Definition
Theorem
A theorem is a mathematical claim whose truth has been established by proof.
Continue from this idea into connected AXIO math.
Defining relationship
A theorem is a claim with a proof behind it.
A theorem is a mathematical statement for which a valid proof has been given, using definitions, axioms, and results already established.
- 1
CLAIM
Even + even is even
Before proof, this can be treated as a conjecture.
- 2
PROOF
Represent any even numbers as 2a and 2b
Their sum still has a factor of 2.
- 3
THEOREM
The claim is established
A valid proof now stands behind the statement.
A theorem is a claim with a proof behind it.
What it means
A theorem is not just a rule that worked in several examples. It comes with a proof showing why it must work in every case covered by the statement.
Remember it
A theorem is a claim with a proof behind it.
Example
“The sum of two even whole numbers is even” is a theorem. Write the numbers as 2a and 2b. Their sum is 2a + 2b = 2(a + b), which is even.
Watch for
A conjecture is a claim believed to be true but not yet proved; after a valid proof, it may become a theorem.
Common mix-up
A named formula is not automatically a theorem. The defining feature is proof.
Exact meaning
A theorem is a mathematical statement for which a valid proof has been given, using definitions, axioms, and results already established.