Mathematics
List five axioms (postulates) used in this book.
Mathematics Proofs
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Answer
Euclid's postulates are:
Axiom 5.1- Things which are equal to the same thing are equal to one another.
Axiom 5.2 - If equals are added to equals, the wholes are equal.
Axiom 5.3 - If equals are subtracted from equals, the remainders are equal.
Axiom 5.4 - Things which coincide with one another are equal to one another.
Axiom 5.5 - The whole is greater than the part.
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Related Questions
Let us look at the triangular numbers figure again. Add two consecutive triangular numbers. For example, T1 + T2 = 4, T2 + T3 = 9, T3 + T4 = 16.
What about T4 + T5 ? Make a conjecture about Tn-1 + Tn.
Look at the following pattern :
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
Make a conjecture about each of the following:
1111112 =
11111112 =
Check if your conjecture is true.
Find counter-examples to disprove the following statements:
(i) If the corresponding angles in two triangles are equal, then the triangles are congruent.
(ii) A quadrilateral with all sides equal is a square.
(iii) A quadrilateral with all angles equal is a square.
(iv) For integers a and b, = a + b
(v) 2n2 + 11 is a prime for all whole numbers n.
(vi) n2 – n + 41 is a prime for all positive integers n.
Take your favourite proof and analyse it step-by-step along the lines discussed in Section A1.5 (what is given, what has been proved, what theorems and axioms have been used, and so on).