Preview

Pythagorean Theorem: Basic trigonometry

Good Essays
Open Document
Open Document
1250 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
Pythagorean Theorem: Basic trigonometry
Pythagorean Theorem:
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I 'll list below some fallacious proofs of the Pythagorean theorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)

Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD, which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid 's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.

Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)

Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown that
(CB = a) + 2r = (AC + AB = b + c).
And if
(1)
a + 2r = b + c then (1)² = (2):
(2)
a² + 4ra + 4r² = b² + 2bc + c².
Now if 4ra + 4r² = 2bc, then a² = b² + c². But 4ra + 4r² is greater than, equal to, or less than 2bc.
If 4ra + 4r² > or < 2bc, then a² + 4ra + 4r² > or < b² + 2bc + c²; i.e. a + 2r < or > b + c, which is absurd. Hence, 4ra + 4r² = 2bc and, therefore, a² = b² + c².
This proof is accompanied by an editors 's Note:
So far as we know, this proof has not been given before. If it has not been published before, it may be properly called a new



References: 1. E. S. Loomis, The Pythagorean Proposition, NCTM, 1968 |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2012 Alexander Bogomolny

You May Also Find These Documents Helpful

  • Powerful Essays

    Logic Exercise 1 and 2

    • 751 Words
    • 3 Pages

    Explanation: The problem is on the second premise. All A are B All C are B All A are C if C and B are interchanged it will be valid.…

    • 751 Words
    • 3 Pages
    Powerful Essays
  • Good Essays

    Pythagorean Theorem Essay

    • 489 Words
    • 2 Pages

    The Pythagorean theorem (A^2 + B^2 = C^2) has been impacting all types of people and careers since it was first realized during Ancient Greece times. It is one of the most widely recognized theorems in the mathematics community, and used much more than the average person knows: whether you need need to know the dimensions of a bag or you need find the distance from location to another, the Pythagorean theorem can be used. Everyone who was taught this theorem in their first year of algebra continues to carry on the knowledge into their real life. So, at least with the Pythagorean theorem when those annoying students ask “When are we going to use this in real life?” they will have an answer.…

    • 489 Words
    • 2 Pages
    Good Essays
  • Good Essays

    Pythagorean Quadratic

    • 644 Words
    • 3 Pages

    The Pythagorean Theorem was termed after Pythagoras, who was a well-known Greek philosopher and mathematician, and the Pythagorean Theorem is one of the first theorems identified in ancient civilizations. “The Pythagorean theorem says that in any right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse” (Dugopolski, 2012, p. 366 para. 8). For this reason, many builders from various times throughout history have used this theorem to assure that their foundations were laid out with right angles. In this assignment, we will use the example of locating a treasure using two different treasure maps as the two points needed to determine how many paces it will take to find the exact location to start digging for treasure.…

    • 644 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    Distributive: a ( b + c ) = a * b + a * c…

    • 407 Words
    • 2 Pages
    Satisfactory Essays
  • Good Essays

    (3 x 7) x 8 = 3 x (7 x 8) = (3 x 8) x 7…

    • 1488 Words
    • 16 Pages
    Good Essays
  • Good Essays

    Pythagorean Triples

    • 1047 Words
    • 5 Pages

    The Pythagorean triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because:…

    • 1047 Words
    • 5 Pages
    Good Essays
  • Good Essays

    Pythagoras Legacy

    • 386 Words
    • 2 Pages

    One of Phytagoras's famous legacy is the Pythagoras theorem, which states that the hypotenuse squared of a right triangle is equal to the sum of the squares of the legs (sides of the elbows). Although the facts in this theorem have been widely known before the birth of Pythagoras, this theorem is credited to Pythagoras because he was the first to prove this observation mathematically.…

    • 386 Words
    • 2 Pages
    Good Essays
  • Satisfactory Essays

    Division by a complex number is a very similar process to ‘rationalising’ surds – we call it ‘realising’…

    • 528 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    This argument is valid. My proof for validity can be found in my appendix at the end of the paper. [And no, I am not going to provide an appendix for a sample paper].…

    • 595 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    Pythagorean Triples

    • 417 Words
    • 2 Pages

    To begin you must understand the Pythagoras theorem is an equation of a2 + b2 = c2. This simply means that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest. Let a, b, and c be the three sides of a right angled triangle. To define, a right angled triangle is a triangle in which any one of the angles is equal to 90 degrees. The longest side of the right angled triangle is called the 'hypotenuse '. Once you have this basic understanding you can apply the understanding that if a, b, and c are positive integers, they are called Pythagorean Triples.…

    • 417 Words
    • 2 Pages
    Satisfactory Essays
  • Better Essays

    Fermat's Last Theorem

    • 3261 Words
    • 14 Pages

    In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.…

    • 3261 Words
    • 14 Pages
    Better Essays
  • Good Essays

    The thing that Pythagoras is probably the most famous for is the Pythagorean Theorem. The Pythagorean Theorem is used in the field of mathematics and it states the following: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides. This means that if one makes a square (with all sides equal in length) out of a triangle with a right angle, the areas of the squares made from the two shorter sides, when added together, equal the area of the square made from the long side. Another geometrical discovery made by Pythagoras is that the diagonal of a square is not a rational multiple of its side. The latter discovery proved the existence of irrational numbers and therefore changed the entire Greek mathematical belief that whole numbers and their ratios could account for geometrical properties. He also discovered a formula to find out how many degrees there are in a polygon. Pythagoras came up with (n-2)180°= the number of degrees in a polygon, where (n) represents the number of sides in the polygon. For example, a triangle has three sides, 3-2=1, 1x180=180, which is the total sum of all the inner angles of a triangle. Along with that he found out that the sum of all the outer angles of a polygon is always equal to three hundred sixty degrees. This is true for every single polygon, regardless of the number of the sides.…

    • 750 Words
    • 3 Pages
    Good Essays
  • Good Essays

    MATHEMATICAL FORMULAE

    • 600 Words
    • 3 Pages

    3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)…

    • 600 Words
    • 3 Pages
    Good Essays
  • Good Essays

    College Algebra Notes

    • 1407 Words
    • 6 Pages

    (a + bi) + (c + di) = (a + c) + (b + d)i…

    • 1407 Words
    • 6 Pages
    Good Essays
  • Good Essays

    Pythagoras

    • 543 Words
    • 2 Pages

    In today’s world, there are a multitude of mathematical theorems and formulas. One theorem that is particularly renowned is the Pythagorean Theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of any right triangle. While most people have heard of or even used the Pythagorean Theorem, many know little of the man who proved it.…

    • 543 Words
    • 2 Pages
    Good Essays