In Euclidean geometry system, the following are some axioms in geometry system:
(1) Equal quantities are equal to each other.
(2) The sum of the same amount plus the same amount is equal.
(3) If the same amount is subtracted from the same amount, the difference is equal.
(4) objects that can overlap each other are congruent.
The following are algebraic expressions of common equality axioms:
① if a=b, then a+c = b+C.
② if a=b, then a-c = b-c.
③ If a=b and c≠0, then ac=bc.
④ if a=b and c≠0, then a/c = b/c.
⑤ If a=b and b=c, then A = C. ..
In mathematics, the word axiom is used in two related but different meanings-logical axiom and illogical axiom. In both senses, axioms are the starting point for deducing other propositions. Unlike theorems, an axiom (unless it is redundant) cannot be deduced from other axioms, otherwise it is not the starting point itself, but some kind of result that can be obtained from the starting point-it can be simply classified as a theorem.
Extended data
The ancient Greeks thought that geometry was also one of several sciences, and regarded geometric theorems as equal to scientific facts. They developed and used logical reasoning as a method to avoid mistakes and used it to construct and transmit knowledge. Aristotle's post-analysis is a decisive exposition of this traditional view.
"Axiom", in traditional terms, refers to a self-evident assumption in many branches of science.
On the basis of various scientific fields, there may be some unconfirmed and accepted additional assumptions, which are called "postulates". Axioms exist in many branches of science, but the postulates in each branch of science are different. The validity of postulate must be based on real-world experience. Indeed, Aristotle once said that if readers doubt the truth of postulate, the content of this science cannot be successfully spread.
Web page link Baidu encyclopedia-axiom