| {"message": {"transcript": [{"chunks": [{"end": 0.32, "start": 0.0, "text": "We're"}, {"end": 0.6, "start": 0.32, "text": "going"}, {"end": 0.6, "start": 0.6, "text": "to"}, {"end": 1.04, "start": 0.6, "text": "talk"}, {"end": 1.36, "start": 1.04, "text": "about"}, {"end": 1.72, "start": 1.36, "text": "E,"}, {"end": 2.44, "start": 1.72, "text": "the"}, {"end": 2.8, "start": 2.44, "text": "big"}, {"end": 3.36, "start": 2.8, "text": "famous"}, {"end": 4.4, "start": 3.36, "text": "constant"}, {"end": 4.48, "start": 4.4, "text": "E."}, {"end": 5.08, "start": 4.48, "text": "Okay,"}, {"end": 5.28, "start": 5.08, "text": "it's"}, {"end": 5.6, "start": 5.28, "text": "one"}, {"end": 5.8, "start": 5.6, "text": "of"}, {"end": 5.84, "start": 5.8, "text": "the"}, {"end": 6.32, "start": 5.84, "text": "famous"}, {"end": 6.88, "start": 6.32, "text": "mathematical"}, {"end": 7.32, "start": 6.88, "text": "constants,"}, {"end": 7.72, "start": 7.32, "text": "one"}, {"end": 7.8, "start": 7.72, "text": "of"}, {"end": 7.8, "start": 7.8, "text": "the"}, {"end": 7.8, "start": 7.8, "text": "most"}, {"end": 8.48, "start": 7.8, "text": "important."}, {"end": 8.56, "start": 8.48, "text": "It"}, {"end": 9.08, "start": 8.56, "text": "goes"}, {"end": 9.52, "start": 9.08, "text": "along"}, {"end": 9.68, "start": 9.52, "text": "with"}, {"end": 10.88, "start": 9.68, "text": "pi"}, {"end": 11.04, "start": 10.88, "text": "and,"}, {"end": 11.44, "start": 11.04, "text": "I"}, {"end": 11.68, "start": 11.44, "text": "don't"}, {"end": 11.68, "start": 11.68, "text": "know,"}, {"end": 12.04, "start": 11.68, "text": "golden"}, {"end": 13.08, "start": 12.04, "text": "ratio"}, {"end": 13.32, "start": 13.08, "text": "and"}, {"end": 14.12, "start": 13.32, "text": "a"}, {"end": 14.76, "start": 14.12, "text": "square"}, {"end": 14.88, "start": 14.76, "text": "root"}, {"end": 15.08, "start": 14.88, "text": "of"}, {"end": 15.16, "start": 15.08, "text": "2."}, {"end": 15.76, "start": 15.16, "text": "There"}, {"end": 15.92, "start": 15.76, "text": "are"}, {"end": 16.4, "start": 15.92, "text": "constants"}, {"end": 16.52, "start": 16.4, "text": "in"}, {"end": 16.96, "start": 16.52, "text": "maths"}, {"end": 17.04, "start": 16.96, "text": "that"}, {"end": 17.2, "start": 17.04, "text": "are"}, {"end": 17.56, "start": 17.2, "text": "the"}, {"end": 17.92, "start": 17.56, "text": "most"}, {"end": 18.36, "start": 17.92, "text": "important"}, {"end": 19.4, "start": 18.36, "text": "constants,"}, {"end": 19.64, "start": 19.4, "text": "and"}, {"end": 20.08, "start": 19.64, "text": "E"}, {"end": 20.28, "start": 20.08, "text": "is"}, {"end": 20.36, "start": 20.28, "text": "one"}, {"end": 20.68, "start": 20.36, "text": "of"}, {"end": 20.96, "start": 20.68, "text": "those"}, {"end": 22.0, "start": 20.96, "text": "constants."}, {"end": 22.24, "start": 22.0, "text": "So"}, {"end": 22.96, "start": 22.24, "text": "E"}, {"end": 23.16, "start": 22.96, "text": "is"}, {"end": 23.24, "start": 23.16, "text": "an"}, {"end": 23.56, "start": 23.24, "text": "irrational"}, {"end": 23.84, "start": 23.56, "text": "number"}, {"end": 24.12, "start": 23.84, "text": "and"}, {"end": 24.32, "start": 24.12, "text": "it's"}, {"end": 24.56, "start": 24.32, "text": "equal"}, {"end": 25.08, "start": 24.56, "text": "to"}, {"end": 28.04, "start": 25.08, "text": "2.718281828"}, {"end": 28.72, "start": 28.04, "text": "something"}, {"end": 29.28, "start": 28.72, "text": "something"}, {"end": 29.96, "start": 29.28, "text": "something"}], "text": " We're going to talk about E, the big famous constant E. Okay, it's one of the famous mathematical constants, one of the most important. It goes along with pi and, I don't know, golden ratio and a square root of 2. There are constants in maths that are the most important constants, and E is one of those constants. So E is an irrational number and it's equal to 2.718281828 something something something"}, {"chunks": [{"end": 30.72, "start": 30.0, "text": "The"}, {"end": 31.48, "start": 30.72, "text": "problem"}, {"end": 31.8, "start": 31.48, "text": "with"}, {"end": 32.72, "start": 31.8, "text": "E"}, {"end": 32.92, "start": 32.72, "text": "is"}, {"end": 33.16, "start": 32.92, "text": "it's"}, {"end": 33.48, "start": 33.16, "text": "not"}, {"end": 34.44, "start": 33.48, "text": "defined"}, {"end": 34.76, "start": 34.44, "text": "by"}, {"end": 35.519999999999996, "start": 34.76, "text": "geometry."