| This article may be too technical for most readers to understand. Please help improve Template:The artipage to make it understandable to non-experts, without removing the technical details. The talk page may contain suggestions. |
| This article may contain an excessive amount of intricate detail that may only interest a specific audience. |
Axiom (or Postulate)[]
A generally-accepted mathematical statement which is used without proof, as it is considered to be self-evident.
For example, in real analysis, the following three statements are assumed as axioms:
- There exists a real number (called '0'), such that for any real number x, x + 0 = x.
- There exists a real number (called '1'), such that for any real number x, 1x = x.
- 0 ≠ 1
Theorem[]
A mathematical statement which can be proven from other mathematical statements (either axioms or other theorems).
For example:
- There are infinitely many prime numbers.
- Pythagoras' Theorem: For any right-angled triangle with side lengths a, b and c (where c represents the side opposite the right angle, and a and b represent the other two sides), a²+b²=c².
- Fundamental Theorem of Arithmetic: Any natural number greater than 1 is either a prime number or a composite number (that is, it can be made by multiplying prime numbers together, e.g. 30 = 2 x 3 x 5). Furthermore, for each composite number, there is only one way to do this (so, 2, 3 and 5 are the only prime numbers that can be multiplied to make 30).
Proposition[]
A theorem of no particular importance.
Examples:
- If x + 3 = 8, then x = 5.
- 561 is not a prime number (it is 3 × 11 × 17).
Lemma[]
A theorem which is used as part of the proof of a larger theorem.
Example:
- Euclid's Lemma (used to prove the Fundamental Theorem of Arithmetic from above):
Suppose that a and b are integers, and p is a prime number. If a × b is divisible by p, then either a is divisible by p or b is divisible by p.
For example, take a = 337 and b = 1341. 337 × 1431 = 451917, which is divisible by 3: 451917 / 3 = 150639. By Euclid's Lemma, either 337 is divisible by 3 or 1431 is divisible by 3. (In fact, it's the latter: 1431 / 3 = 477).
Corollary[]
A statement which can easily be deduced from one other theorem or definition.
Example:
- There is no way to make a right-angled triangle with side lengths 3, 5 and 6.
(This can be deduced from Pythagoras' Theorem: if it were possible, then 3² + 5² would equal 6². But, 3² + 5² = 34 (9 + 25), and 6² = 36 - so 3² + 5² ≠ 6²).
Conjecture[]
A statement which appears to be true, but cannot (yet) be proven.
Example:
- The Twin Prime Conjecture: There are infinitely many pairs of prime numbers separated by 2 (e.g. (3,5), (5,7), (11,13), (17,19),...).
This is believed to be true - but, as of February 2015, nobody knows for sure.