Key Difference – Postulate vs Theorem
Postulates and theorems are two common terms that are often used in mathematics. A postulate is a statement that is assumed to be true, without proof. A theorem is a statement that can be proven true. This is the key difference between postulate and theorem. Theorems are often based on postulates.
What is a Postulate?
A postulate is a statement that is assumed to be true without any proof. Postulate is defined by the Oxford dictionary as “thing suggested or assumed as true as the basis for reasoning, discussion, or belief” and by the American Heritage dictionary as “something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument”.
Postulates are also known as axioms. Postulates do not have to be proven since they are visibly correct. For example, the statement that two points make a line is a postulate. Postulates are the basis from which theorems and lemmas are created. A theorem can be derived from one or more postulates.
Given below are some basic characteristics that all postulates have:
- Postulates should be easy to understand – they should not have a lot of words that are difficult to understand.
- They should be consistent when combined with other postulates.
- They should have the ability to be used independently.
However, some postulates – such as Einstein’s postulate that the universe is homogenous – are not always correct. A postulate may become obviously incorrect after a new discovery.
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
What is a Theorem?
A theorem is a statement that can be proven as true. The Oxford dictionary defines theorem as a “general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths” and Merriam-Webster defines it as “a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions”.
Theorems can be proven by logical reasoning or by using other theorems which have already been proven true. A theorem that has to be proved in order to prove another theorem is called a lemma. Both lemmas and theorems are based on postulates. A theorem typically has two parts known as hypothesis and conclusions. Pythagorean theorem, four color theorem, and Fermat’s Last Theorem are some examples of theorems.
Visualization of Pythagorean theorem
What is the difference between Postulate and Theorem?
Definition:
Postulate: Postulate is defined as “a statement accepted as true as the basis for argument or inference.”
Theorem: Theorem is defined as “general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths”.
Proof:
Postulate: A postulate is a statement that is assumed to be true without any proof.
Theorem: A theorem is a statement that can be proven as true.
Relation:
Postulate: Postulates are the basis for theorems and lemmas.
Theorem: Theorems are based on postulates.
Need to Prove:
Postulate: Postulates don’t need to be proven since they state the obvious.
Theorem: Theorems can be proven by logical reasoning or by using other theorems which have been proven true.
Image Courtesy:
“Pythagorean theorem abc” By Pythagoras abc.png: nl:Gebruiker:Andre_Engels – Pythagoras abc.png (CC BY-SA 3.0) via Commons Wikimedia
“Parallel postulate en” By 6054 – Edit of http://pl.wikipedia.org/wiki/Grafika:Parallel_postulate.svg by User: Harkonnen2 (CC BY-SA 3.0) via Commons Wikimedia