}, {"end": 35.76, "start": 35.519999999999996, "text": "Now,"}, {"end": 36.76, "start": 35.76, "text": "pi"}, {"end": 36.92, "start": 36.76, "text": "is"}, {"end": 37.16, "start": 36.92, "text": "something"}, {"end": 37.36, "start": 37.16, "text": "that's"}, {"end": 37.72, "start": 37.36, "text": "defined"}, {"end": 37.92, "start": 37.72, "text": "by"}, {"end": 38.64, "start": 37.92, "text": "geometry,"}, {"end": 38.84, "start": 38.64, "text": "right?"}, {"end": 38.96, "start": 38.84, "text": "It's"}, {"end": 39.0, "start": 38.96, "text": "the"}, {"end": 40.16, "start": 39.0, "text": "ratio"}, {"end": 40.28, "start": 40.16, "text": "of"}, {"end": 40.44, "start": 40.28, "text": "a"}, {"end": 41.16, "start": 40.44, "text": "circle"}, {"end": 41.96, "start": 41.16, "text": "circumference"}, {"end": 42.08, "start": 41.96, "text": "and"}, {"end": 42.32, "start": 42.08, "text": "its"}, {"end": 42.8, "start": 42.32, "text": "diameter,"}, {"end": 43.24, "start": 42.8, "text": "and"}, {"end": 43.519999999999996, "start": 43.24, "text": "it's"}, {"end": 43.92, "start": 43.519999999999996, "text": "something"}, {"end": 44.0, "start": 43.92, "text": "the"}, {"end": 44.32, "start": 44.0, "text": "ancient"}, {"end": 45.0, "start": 44.32, "text": "Greeks"}, {"end": 45.28, "start": 45.0, "text": "knew"}, {"end": 46.0, "start": 45.28, "text": "about."}, {"end": 46.16, "start": 46.0, "text": "And"}, {"end": 46.4, "start": 46.16, "text": "a"}, {"end": 46.84, "start": 46.4, "text": "lot"}, {"end": 46.92, "start": 46.84, "text": "of"}, {"end": 47.36, "start": 46.92, "text": "mathematical"}, {"end": 47.92, "start": 47.36, "text": "constants"}, {"end": 48.2, "start": 47.92, "text": "go"}, {"end": 48.44, "start": 48.2, "text": "back"}, {"end": 48.6, "start": 48.44, "text": "to"}, {"end": 49.04, "start": 48.6, "text": "the"}, {"end": 49.239999999999995, "start": 49.04, "text": "ancient"}, {"end": 49.68, "start": 49.239999999999995, "text": "Greeks."}, {"end": 49.8, "start": 49.68, "text": "But"}, {"end": 49.96, "start": 49.8, "text": "E"}, {"end": 50.28, "start": 49.96, "text": "is"}, {"end": 51.36, "start": 50.28, "text": "different."}, {"end": 51.56, "start": 51.36, "text": "E"}, {"end": 51.760000000000005, "start": 51.56, "text": "is"}, {"end": 52.16, "start": 51.760000000000005, "text": "not"}, {"end": 52.6, "start": 52.16, "text": "based"}, {"end": 52.8, "start": 52.6, "text": "on"}, {"end": 52.879999999999995, "start": 52.8, "text": "a"}, {"end": 53.44, "start": 52.879999999999995, "text": "shape."}, {"end": 54.16, "start": 53.44, "text": "It's"}, {"end": 54.44, "start": 54.16, "text": "not"}, {"end": 54.72, "start": 54.44, "text": "based"}, {"end": 54.92, "start": 54.72, "text": "on"}, {"end": 56.2, "start": 54.92, "text": "geometry."}, {"end": 56.64, "start": 56.2, "text": "It's"}, {"end": 56.760000000000005, "start": 56.64, "text": "a"}, {"end": 57.36, "start": 56.760000000000005, "text": "mathematical"}, {"end": 57.64, "start": 57.36, "text": "constant"}, {"end": 57.879999999999995, "start": 57.64, "text": "that"}, {"end": 58.08, "start": 57.879999999999995, "text": "is"}, {"end": 58.519999999999996, "start": 58.08, "text": "related"}, {"end": 58.6, "start": 58.519999999999996, "text": "to"}, {"end": 59.480000000000004, "start": 58.6, "text": "growth"}, {"end": 59.6, "start": 59.480000000000004, "text": "and"}, {"end": 59.96, "start": 59.6, "text": "rate"}], "text": " The problem with E is it's not defined by geometry. Now, pi is something that's defined by geometry, right? It's the ratio of a circle circumference and its diameter, and