS BY mathematical inductionVocabulary
Collocations
Examples
Examples of “mathematical induction” in a Sentence
verb
British English
- We shall inductively prove the formula.
- The theorem is proved by inducting on n.
American English
- We will induct on the size of the set.
- The solution was found by inducting from the base case.
adverb
British English
- The result follows inductively.
- One proceeds inductively to show the property holds for all n.
American English
- The formula can be derived inductively.
- The process builds the solution inductively.
adjective
British English
- An inductive proof requires careful setup.
- The inductive hypothesis is central to the argument.
American English
- The inductive step is often the most challenging.
- He presented an inductive argument for the lemma.
Usage
Meaning in Context
Business
Never used.
Academic
Core term in university-level mathematics, computer science, and logic courses.
Everyday
Virtually never used.
Technical
Fundamental proof technique in discrete mathematics, number theory, algorithm analysis, and formal logic.
Vocabulary
Synonyms of “mathematical induction”
Strong
Neutral
Weak
Vocabulary
Antonyms of “mathematical induction”
Watch out
Common Mistakes When Using “mathematical induction”
- Confusing it with scientific induction.
- Forgetting to prove the base case.
- Misapplying the inductive step (e.g., assuming P(k+1) is true to prove itself).
- Using it to prove statements about real numbers without modification.
FAQ
Frequently Asked Questions
No, they are opposites in a sense. Inductive reasoning in science draws probable general conclusions from specific observations. Mathematical induction is a deductive proof technique that guarantees the truth of a statement for an infinite set of numbers.
No, it can only be used to prove statements parameterised by the natural numbers (e.g., 'for all integers n ≥ 1, P(n) is true'). It is not applicable to statements about real numbers or continuous domains without significant modification.
The proof becomes invalid. You might 'prove' false statements. For example, you could 'prove' that all horses are the same colour if you only assume the inductive step without a solid base.
Strong induction is a variant where the inductive hypothesis assumes the statement is true for *all* natural numbers less than or equal to k, not just for a single k. This is logically equivalent to simple induction but is often more convenient for certain proofs (e.g., involving recursive definitions).
A rigorous proof technique in mathematics used to prove statements that are claimed to be true for all natural numbers. It consists of two steps: proving the base case (usually for n=1) and proving that if the statement holds for an arbitrary integer k (the inductive hypothesis), then it also holds for k+1.
Mathematical induction is usually academic, technical in register.
Mathematical induction: in British English it is pronounced /ˌmæθəˌmætɪkl ɪnˈdʌkʃn/, and in American English it is pronounced /ˌmæθəˈmætɪkəl ɪnˈdəkʃən/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “The domino effect (used as a common analogy for induction)”
Learning
Memory Aids
Mnemonic
Think of a line of dominoes. The base case is knocking over the first domino. The inductive step proves that if one domino falls (k), it will knock over the next one (k+1). Therefore, all dominoes will fall.
Conceptual Metaphor
A LADDER (you get on the first rung, and each rung lets you reach the next) or a DOMINO CHAIN (as above).
Practice
Quiz
What is the primary purpose of mathematical induction?