Geometry for dummies - A book review

 

In high school, I remember enjoying geometry class. It was in this class where most students first learn about proofs, and I liked the satisfaction of putting together solid proofs for theorems based a collection of given facts. Recently, I searched out for books to re-learn these skills. My first book selected from the local library was too advanced. I felt hugely disappointed in not being able to follow the first proof presented. I returned that book and found Mark Ryan’s Geometry for dummies in the same section.

Ryan’s approach is to introduce a concept and show a proof based on it. Then, he gives the reader a chance to apply the concept to another proof. He then walks the reader through longer proofs using multiple concepts earlier introduced. The author has a very clear way of explaining the ideas that makes this book easier to read (especially compared to the first book I checked out).

The book gives the basics and some more advance concepts on angles, triangles, polygons, circles and has an introduction to 3-dimensional geometry.

The author's biography says Mark Ryan practiced law for four years before deciding he should do something he enjoys. To me, this isn't surprising. In another post, I introduced a famous lawyer who improved his debating skill by studying Euclid: Math Vacation: Abraham Lincoln - A President Trained by Euclid (jamesmacmath.blogspot.com)

Book Website: Geometry For Dummies, 3rd Edition | Wiley

Pythagorean Triple Generators - Part III

 

 

(Image: Iconfinder - Dea Jae on Iconfinder)

Two prior posts were made about Pythagorean Triples and an algorithm for producing triples (Part I, Part II).

I recently came across two additional algorithms for producing Pythagorean triples.

For reference my earlier post gave the first algorithm:

Let n and m be any positive integers where n>m. The Pythagorean Triple generated is formed by the numbers:
2nm
n² - m²
n² +m²

Below is another method.

Start with any odd number and square it.
Example: start with 3; 3x3 = 9

Next, find the two consecutive numbers that add up to the square of the starting number.

In this example, 4 + 5 = 9

The starting number, 3, and the two consecutive numbers, 4 and 5, will be a Pythagorean Triple (3,4,5).

Here’s another example:

Start with the odd number, 5. Square it: 25

Find the two consecutive numbers that add up to 25: 12 and 13

The Pythagorean Triple is (5, 12, 13)

Next is a third method which starts with any multiple of 4.

Example: 8
Take half of the starting number, 8 --> 4

Square this result, 4x4 = 16

Finally, use the two odd numbers immediately before and after this square: 15, 17

These two odd numbers and the original starting number will be a Pythagorean Triple (8, 15, 17)

One may use this method with even starting numbers that are not multiples of 4; however, the Pythagorean Triples produced can be reduced by a common denominator and will be found to be similar to a primitive triple. A primitive triple is one that cannot be reduced. For example, starting with 6; square half of 6 to get 3x3 = 9; use the numbers immediately before and after to get 8 and 10. The triple (6,8,10) is similar to the triangle (3,4,5). So, if you are trying to produce primitive triples, starting with multiples of 4, one will get triples that are irreducible triples. 

 

Here is a link to a spreadsheet I made for these different Pythagorean Triple generators: Pythagorean generator.  

Pythagorean Triples can also be found in the Online Encyclopedia of Integer Sequences: A001844 - OEIS

Reference: I read about these two additional Pythagorean Triple generators in the book Geometry for dummies by Mark Ryan. 

The site Math is Fun has good illustrations and further information on triples: Pythagorean Triples - Advanced (mathsisfun.com)

Additional formulae for producing triples are found on Wikipedia.

More interesting facts about Pythagorean Triples on Dr Ron Knott's archive.

 

Lucky Numbers

 

(Image: Icon Finder - Alpar-Etele Meder)

Lucky numbers are usually associated with people’s association of a number with something like a birthday, a sport’s jersey number, or a superstition about a particular number. There is another definition of lucky numbers. Mathematician Stanislaw Ulam defined a sequence he described as “lucky numbers.” (A prior post featured Ulam: Math Vacation: Spirals of Prime Numbers (jamesmacmath.blogspot.com))

The sequence is produced using a sieve in which one begins with the natural numbers and this list is reduced by eliminating numbers using a set of rules. The numbers remaining are termed the lucky numbers. (Another numerical sieve is the sieve of Eratosthenes, which produces prime numbers.)

The first element of the sequence is 1, the first member of the sequence of natural numbers. Next, in the first application of the sieve, every second member of the sequence is removed. This eliminates all the even numbers. The remaining sequence is 1, 3, 5, 7, 9, 11…The next surviving, or “lucky,” element is 3. Next, the sieve removes every third member of the remaining sequence.

Removing every third member leaves us with 1, 3, 5, 7, 9, 11, 13, 15, 17, 19…

The next surviving number is 7, so now every seventh member of the remaining sequence is eliminated, so we have: 1, 3, 7, 9, 13, 15, 19…

Members of the sequence under 100 are: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99

The lucky number sequence has been studied extensively and has been found to be similar to the prime number sequence.

•      Many, but not all, of the lucky number sequence’s members are prime.
•       In both sequences, the pattern is irregular.
•       As with the primes, the lucky numbers can appear as twins. Examples in the primes are: (5,7), (17,19), (41, 43). Examples in the lucky number sequence are: (7,9), (13,15), (31,33). Twin primes are discussed in the post: Math Vacation: Prime Numbers - a property rediscovered (jamesmacmath.blogspot.com)
•       The frequency of prime numbers and lucky numbers are similar.
•       The frequency of twins is similar in the prime and lucky sequences. A table comparing these frequencies is found here: MATHEWS: Lucky Numbers (archive.org)
• There exists a conjecture for the lucky numbers that is analogous to the Goldbach Conjecture for primes. That is, every even number can be expressed as the sum of two lucky numbers. 

The lucky numbers are sequence A000959 is the On-Line Encyclopedia of Integer Sequences.

A listing of the first 200,00 members of this sequence can be found here:  oeis.org/A000959/b000959.txt