it's something the ancient Greeks knew about. And a lot of mathematical constants go back to the ancient Greeks. But E is different. E is not based on a shape. It's not based on geometry. It's a mathematical constant that is related to growth and rate"}, {"chunks": [{"end": 60.08, "start": 60.0, "text": "of"}, {"end": 61.32, "start": 60.08, "text": "change,"}, {"end": 62.28, "start": 61.32, "text": "but"}, {"end": 62.6, "start": 62.28, "text": "why"}, {"end": 62.8, "start": 62.6, "text": "is"}, {"end": 62.84, "start": 62.8, "text": "it"}, {"end": 63.2, "start": 62.84, "text": "related"}, {"end": 63.4, "start": 63.2, "text": "to"}, {"end": 63.88, "start": 63.4, "text": "growth"}, {"end": 64.08, "start": 63.88, "text": "and"}, {"end": 64.32, "start": 64.08, "text": "rate"}, {"end": 64.32, "start": 64.32, "text": "of"}, {"end": 64.72, "start": 64.32, "text": "change?"}, {"end": 64.88, "start": 64.72, "text": "So"}, {"end": 65.08, "start": 64.88, "text": "let's"}, {"end": 65.28, "start": 65.08, "text": "look"}, {"end": 65.28, "start": 65.28, "text": "at"}, {"end": 65.36, "start": 65.28, "text": "the"}, {"end": 65.92, "start": 65.36, "text": "original"}, {"end": 67.04, "start": 65.92, "text": "problem"}, {"end": 67.4, "start": 67.04, "text": "where"}, {"end": 67.68, "start": 67.4, "text": "E"}, {"end": 67.96, "start": 67.68, "text": "was"}, {"end": 68.36, "start": 67.96, "text": "first"}, {"end": 68.88, "start": 68.36, "text": "used."}, {"end": 69.08, "start": 68.88, "text": "So"}, {"end": 69.36, "start": 69.08, "text": "we're"}, {"end": 69.68, "start": 69.36, "text": "going"}, {"end": 70.24, "start": 69.68, "text": "to"}, {"end": 70.6, "start": 70.24, "text": "go"}, {"end": 70.6, "start": 70.6, "text": "back"}, {"end": 70.6, "start": 70.6, "text": "to"}, {"end": 70.64, "start": 70.6, "text": "the"}, {"end": 71.12, "start": 70.64, "text": "17th"}, {"end": 71.64, "start": 71.12, "text": "century,"}, {"end": 71.88, "start": 71.64, "text": "and"}, {"end": 72.28, "start": 71.88, "text": "this"}, {"end": 72.36, "start": 72.28, "text": "is"}, {"end": 72.76, "start": 72.36, "text": "Jacob"}, {"end": 73.2, "start": 72.76, "text": "Bernoulli,"}, {"end": 73.24, "start": 73.2, "text": "and"}, {"end": 73.32, "start": 73.24, "text": "he"}, {"end": 73.76, "start": 73.32, "text": "was"}, {"end": 74.28, "start": 73.76, "text": "interested"}, {"end": 74.76, "start": 74.28, "text": "in"}, {"end": 75.52, "start": 74.76, "text": "compound"}, {"end": 76.64, "start": 75.52, "text": "interest,"}, {"end": 76.92, "start": 76.64, "text": "so"}, {"end": 77.24, "start": 76.92, "text": "earning"}, {"end": 77.8, "start": 77.24, "text": "interest"}, {"end": 78.28, "start": 77.8, "text": "on"}, {"end": 78.64, "start": 78.28, "text": "your"}, {"end": 78.92, "start": 78.64, "text": "money."}, {"end": 79.2, "start": 78.92, "text": "So"}, {"end": 79.68, "start": 79.2, "text": "imagine"}, {"end": 80.0, "start": 79.68, "text": "you've"}, {"end": 80.64, "start": 80.0, "text": "got"}, {"end": 80.96000000000001, "start": 80.64, "text": "one"}, {"end": 81.64, "start": 80.96000000000001, "text": "pound"}, {"end": 81.64, "start": 81.64, "text": "in"}, {"end": 81.64, "start": 81.64, "text": "the"}, {"end": 81.88, "start": 81.64, "text": "bank,"}, {"end": 82.2, "start": 81.88, "text": "and"}, {"end": 82.4, "start": 82.2, "text": "you"}, {"end": 82.56, "start": 82.4, "text": "have"}, {"end": 82.72, "start": 82.56, "text": "a"}, {"end": 83.08, "start": 82.72, "text": "very"}, {"end": 83.6, "start": 83.08, "text": "generous"}, {"end": 83.96000000000001, "start": 83.6, "text": "bank,"}, {"end": 84.4, "start": 83.96000000000001, "text": "and"}, {"end": 84.96000000000001, "start": 84.4, "text": "they're"}, {"end": 85.36, "start": 84.96000000000001, "text": "going"}, {"end": 85.36, "start": 85.36, "text": "to"}, {"end": 85.36, "start": 85.36, "text": "offer"}, {"end": 85.76, "start": 85.36, "text": "you"}, {"end": 87.08, "start": 85.76, "text": "100%"}, {"end": 88.16, "start": 87.08, "text": "interest"}, {"end": 88.52, "start": 88.16, "text": "every"}, {"end": 89.03999999999999, "start": 88.52, "text": "year."