proof

In this episode of “How To Prove It”, we consider two similar sounding terms which have great significance in higher mathematics: contradiction and contrapositive. Both of them arise in connection with proofs. We give several examples of proofs of both these kinds.

In our last Low Floor High Ceiling article, we had looked at Squaring the Dots... a series of questions on counting the dots inside squares of different sizes and orientations drawn on dotted paper with the dots as lattice points. The focus of the activity was to tilt squares and try to find a general formula for the number of dots inside the square of a particular tilt, as the side of the square changed.

Pure geometry or Euclidean geometry is a body of theorems and corollaries logically derived from certain axioms and postulates as presented in Euclid’s Elements. Later geometers, both Greek and others, have added to this. Occasionally some algebra is brought in but not trigonometry. Abraham Lincoln is said to have read the Elements just for the reasoning.

This article describes an activity where students created different geometrical shapes using a closed-loop string and developed conceptual understanding by engaging with properties of the shapes. The activity encouraged them to think deeply about the meaning of points, straight lines, edges, faces, and angles of geometrical shapes. Using standard models which are generally available, students only get to view geometrical shapes or build them by following a set of instructions.

Check this Student Corner – Featuring articles written by students.

Elegant proofs of mathematical statements present the beauty of mathematics and enhance our learning pleasure. This is particularly so for ‘proofs without words’ which significantly improve the visual proof capability. In this article, we present three largely visual proofs which carry a great deal of elegance. In a strict sense they are not pure proofs without words because of the mathematical expressions and formulas that appear in them. Nevertheless, they carry a lot of appeal.

In an article written in the February 1999 issue of the American Mathematical Monthly, titled appropriately “Magic Squares Indeed!”, the authors point out a truly remarkable property of this magic square; namely:
8162 + 3572 + 4922 = 6182 + 7532 + 2942,
In the 3rd part of the series, we are trying to find all triples (a, b, c) of coprime positive integers satisfying the property a2 = b(b + c). What solutions does the equation have (in coprime positive integers) other than (a, b, c) = (6, 4, 5)?
In the accompanying article on Tangrams, a claim was made that it is not possible to find integers a and b which make any of the following equalities true: √6 = a+b√2, √7 = a+b√2, √12 = a+b√2, and so on. However, the proofs may not be obvious.
How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

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