Project
Introduction
Mathematics consists of many implications that deal not only with abstract matters, but with the real world. One of such wonders, that can be explained mathematically, is the existence of a Pythagorean triangle. Invented many years ago, the Pythagorean Theorem describes the relationship of three integers in every right triangle, when the length of hypotenuse squared equals to the sum of the length of the sides: c² = a² + b²
Body
A Pythagorean triple is simply a set of three integers that are the sides of a perfect triangle, when one of the angles is 90 degrees, the right angle.
One of the prime triples is 3,4,5.
9+16 = 25 and we see that this theorem works.
One of the common formulas to generate the set of Pythagorean triples is to multiply or divide each by the same number. To verify this, I will multiply the given set by 6:
18²+24²=30²
There is also another fundamental formula to create more triples, called Euclid’s formula: a = 2×m×n, b = m² - n², and c = m² + n². It works only in the case if m an n are coprime and m-n is odd.
For example, assume than m is 7 and n is 5.
a=2×7×5=70
b= 49-25=24
c=49+25=74
So, this formula works, because 5476 = 576 + 4900
Let’s generate five other sets of Pythagorean triples using Euclid’s formula:
1) Multiplying given set by 3: 72, 210, 222
2) Using Euclid’s formula (m=5, n=3): 30, 16, 34
3) Using Euclid’s formula (m=9, n=5): 90, 56, 106
4) Using Euclid’s formula (m=25, n = 3): 150, 616, 634
5) Using Euclid’s formula (m=38, n=32): 2432, 420, 2468
Conclusion
Pythagorean triples can be considered a mathematical wonder along with several other concepts and mathematical rows that occur in the real life world. Perhaps even early builders used this knowledge when constructing monuments. According to some researches a set 3-4-5 was used in pyramids and temples. In this assignment I have used two different formulas to generate a set of such triples; one is about multiplying every integer by some number k. The second Euclid’s formula is based on creating two coprime numbers, the difference between which is an odd number and calculating a,b and c. There are more ways to compute triples, however, those are the most common. Pythagorean triples in the real world are used in construction, engineering and other similar activities. In everyday life it is useful to know c² = a² + b² when you want to find a diagonal line (hypotenuse) of a triangle and calculate the distance.
References
http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Edenfield/Pythtriples/Pythriples.html
http://www.math.brown.edu/~jhs/frintch2ch3.pdf