}, {"end": 89.48, "start": 89.03999999999999, "text": "Wow,"}, {"end": 89.52, "start": 89.48, "text": "thanks"}, {"end": 89.64, "start": 89.52, "text": "a"}, {"end": 89.96000000000001, "start": 89.64, "text": "lot."}], "text": " of change, but why is it related to growth and rate of change? So let's look at the original problem where E was first used. So we're going to go back to the 17th century, and this is Jacob Bernoulli, and he was interested in compound interest, so earning interest on your money. So imagine you've got one pound in the bank, and you have a very generous bank, and they're going to offer you 100% interest every year. Wow, thanks a lot."}, {"chunks": [{"end": 90.32, "start": 90.0, "text": "So"}, {"end": 91.64, "start": 90.32, "text": "100%"}, {"end": 92.32, "start": 91.64, "text": "interest,"}, {"end": 92.52, "start": 92.32, "text": "so"}, {"end": 92.56, "start": 92.52, "text": "it"}, {"end": 92.76, "start": 92.56, "text": "means"}, {"end": 92.96, "start": 92.76, "text": "after"}, {"end": 93.24, "start": 92.96, "text": "one"}, {"end": 93.76, "start": 93.24, "text": "year"}, {"end": 94.16, "start": 93.76, "text": "you'll"}, {"end": 94.6, "start": 94.16, "text": "have"}, {"end": 94.76, "start": 94.6, "text": "two"}, {"end": 95.08, "start": 94.76, "text": "pounds."}, {"end": 95.36, "start": 95.08, "text": "So"}, {"end": 95.52, "start": 95.36, "text": "you've"}, {"end": 95.84, "start": 95.52, "text": "earned"}, {"end": 96.04, "start": 95.84, "text": "one"}, {"end": 96.48, "start": 96.04, "text": "pound"}, {"end": 97.12, "start": 96.48, "text": "interest"}, {"end": 97.36, "start": 97.12, "text": "and"}, {"end": 97.36, "start": 97.36, "text": "you've"}, {"end": 97.36, "start": 97.36, "text": "got"}, {"end": 97.6, "start": 97.36, "text": "your"}, {"end": 98.08, "start": 97.6, "text": "original"}, {"end": 98.64, "start": 98.08, "text": "pound."}, {"end": 98.84, "start": 98.64, "text": "So"}, {"end": 98.88, "start": 98.84, "text": "you"}, {"end": 99.12, "start": 98.88, "text": "now"}, {"end": 99.2, "start": 99.12, "text": "have"}, {"end": 99.36, "start": 99.2, "text": "two"}, {"end": 100.68, "start": 99.36, "text": "pounds."}, {"end": 101.0, "start": 100.68, "text": "What"}, {"end": 101.24, "start": 101.0, "text": "if"}, {"end": 101.44, "start": 101.24, "text": "I"}, {"end": 101.68, "start": 101.44, "text": "offered"}, {"end": 101.72, "start": 101.68, "text": "you"}, {"end": 102.92, "start": 101.72, "text": "instead"}, {"end": 103.76, "start": 102.92, "text": "50%"}, {"end": 104.24, "start": 103.76, "text": "interest"}, {"end": 104.48, "start": 104.24, "text": "every"}, {"end": 104.72, "start": 104.48, "text": "six"}, {"end": 104.84, "start": 104.72, "text": "months?"}, {"end": 105.2, "start": 104.84, "text": "Now"}, {"end": 105.88, "start": 105.2, "text": "is"}, {"end": 106.24, "start": 105.88, "text": "that"}, {"end": 106.48, "start": 106.24, "text": "better"}, {"end": 106.92, "start": 106.48, "text": "or"}, {"end": 107.84, "start": 106.92, "text": "worse?"}, {"end": 107.92, "start": 107.84, "text": "Well"}, {"end": 108.16, "start": 107.92, "text": "let's"}, {"end": 108.44, "start": 108.16, "text": "think"}, {"end": 108.68, "start": 108.44, "text": "about"}, {"end": 108.8, "start": 108.68, "text": "it."}, {"end": 108.96000000000001, "start": 108.8, "text": "Okay"}, {"end": 109.16, "start": 108.96000000000001, "text": "you're"}, {"end": 109.52, "start": 109.16, "text": "starting"}, {"end": 109.56, "start": 109.52, "text": "with"}, {"end": 109.84, "start": 109.56, "text": "one"}, {"end": 110.24, "start": 109.84, "text": "pound"}, {"end": 110.28, "start": 110.24, "text": "and"}, {"end": 110.32, "start": 110.28, "text": "then"}, {"end": 110.36, "start": 110.32, "text": "I'm"}, {"end": 111.0, "start": 110.36, "text": "gonna"}, {"end": 111.28, "start": 111.0, "text": "offer"}, {"end": 111.44, "start": 111.28, "text": "you"}, {"end": 111.68, "start": 111.44, "text": "50%"}, {"end": 111.92, "start": 111.68, "text": "interest"}, {"end": 112.24, "start": 111.92, "text": "every"}, {"end": 112.68, "start": 112.24, "text": "six"}, {"end": 113.12, "start": 112.68, "text": "months."