1 00:00:00,000 --> 00:00:14,640 What do Euclid, 12-year-old Einstein, and American President James Garfield have in 2 00:00:14,640 --> 00:00:15,640 common? 3 00:00:15,640 --> 00:00:20,600 They all came up with elegant proofs for the famous Pythagorean theorem, the rule that 4 00:00:20,600 --> 00:00:26,540 says for a right triangle, the square of one side plus the square of the other side is 5 00:00:26,540 --> 00:00:29,200 equal to the square of the hypotenuse. 6 00:00:29,200 --> 00:00:33,900 In other words, a-squared plus b-squared equals c-squared. 7 00:00:33,900 --> 00:00:38,780 This statement is one of the most fundamental rules of geometry and the basis for practical 8 00:00:38,780 --> 00:00:45,360 applications like constructing stable buildings and triangulating GPS coordinates. 9 00:00:45,360 --> 00:00:51,020 The theorem is named for Pythagoras, a Greek philosopher and mathematician in the 6th century 10 00:00:51,020 --> 00:00:55,220 BC, but it was known more than a thousand years earlier. 11 00:00:55,220 --> 00:01:03,180 A Babylonian tablet from around 1800 BC lists 15 sets of numbers that satisfy the theorem. 12 00:01:03,180 --> 00:01:09,640 Some historians speculate that ancient Egyptian surveyors used one such set of numbers, 3, 13 00:01:09,640 --> 00:01:12,760 4, 5, to make square corners. 14 00:01:12,760 --> 00:01:17,740 The theory is that surveyors could stretch a knotted rope with 12 equal segments to form 15 00:01:17,740 --> 00:01:22,280 a triangle with sides of length 3, 4, and 5. 16 00:01:22,360 --> 00:01:27,600 According to the converse of the Pythagorean theorem, that has to make a right triangle, 17 00:01:27,600 --> 00:01:29,720 and therefore a square corner. 18 00:01:29,720 --> 00:01:36,480 And the earliest known Indian mathematical texts, written between 800 and 600 BC, state 19 00:01:36,480 --> 00:01:42,320 that a rope stretched across the diagonal of a square produces a square twice as large 20 00:01:42,320 --> 00:01:43,880 as the original one. 21 00:01:43,880 --> 00:01:49,560 That relationship can be derived from the Pythagorean theorem. 22 00:01:49,560 --> 00:01:54,080 But how do we know that the theorem is true for every right triangle on a flat surface, 23 00:01:54,080 --> 00:01:57,920 not just the ones these mathematicians and surveyors knew about? 24 00:01:57,920 --> 00:01:59,240 Because we can prove it. 25 00:01:59,240 --> 00:02:04,040 Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold 26 00:02:04,040 --> 00:02:06,820 true all the time. 27 00:02:06,820 --> 00:02:13,800 One classic proof, often attributed to Pythagoras himself, uses a strategy called proof by rearrangement. 28 00:02:13,800 --> 00:02:20,120 Take four identical right triangles, with side lengths a and b, and hypotenuse length 29 00:02:20,120 --> 00:02:21,160 c. 30 00:02:21,160 --> 00:02:25,280 Arrange them so that their hypotenuses form a tilted square. 31 00:02:25,280 --> 00:02:28,800 The area of that square is c squared. 32 00:02:28,800 --> 00:02:35,080 Now rearrange the triangles into two rectangles, leaving smaller squares on either side. 33 00:02:35,080 --> 00:02:39,560 The areas of those squares are a squared and b squared. 34 00:02:39,560 --> 00:02:40,700 Here's the key. 35 00:02:40,700 --> 00:02:47,480 The total area of the figure didn't change, and the areas of the triangles didn't change. 36 00:02:47,480 --> 00:02:53,420 So the empty space in one, c squared, must be equal to the empty space in the other, 37 00:02:53,420 --> 00:02:57,660 a squared plus b squared. 38 00:02:57,660 --> 00:03:02,940 Another proof comes from a fellow Greek mathematician, Euclid, and was also stumbled upon almost 39 00:03:02,940 --> 00:03:06,560 2,000 years later by 12-year-old Einstein. 40 00:03:06,560 --> 00:03:11,960 This proof divides one right triangle into two others, and uses the principle that if 41 00:03:11,960 --> 00:03:16,960 the corresponding angles of two triangles are the same, the ratio of their sides is 42 00:03:16,960 --> 00:03:18,140 the same, too. 43 00:03:18,140 --> 00:03:27,560 So for these three similar triangles, you can write these expressions for their sides. 44 00:03:27,560 --> 00:03:38,560 Next, rearrange the terms. 45 00:03:38,560 --> 00:03:46,640 And finally, add the two equations together and simplify to get a b squared plus a c squared 46 00:03:46,640 --> 00:03:57,000 equals b c squared, or a squared plus b squared equals c squared. 47 00:03:57,000 --> 00:04:02,920 Here's one that uses tessellation, a repeating geometric pattern, for a more visual proof. 48 00:04:02,920 --> 00:04:04,680 Can you see how it works? 49 00:04:04,680 --> 00:04:09,480 Pause the video if you'd like some time to think about it. 50 00:04:09,480 --> 00:04:10,640 Here's the answer. 51 00:04:10,640 --> 00:04:15,680 The dark gray square is a squared, and the light gray one is b squared. 52 00:04:15,680 --> 00:04:18,480 The one outlined in blue is c squared. 53 00:04:18,480 --> 00:04:25,080 Each blue outlined square contains the pieces of exactly one dark and one light gray square, 54 00:04:25,080 --> 00:04:28,120 proving the Pythagorean theorem again. 55 00:04:28,120 --> 00:04:32,160 And if you'd really like to convince yourself, you could build a turntable with three square 56 00:04:32,160 --> 00:04:36,300 boxes of equal depth connected to each other around a right triangle. 57 00:04:36,300 --> 00:04:40,520 If you fill the largest square with water and spin the turntable, the water from the 58 00:04:40,520 --> 00:04:45,240 large square will perfectly fill the two smaller ones. 59 00:04:45,240 --> 00:04:51,280 The Pythagorean theorem has more than 350 proofs and counting, ranging from brilliant 60 00:04:51,280 --> 00:04:52,280 to obscure. 61 00:04:52,280 --> 00:04:53,960 Can you add your own to the mix?