Pythagorean Triples Essay Example
2
Assignment: Following completion of your readings, complete exercise 4 in the “Projects” section on page 620 of Mathematics in Our World. 1. Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this. 2. After building your triples, verify each of them in the Pythagorean Theorem equation.
Exercise #4: The numbers 3, 4, and 5 are called Pythagorean triples since 32+42=52. The numbers 5, 12, and 13 are also Pythagorean triples since 52+122=132. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples. Research the topic of Pythagorean triples and write a brief report on the subject.
You can generate Pythagorean triples using the following expressions:
Pick two positive integers, m and n, with m less than n.
Then the three numbers that form the Pythagorean triple can be calculated from:
Running Head: PYTHAGOREAN 3
n² - m²
2mn
n² + m²
Examples:
1) m = 3, n = 4 n² - m² = (4)² - (3)² = 16 - 9 = 7
2mn = 2(3)(4) = 24 n² + m² = (4)² + (3)² = 16 + 9 = 25
Triple: 7, 24, 25
Check:
(7)² + (24)² = (25)²
49 + 576 = 625
625 = 625
Running Head: PYTHAGOREAN 4
2) m = 1, n = 3 n² - m² = (3)² - (1)² = 9 - 1 = 8
2mn = 2(1)(3) = 6 n² + m² = (3)² + (1)² = 9 + 1 = 10
Triple: 6, 8, 10
Check:
(6)² + (8)² = (10)²
36 + 64 = 100
100 = 100
3) m = 4, n = 5 n² - m² = (5)² - (4)² = 25 - 16 = 9
2mn = 2(4)(5) = 40 n² + m² = (5)² + (4)² = 25 + 16 = 41
Triple: 9, 40, 41
Running Head: PYTHAGOREAN 5
Check:
(9)² + (40)² = (41)²
1600 = 1681
1681 = 1681
4) m = 5, n = 6 n² - m² = (6)² - (5)² =
Assignment: Following completion of your readings, complete exercise 4 in the “Projects” section on page 620 of Mathematics in Our World. 1. Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this. 2. After building your triples, verify each of them in the Pythagorean Theorem equation.
Exercise #4: The numbers 3, 4, and 5 are called Pythagorean triples since 32+42=52. The numbers 5, 12, and 13 are also Pythagorean triples since 52+122=132. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples. Research the topic of Pythagorean triples and write a brief report on the subject.
You can generate Pythagorean triples using the following expressions:
Pick two positive integers, m and n, with m less than n.
Then the three numbers that form the Pythagorean triple can be calculated from:
Running Head: PYTHAGOREAN 3
n² - m²
2mn
n² + m²
Examples:
1) m = 3, n = 4 n² - m² = (4)² - (3)² = 16 - 9 = 7
2mn = 2(3)(4) = 24 n² + m² = (4)² + (3)² = 16 + 9 = 25
Triple: 7, 24, 25
Check:
(7)² + (24)² = (25)²
49 + 576 = 625
625 = 625
Running Head: PYTHAGOREAN 4
2) m = 1, n = 3 n² - m² = (3)² - (1)² = 9 - 1 = 8
2mn = 2(1)(3) = 6 n² + m² = (3)² + (1)² = 9 + 1 = 10
Triple: 6, 8, 10
Check:
(6)² + (8)² = (10)²
36 + 64 = 100
100 = 100
3) m = 4, n = 5 n² - m² = (5)² - (4)² = 25 - 16 = 9
2mn = 2(4)(5) = 40 n² + m² = (5)² + (4)² = 25 + 16 = 41
Triple: 9, 40, 41
Running Head: PYTHAGOREAN 5
Check:
(9)² + (40)² = (41)²
1600 = 1681
1681 = 1681
4) m = 5, n = 6 n² - m² = (6)² - (5)² =