}, {"end": 113.44, "start": 113.12, "text": "So"}, {"end": 113.84, "start": 113.44, "text": "after"}, {"end": 114.28, "start": 113.84, "text": "six"}, {"end": 114.8, "start": 114.28, "text": "months"}, {"end": 114.96000000000001, "start": 114.8, "text": "you"}, {"end": 115.48, "start": 114.96000000000001, "text": "now"}, {"end": 115.52, "start": 115.48, "text": "have"}, {"end": 115.84, "start": 115.52, "text": "one"}, {"end": 116.32, "start": 115.84, "text": "pound"}, {"end": 116.52, "start": 116.32, "text": "fifty."}, {"end": 116.52, "start": 116.52, "text": "And"}, {"end": 116.8, "start": 116.52, "text": "then"}, {"end": 117.03999999999999, "start": 116.8, "text": "you"}, {"end": 117.56, "start": 117.03999999999999, "text": "wait"}, {"end": 117.76, "start": 117.56, "text": "another"}, {"end": 118.12, "start": 117.76, "text": "six"}, {"end": 118.76, "start": 118.12, "text": "months"}, {"end": 118.88, "start": 118.76, "text": "and"}, {"end": 119.32, "start": 118.88, "text": "you're"}, {"end": 119.96000000000001, "start": 119.32, "text": "earning"}], "text": " So 100% interest, so it means after one year you'll have two pounds. So you've earned one pound interest and you've got your original pound. So you now have two pounds. What if I offered you instead 50% interest every six months? Now is that better or worse? Well let's think about it. Okay you're starting with one pound and then I'm gonna offer you 50% interest every six months. So after six months you now have one pound fifty. And then you wait another six months and you're earning"}, {"chunks": [{"end": 120.48, "start": 120.0, "text": "50%"}, {"end": 121.28, "start": 120.48, "text": "interest"}, {"end": 121.44, "start": 121.28, "text": "on"}, {"end": 121.64, "start": 121.44, "text": "your"}, {"end": 122.04, "start": 121.64, "text": "total,"}, {"end": 122.64, "start": 122.04, "text": "which"}, {"end": 122.92, "start": 122.64, "text": "is"}, {"end": 123.12, "start": 122.92, "text": "another"}, {"end": 123.52, "start": 123.12, "text": "75p."}, {"end": 123.68, "start": 123.52, "text": "And"}, {"end": 124.48, "start": 123.68, "text": "then"}, {"end": 124.8, "start": 124.48, "text": "you"}, {"end": 124.96, "start": 124.8, "text": "add"}, {"end": 125.24, "start": 124.96, "text": "that"}, {"end": 125.48, "start": 125.24, "text": "on"}, {"end": 125.76, "start": 125.48, "text": "to"}, {"end": 126.2, "start": 125.76, "text": "what"}, {"end": 126.32, "start": 126.2, "text": "you"}, {"end": 126.44, "start": 126.32, "text": "had."}, {"end": 126.76, "start": 126.44, "text": "So"}, {"end": 126.88, "start": 126.76, "text": "it's"}, {"end": 127.56, "start": 126.88, "text": "\u00a32.25."}, {"end": 128.12, "start": 127.56, "text": "Better."}, {"end": 128.52, "start": 128.12, "text": "It's"}, {"end": 129.44, "start": 128.52, "text": "better."}, {"end": 129.72, "start": 129.44, "text": "So"}, {"end": 130.08, "start": 129.72, "text": "what"}, {"end": 130.64, "start": 130.08, "text": "happens"}, {"end": 130.64, "start": 130.64, "text": "if"}, {"end": 130.68, "start": 130.64, "text": "I"}, {"end": 130.96, "start": 130.68, "text": "do"}, {"end": 131.2, "start": 130.96, "text": "this"}, {"end": 131.48, "start": 131.2, "text": "more"}, {"end": 131.92, "start": 131.48, "text": "regularly?"}, {"end": 132.36, "start": 131.92, "text": "What"}, {"end": 132.76, "start": 132.36, "text": "if"}, {"end": 132.92, "start": 132.76, "text": "I"}, {"end": 132.92, "start": 132.92, "text": "do"}, {"end": 133.44, "start": 132.92, "text": "it"}, {"end": 133.52, "start": 133.44, "text": "every"}, {"end": 133.6, "start": 133.52, "text": "month?"