1. We Should Not Call It the Pythagorean Theorem!
Presented by Eric Hutchinson and Aminul KM

2. The Pythagorean Theorem
The sum of the squares of each leg of a right angled triangle is equal to the square of the hypotenuse
a² + b² = c²

3. The Pythagorean Theorem
The Pythagorean Theorem takes its name from the ancient Greek Mathematician Pythagoras
Purportedly first to offer a proof of the theorem
However at this time proofs were spoken and not written

4. Pythagoras
Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem.
He was born around 572 B.C. on the island of Samos, in Greece.
For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself.

5. Pythagoras
At about 530 B.C., Pythagoras settled in a Greek town in southern Italy called Crotona.
Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagoreans.

6. Pythagoras
The Pythagorean School excelled in many subjects, such as music, medicine and mathematics
In the society, members were known as mathematikoi, which is Greek for mathematicians

7. Here comes Euclid
This is the first known written proof of this special right triangle property.
It was written in his book “Thirteen Books of Euclid’s Elements”
In book I Proposition 47, we see the actual famous proof.

8. Proof (Euclid)

9. Proof (Euclid)

10. Proof (Euclid)

11. Multiple Discoveries
Long before Pythagoras people noticed right triangle special relationships.
History tells us that this theorem has been introduced through drawings, texts, legends, and stories from Babylon, Egypt and China, dating back to BC

12. Visual Proof

13. The Babylonians
Ancient clay tablets from Babylonia reveal that the Babylonians, 1000 years before Pythagoras, knew the Pythagorean relationships.
Tablets reveal Babylonians used the Pythagorean Theorem to approximate the square root of 2.

14. Babylonian Theorem

15. Babylonian Tablets (1800 BC)

16. Babylonian Tablets (1800 BC)
Last two columns:
Column heading is translated loosely as “number”
Contains row numbers
Green numbers represent missing data

17. Babylonian Tablets (1800 BC)
Second column:
Column heading translated as “width”
Error corrections in red
Sexagesimal numbers For example 1,59 is *60+59=119 and ,19 = 5*60+19=319

18. Babylonian Tablets (1800 BC)
Third column
Column heading translated “diagonal”
Sexagesimal #s
The number 3,12, represents 3* * = 11521

19. Babylonian Tablets (1800 BC)
This tablet contains the expression d2 / l2 where l2 = d2 - w2

20. Babylonian Tablets (1800 BC)
Translating the tablet numbers, we see that these numbers are Pythagorean triples
The Babylonians did understand the Right Triangle Properties, but no written records of the proof was discovered.
Width
Length
Diagonal
119
120
169
3367
11018
11521
4601
4800
6649
12709
13500
18541 65 72 97
319
360
481
2291
2700
3541
799
960
1249
541
546
769
4961
6480
8161 45 60 75
1679
2400
2929
1771
3229

21. Babylonian Tablets (1800 BC)
We know something about how the tablet was constructed
We do not know exactly why it was created
Ordering of rows indicate it may have been used in an early form of trigonometry in order to make the arithmetic easier

22. Babylonian Tablets (1800 BC)
Another Babylonian tablet states the following which indicates an understanding of Pythagorean Theorem:
“4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times is is the breadth.”

23. The Mathematics of India
The Vedic people entered India about 1500BC from the region that today is Iran
The word Vedic describes the religion of these people
The name comes from their collections of sacred texts known as the Vedas.
The Sulbasutras are appendices to the Vedas

24. Sulbasutras (800BC – 200BC)
The Sulbasutras are Ancient Hindu manuals of geometrical constructions used by Vedic priests

25. Hindu Calculation of Square Root Using the Babylonian Theorem

26. Construction of Altars
Geometry of Shulba Sutras was used in the construction of altars required for sacrificial ritual

27. Construction of Altars
Geometry of Shulba Sutras was used in the construction of altars required for sacrificial ritual

28. Sulbasutras (800BC – 200BC)
The Baudhayana Sulbasutra gives one case of the theorem explicitly: “The rope which is stretched across the diagonal of a square produces an area double the size of the original square”

29. Sulbasutras (800BC – 200BC)
The Katyayana Sulbasutra gives a more general case: “The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together”

30. Sulbasutras (800BC – 200BC)
Results are stated in terms of “ropes”. A rope is an instrument used for measuring an altar
No formal proofs are shown because this text is meant to be a construction manual
However, there are many examples of Pythagorean triples found in the Sulbasutras
(5,12,13), (12,16,20), (8,15,17), and more all occur in the text.

31. Sulbasutras (800BC – 200BC)
The following construction occurs in most of the different Sulbasutras.
ABCD and PQRS are given squares
Mark X such that PX is equal to AB
Square SXYZ has an area equal to the sum of the areas of ABCD and PQRS Then

32. China The Qin emerge victorious from the Warring States Period.
Emperor Qin Shi Huangdi, ordered the burning of many books in 211 BC.
We do know by 500 BC they were aware of Pythagorean theorem, if which they referred to as the Gougu Theorem.

33. Zhou Bi Suan Jing One of the oldest Chinese mathematical books
Dedicated to astronomical observation and calculation
From Zhoa Dynasty (1046 BC to 256 BC)
At the end of each Chinese dynasty, libraries were burned, however the Zhoa Dynasty works were saved.

34. Zhou Bi Suan Jing
Anonymous collection of 245 problems encountered by the Duke of Zhou & Shang Goa (astronomer, mathematician)
Contains one of the first recorded proofs of the Pythagorean Theorem

35. Proof (Zhou Bi Suan Jing)
Here is the Chinese original drawing

36. Proof (Zhou Bi Suan Jing)
Here is a redrawing of the original

37. Proof (Zhou Bi Suan Jing)

38. President Garfield 20th President of the United States
Served from March 4, 1881 to September 19, 1881 (assassinated)
Before politics, Garfield wanted to become a mathematics professor
While in the House of Representatives, he came up with a proof of the Pythagorean Theorem (1876)

39. Proof (Garfield)
The two key facts that are needed for Garfield’s proof are:
1.) The sum of the angles of any triangle is 180 degrees
2.) The area of a trapezoid formula:

40. Proof (Garfield)
Start with the following diagram made up of three right triangles. These form a trapezoid with bases a and b with a height of a + b. The area of the 3 triangles equals the area of the trapezoid.

41. Proof (Garfield)
Trapezoidal Area = Area of 3 triangles

42. Conclusion
Pythagoras was not the first person to understand this special right triangle relationship
There is evidence that the Babylonians, Indians, and Chinese knew of this relationship before Pythagoras
We could call it Gougu Theorem or Babylonian Theorem or just simply the Right Triangle Theorem

43. Thank You!
Eric Hutchinson, College of Southern Nevada, Las Vegas
Aminul KM, College of Southern Nevada, Las Vegas