The Infinitude of Pythagorean Triples

In the Understanding the Fermat’s Last Theorem post, I have mentioned about Pythagorean Triples.  In this post, we will show that there are infinitely many of them. We will use intuitive reasoning to prove the theorem.

For the 100th time (kidding), recall that the Pythagorean Theorem states that in a right triangle with side lengths a, b and c, where c is the hypotenuse, the equation c^2 = a^2 + b^2  is satisfied. For example, if we have a triangle with side lengths 2 and 3 units, then the hypotenuse is \sqrt{13}. The converse of the Pythagorean theorem is also true: If you have side lengths, a, b and c, which satisfies the equation above, we are sure that the angle opposite to the longest side is a right angle.

We are familiar with right triangles with integral sides. The triangle with sides (3, 4, 5) units, for instance, is a right triangle.  This is also the same with (5, 12, 13) and (8, 15, 17).  We will call this triples, the Pythagorean triples ,or geometrically, right triangles having integral side lengths. » Read more

Understanding the Fermat’s Last Theorem

The Fermat’s Last Theorem is one of the hardest problems in the history of mathematics.  The problem was written by Pierre de Fermat in 1637, and it was only solved more than 300 years later —  in 1995 by Professor Andrew Wiles.

But what is exactly the Fermat’s Last Theorem?

The Fermat’s Last Theorem is an extension of the Pythagorean Theorem.  Recall that the Pythagorean Theorem states that given a right triangle whose side lengths are x, y and hypotenuse z, x^2 + y^2 = z^2 is satisfied. For example, a right triangle with side lengths 2, and 3 has hypotenuse \sqrt{13}.

There are some interesting things that we can see if we examine the side lengths of right triangles.  For instance, if we let the triples (x,y,z) be the side lengths of a right triangle, where z is the hypotenuse, we can find triples such that all lengths are integers. The triples  (3, 4, 5), (5,12,13), (8,15,17) are integer triples, and they satisfy the Pythagorean Theorem. These triples are called Pythagorean Triples. It is not also difficult to see that there are infinitely many Pythagorean Triples (Can you see why?). » Read more

WordPress Blogging Tutorial 3 – Publishing Your First Blog

In the previous article, we have discussed the basics of blogging. We have typed our draft, inserted pictures and saved it. In this post, we are going to discuss the finishing touches of our first post and then publish it.

Before we publish our first blog, we must do two more things: We will add categories and then add tags.  We want search engines (Google, Yahoo, MSN) to see our post.  The best way of doing this is placing  categories and tags in each of our post.  Of course, at first, search engines will not place our blog at the top list during a search, but as more visitors visit our blog, and as more articles are being posted, it will increase its search engine rankings, and will increase the probability of being listed when a search is performed.

A category is a content specification of a post. For instance, since our blog is about high school mathematics, we might want include categories such as Algebra, Geometry, Trigonometry, Probability, Statistics and Calculus. Of course, we can add more categories such as Math Games, Famous Mathematicians, or anything related to high school mathematics. Our first post, the Pythagorean Theorem falls under Geometry and Trigonometry categories.

Tags are the keywords that are related to our article.  The possible tags for our post The Pythagorean Theorem are Pythagorean theorem, proof without words, Pythagoras, and right triangles. Notice that tags are more content-specific than categories. » Read more

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