}, {"end": 133.92, "start": 133.6, "text": "I"}, {"end": 134.28, "start": 133.92, "text": "offer"}, {"end": 134.52, "start": 134.28, "text": "you"}, {"end": 134.8, "start": 134.52, "text": "1"}, {"end": 135.44, "start": 134.8, "text": "12th"}, {"end": 136.92000000000002, "start": 135.44, "text": "interest"}, {"end": 137.36, "start": 136.92000000000002, "text": "every"}, {"end": 138.36, "start": 137.36, "text": "month."}, {"end": 138.48, "start": 138.36, "text": "Is"}, {"end": 138.76, "start": 138.48, "text": "that"}, {"end": 139.24, "start": 138.76, "text": "better?"}, {"end": 139.32, "start": 139.24, "text": "So"}, {"end": 139.52, "start": 139.32, "text": "let's"}, {"end": 139.8, "start": 139.52, "text": "think"}, {"end": 140.16, "start": 139.8, "text": "about"}, {"end": 140.28, "start": 140.16, "text": "that."}, {"end": 140.72, "start": 140.28, "text": "So"}, {"end": 140.76, "start": 140.72, "text": "after"}, {"end": 140.96, "start": 140.76, "text": "the"}, {"end": 141.24, "start": 140.96, "text": "first"}, {"end": 142.4, "start": 141.24, "text": "month,"}, {"end": 142.8, "start": 142.4, "text": "it's"}, {"end": 143.07999999999998, "start": 142.8, "text": "going"}, {"end": 143.07999999999998, "start": 143.07999999999998, "text": "to"}, {"end": 143.16, "start": 143.07999999999998, "text": "be"}, {"end": 143.68, "start": 143.16, "text": "multiplied"}, {"end": 144.0, "start": 143.68, "text": "by"}, {"end": 144.2, "start": 144.0, "text": "this."}, {"end": 145.12, "start": 144.2, "text": "1"}, {"end": 147.28, "start": 145.12, "text": "plus"}, {"end": 147.4, "start": 147.28, "text": "1"}, {"end": 148.2, "start": 147.4, "text": "12th."}, {"end": 148.36, "start": 148.2, "text": "So"}, {"end": 148.48, "start": 148.36, "text": "1"}, {"end": 148.76, "start": 148.48, "text": "12th,"}, {"end": 148.96, "start": 148.76, "text": "that's"}, {"end": 149.12, "start": 148.96, "text": "your"}, {"end": 149.96, "start": 149.12, "text": "interest."}], "text": " 50% interest on your total, which is another 75p. And then you add that on to what you had. So it's \u00a32.25. Better. It's better. So what happens if I do this more regularly? What if I do it every month? I offer you 1 12th interest every month. Is that better? So let's think about that. So after the first month, it's going to be multiplied by this. 1 plus 1 12th. So 1 12th, that's your interest."}, {"chunks": [{"end": 150.2, "start": 150.0, "text": "And"}, {"end": 150.44, "start": 150.2, "text": "then"}, {"end": 150.92, "start": 150.44, "text": "you're"}, {"end": 151.48, "start": 150.92, "text": "adding"}, {"end": 151.48, "start": 151.48, "text": "that"}, {"end": 151.48, "start": 151.48, "text": "onto"}, {"end": 151.76, "start": 151.48, "text": "the"}, {"end": 152.24, "start": 151.76, "text": "original"}, {"end": 152.76, "start": 152.24, "text": "pound"}, {"end": 152.88, "start": 152.76, "text": "that"}, {"end": 153.12, "start": 152.88, "text": "you've"}, {"end": 153.4, "start": 153.12, "text": "got."}, {"end": 154.08, "start": 153.4, "text": "So"}, {"end": 154.12, "start": 154.08, "text": "you"}, {"end": 154.2, "start": 154.12, "text": "do"}, {"end": 154.48, "start": 154.2, "text": "that,"}, {"end": 154.84, "start": 154.48, "text": "that's"}, {"end": 155.08, "start": 154.84, "text": "your"}, {"end": 155.36, "start": 155.08, "text": "first"}, {"end": 156.32, "start": 155.36, "text": "month."}, {"end": 156.64, "start": 156.32, "text": "Then"}, {"end": 156.92, "start": 156.64, "text": "for"}, {"end": 157.16, "start": 156.92, "text": "your"}, {"end": 157.16, "start": 157.16, "text": "second"}, {"end": 157.32, "start": 157.16, "text": "month,"}, {"end": 157.4, "start": 157.32, "text": "you"}, {"end": 157.6, "start": 157.4, "text": "take"}, {"end": 158.0, "start": 157.6, "text": "that"}, {"end": 158.16, "start": 158.0, "text": "and"}, {"end": 158.44, "start": 158.16, "text": "multiply"}, {"end": 158.52, "start": 158.44, "text": "it"}, {"end": 158.8, "start": 158.52, "text": "again"}, {"end": 159.24, "start": 158.8, "text": "by"}, {"end": 159.44, "start": 159.24, "text": "the"}, {"end": 159.68, "start": 159.44, "text": "same"}, {"end": 160.44, "start": 159.68, "text": "value."