# what is postulates and axioms

Axioms and postulates are the basic assumptions and are accepted without demonstration. All other theorems must be proven with the aid of these basic assumptions. At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. For Example:

Axiom - Things which are equal to the same thing are also equal to one another(common to many sciences).

Postulate - It is possible to draw a straight line from any point to any other point(particularly in Geometry).

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Euclid describes Axioms and Postulates, given below. The Postulates talk about straight lines, circles, right angles and parallel lines (these have been defined already, but here is more information about them). The Axioms are about relationships; what does equal mean, how do you add or subtract things, and so on. Remember that we are talking about lines and angles, not numbers, so adding and subtracting need to be thought about.

In modern mathematics, the first principles of any formal deductive system are 'axioms', so perhaps the Postulates, Axioms and Definitions should all be considered axioms.

Euclid's Axioms (or Common Notions)
Axiom 1 - Things which equal the same thing also equal one another.
Axiom 2 - If equals are added to equals, then the wholes are equal.
Axiom 3 - If equals are subtracted from equals, then the remainders are equal.
Axiom 4 - Things which coincide with one another equal one another.
Axiom 5 - The whole is greater than the part.