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Axioms, Postulates And Theorems (view source)
Revision as of 13:49, 24 April 2019
, 13:49, 24 April 2019→Concept Map
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[[Introduction to angles]]
−Introduction to pairs of angles
−Solved problems
−Concept 3 - Points, lines and angles
−===== Activities =====
==== Concept 1 - Introduction to geometry ====
==== Concept 1 - Introduction to geometry ====
+One interesting question about the assumptions for Euclid's system of geometry is the ''difference'' between the "axioms" and the "postulates." "Axiom" is from Greek ''axíôma'', "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply ''postulated'', e.g. "let" this be true. There need not even be a claim to truth, just the notion that we are going to do it this way and see what happens. Euclid's postulates, indeed, could be thought of as those assumptions that were necessary and sufficient to derive truths of geometry, of some of which we might otherwise already be intuitively persuaded. As first principles of geometry, however, both axioms and postulates, on Aristotle's understanding, would have to be self-evident. This never seemed entirely quite right, at least for the Fifth Postulate -- hence many centuries of trying to derive it as a Theorem. In the modern practice, as in Hilbert's geometry, the first principles of any formal deductive system are "axioms," regardless of what we think about their truth -- which in many cases has been a purely conventionalistic attitude. Given Kant's view of geometry, however, the Euclidean distinction could be restored: "axioms" would be ''analytic'' propositions, and "postulates" ''synthetic''. Whether any of Euclid's original axioms ''are'' analytic is a good question.
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It is useful to discuss with students about Euclid and his great contribution to Mathematics. The below two statements helps to understand and prove the theorems in geometry. Also, through a combination of activities, help the students understand results in the nature of axioms and postulates.
It is useful to discuss with students about Euclid and his great contribution to Mathematics. The below two statements helps to understand and prove the theorems in geometry. Also, through a combination of activities, help the students understand results in the nature of axioms and postulates.
# Certain statements which are valid in all branches of mathematics whose validity is taken for granted without seeking mathematical proofs is called axioms
# Certain statements which are valid in all branches of mathematics whose validity is taken for granted without seeking mathematical proofs is called axioms
# Some statement which are taken for granted in a particular branches of mathematics is called postulates.
# Some statement which are taken for granted in a particular branches of mathematics is called postulates.
−==== Concept 2 - Euclid' Axioms ====
+==== Concept 2 - Euclid's Axioms and Postulates ====
+* ''First Axiom'': Things which are equal to the same thing are also equal to one another.
+* ''Second Axiom'': If equals are added to equals, the whole are equal.
+* ''Third Axiom'': If equals be subtracted from equals, the remainders are equal.
+* ''Fourth Axiom'': Things which coincide with one another are equal to one another.
+* ''Fifth Axiom'': The whole is greater than the part.
+* ''First Postulate'': To draw a line from any point to any point.
+* ''Second Postulate'': To produce a finite straight line continuously in a straight line.
+* ''Third Postulate'': To describe a circle with any center and distance.
+* ''Fourth Postulate'': That all right angles are equal to one another.
+* ''Fifth Postulate'': That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side of which are the angles less than the two right angles.
===== Activities =====
===== Activities =====
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====== [[Axiom 5: The whole is greater than the part|The whole is greater than the part]] ======
====== [[Axiom 5: The whole is greater than the part|The whole is greater than the part]] ======
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