}, {"end": 160.68, "start": 160.44, "text": "And"}, {"end": 160.96, "start": 160.68, "text": "then"}, {"end": 161.12, "start": 160.96, "text": "your"}, {"end": 161.12, "start": 161.12, "text": "third"}, {"end": 161.16, "start": 161.12, "text": "month,"}, {"end": 161.4, "start": 161.16, "text": "you"}, {"end": 161.72, "start": 161.4, "text": "would"}, {"end": 162.0, "start": 161.72, "text": "multiply"}, {"end": 162.0, "start": 162.0, "text": "it"}, {"end": 162.44, "start": 162.0, "text": "again"}, {"end": 162.64, "start": 162.44, "text": "and"}, {"end": 163.08, "start": 162.64, "text": "again."}, {"end": 163.16, "start": 163.08, "text": "You"}, {"end": 163.64, "start": 163.16, "text": "actually"}, {"end": 163.64, "start": 163.64, "text": "do"}, {"end": 163.64, "start": 163.64, "text": "that"}, {"end": 163.92, "start": 163.64, "text": "12"}, {"end": 164.36, "start": 163.92, "text": "times"}, {"end": 164.52, "start": 164.36, "text": "in"}, {"end": 164.92, "start": 164.52, "text": "a"}, {"end": 165.08, "start": 164.92, "text": "year."}, {"end": 165.4, "start": 165.08, "text": "So"}, {"end": 165.4, "start": 165.4, "text": "in"}, {"end": 165.44, "start": 165.4, "text": "a"}, {"end": 165.48, "start": 165.44, "text": "year,"}, {"end": 166.88, "start": 165.48, "text": "you"}, {"end": 167.0, "start": 166.88, "text": "would"}, {"end": 167.2, "start": 167.0, "text": "raise"}, {"end": 167.4, "start": 167.2, "text": "that"}, {"end": 167.4, "start": 167.4, "text": "to"}, {"end": 167.4, "start": 167.4, "text": "the"}, {"end": 167.76, "start": 167.4, "text": "power"}, {"end": 168.16, "start": 167.76, "text": "of"}, {"end": 169.52, "start": 168.16, "text": "12,"}, {"end": 169.72, "start": 169.52, "text": "and"}, {"end": 169.72, "start": 169.72, "text": "you"}, {"end": 170.16, "start": 169.72, "text": "would"}, {"end": 170.68, "start": 170.16, "text": "get"}, {"end": 172.48, "start": 170.68, "text": "\u00a32.61."}, {"end": 172.56, "start": 172.48, "text": "So"}, {"end": 172.68, "start": 172.56, "text": "it's"}, {"end": 172.96, "start": 172.68, "text": "actually"}, {"end": 173.36, "start": 172.96, "text": "better."}, {"end": 173.44, "start": 173.36, "text": "In"}, {"end": 173.76, "start": 173.44, "text": "fact,"}, {"end": 173.8, "start": 173.76, "text": "the"}, {"end": 174.32, "start": 173.8, "text": "more"}, {"end": 175.52, "start": 174.32, "text": "frequent"}, {"end": 175.76, "start": 175.52, "text": "your"}, {"end": 176.28, "start": 175.76, "text": "interest"}, {"end": 177.0, "start": 176.28, "text": "is,"}, {"end": 177.24, "start": 177.0, "text": "the"}, {"end": 177.56, "start": 177.24, "text": "better"}, {"end": 177.64, "start": 177.56, "text": "the"}, {"end": 178.36, "start": 177.64, "text": "result."}, {"end": 178.6, "start": 178.36, "text": "Let's"}, {"end": 179.04, "start": 178.6, "text": "start"}, {"end": 179.52, "start": 179.04, "text": "with"}, {"end": 179.96, "start": 179.52, "text": "every"}], "text": " And then you're adding that onto the original pound that you've got. So you do that, that's your first month. Then for your second month, you take that and multiply it again by the same value. And then your third month, you would multiply it again and again. You actually do that 12 times in a year. So in a year, you would raise that to the power of 12, and you would get \u00a32.61. So it's actually better. In fact, the more frequent your interest is, the better the result. 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"12"}, {"end": 208.52, "start": 208.52, "text": "if"}, {"end": 208.52, "start": 208.52, "text": "you"}, {"end": 208.68, "start": 208.52, "text": "do"}, {"end": 208.88, "start": 208.68, "text": "it"}, {"end": 209.16, "start": 208.88, "text": "every"}, {"end": 209.96, "start": 209.16, "text": "month."