Theorem
Theorem
An assertion that can be proved true using the rules of logic. A theorem is proven from axioms, postulates, or other theorems already known to be true.
See also
Example
Problem: Demonstrate the structure of a theorem and its proof by proving the following: 'If n is an even integer, then n² is also an even integer.'
State the theorem: Theorem: For any integer n, if n is even, then n² is even. A theorem always has a clear hypothesis (the 'if' part) and a conclusion (the 'then' part).
If n is even, then n2 is even.
Begin the proof — use the definition: Since n is even, by definition there exists an integer k such that n = 2k. This starting point comes from an accepted definition, which functions like an axiom.
n=2kfor some integer k
Apply logical reasoning: Square both sides of the equation and simplify using standard algebra rules.
n2=(2k)2=4k2=2(2k2)
Draw the conclusion: Since 2k² is an integer (call it m), we have n² = 2m, which means n² is even by definition. The proof is complete because every step followed logically from accepted truths.
n2=2mwhere m=2k2 is an integer
Answer: The statement 'if n is even, then n² is even' is now a theorem because it has been proven true using definitions and logical deduction. Before the proof, it was merely a conjecture.
Frequently Asked Questions
What is the difference between a theorem and a theory?
A theorem is a single statement proven true by logical deduction within mathematics. A theory, in science, is a broad explanatory framework supported by evidence but not 'proven' in the mathematical sense. In mathematics, a 'theory' refers to an entire branch of study (like number theory), not a single proven statement.
What is the difference between a theorem, a lemma, and a corollary?
All three are proven statements, but they differ in role. A theorem is a major result considered important on its own. A lemma is a smaller helper result proven mainly to assist in proving a theorem. A corollary is a statement that follows easily and directly from a theorem that has already been proven.
Theorem vs. Conjecture
A theorem is a statement that has been rigorously proven to be true. A conjecture is a statement believed to be true based on evidence or patterns, but not yet proven. For example, the Pythagorean theorem was once a conjecture until it was first proven. Goldbach's conjecture — that every even integer greater than 2 is the sum of two primes — remains unproven, so it is not a theorem despite centuries of supporting evidence.
Why It Matters
Theorems are the building blocks of all mathematical knowledge. Once a theorem is proven, it is true permanently and can be used to prove further results, creating an ever-growing chain of reliable knowledge. Famous theorems like the Pythagorean theorem, the fundamental theorem of calculus, and the prime number theorem underpin everything from engineering and physics to computer science and cryptography.
Common Mistakes
Mistake: Thinking that many examples are enough to prove a theorem.
Correction: Checking specific cases — even thousands of them — does not constitute a proof. A theorem requires a logical argument that covers all possible cases. For instance, a pattern that holds for the first million integers might still fail for integer 1,000,001.
Mistake: Using the words 'theorem' and 'conjecture' interchangeably.
Correction: A conjecture is an unproven claim; a theorem is a proven one. Calling an unproven statement a theorem is incorrect no matter how likely it seems to be true.
Related Terms
- Axiom — Starting assumptions theorems are built upon
- Postulate — Accepted truths, especially in geometry
- Lemma — Helper result used to prove a theorem
- Corollary — Result that follows directly from a theorem
- Proof — Logical argument that establishes a theorem
- Conjecture — Unproven statement that may become a theorem
- Pythagorean Theorem — Famous example of a geometric theorem