}], "text": " OK, so if we do it for every week, how much better is that? OK, what I'm saying is you're earning 1 over 52 interest every week. And then after the end of the year, you've got 52 weeks, and you would have \u00a32.69. So it's getting better and better and better. In general, you might be able to see a pattern happening here. In general, it would look like this. You'd be multiplying by 1 plus 1 over n to the power n. Hopefully, you can see that pattern happening. So here, n is equal to 12 if you do it every month."}, {"chunks": [{"end": 210.2, "start": 210.0, "text": "52"}, {"end": 210.44, "start": 210.2, "text": "if"}, {"end": 210.72, "start": 210.44, "text": "you"}, {"end": 210.76, "start": 210.72, "text": "do"}, {"end": 210.92, "start": 210.76, "text": "it"}, {"end": 211.24, "start": 210.92, "text": "every"}, {"end": 211.44, "start": 211.24, "text": "week."}, {"end": 211.48, "start": 211.44, "text": "If"}, {"end": 211.6, "start": 211.48, "text": "you"}, {"end": 211.64, "start": 211.6, "text": "did"}, {"end": 211.88, "start": 211.64, "text": "it"}, {"end": 212.24, "start": 211.88, "text": "every"}, {"end": 212.6, "start": 212.24, "text": "day,"}, {"end": 213.0, "start": 212.6, "text": "it'd"}, {"end": 213.12, "start": 213.0, "text": "be"}, {"end": 213.32, "start": 213.12, "text": "one"}, {"end": 213.76, "start": 213.32, "text": "pound"}, {"end": 214.28, "start": 213.76, "text": "multiplied"}, {"end": 214.52, "start": 214.28, "text": "by"}, {"end": 214.76, "start": 214.52, "text": "one"}, {"end": 215.56, "start": 214.76, "text": "over"}, {"end": 216.0, "start": 215.56, "text": "one"}, {"end": 216.28, "start": 216.0, "text": "over"}, {"end": 217.24, "start": 216.28, "text": "365"}, {"end": 217.76, "start": 217.24, "text": "to"}, {"end": 217.8, "start": 217.76, "text": "the"}, {"end": 218.08, "start": 217.8, "text": "power"}, {"end": 218.12, "start": 218.08, "text": "of"}, {"end": 219.04, "start": 218.12, "text": "365."}, {"end": 220.24, "start": 219.04, "text": "And"}, {"end": 220.88, "start": 220.24, "text": "that's"}, {"end": 221.0, "start": 220.88, "text": "equal"}, {"end": 221.0, "start": 221.0, "text": "to"}, {"end": 221.36, "start": 221.0, "text": "two"}, {"end": 222.28, "start": 221.36, "text": "pounds"}, {"end": 224.56, "start": 222.28, "text": "71."}, {"end": 225.0, "start": 224.56, "text": "And"}, {"end": 225.4, "start": 225.0, "text": "so"}, {"end": 225.4, "start": 225.4, "text": "it"}, {"end": 225.96, "start": 225.4, "text": "would"}, {"end": 226.36, "start": 225.96, "text": "get"}, {"end": 226.72, "start": 226.36, "text": "better"}, {"end": 226.84, "start": 226.72, "text": "if"}, {"end": 227.07999999999998, "start": 226.84, "text": "you"}, {"end": 227.16, "start": 227.07999999999998, "text": "did"}, {"end": 227.56, "start": 227.16, "text": "it"}, {"end": 227.88, "start": 227.56, "text": "every"}, {"end": 228.36, "start": 227.88, "text": "second"}, {"end": 228.72, "start": 228.36, "text": "or"}, {"end": 228.84, "start": 228.72, "text": "every"}, {"end": 230.04, "start": 228.84, "text": "nanosecond."}, {"end": 230.28, "start": 230.04, "text": "What"}, {"end": 230.44, "start": 230.28, "text": "if"}, {"end": 230.68, "start": 230.44, "text": "I"}, {"end": 230.84, "start": 230.68, "text": "could"}, {"end": 230.88, "start": 230.84, "text": "do"}, {"end": 231.0, "start": 230.88, "text": "it"}, {"end": 231.76, "start": 231.0, "text": "continuously?"}, {"end": 232.16, "start": 231.76, "text": "Every"}, {"end": 232.92000000000002, "start": 232.16, "text": "instant"}, {"end": 233.16, "start": 232.92000000000002, "text": "I'm"}, {"end": 233.48, "start": 233.16, "text": "earning"}, {"end": 233.84, "start": 233.48, "text": "interest,"}, {"end": 234.8, "start": 233.84, "text": "continuous"}, {"end": 235.84, "start": 234.8, "text": "interest."}, {"end": 236.04, "start": 235.84, "text": "What"}, {"end": 236.07999999999998, "start": 236.04, "text": "does"}, {"end": 236.32, "start": 236.07999999999998, "text": "that"}, {"end": 236.52, "start": 236.32, "text": "look"}, {"end": 236.76, "start": 236.52, "text": "like?"}, {"end": 237.04, "start": 236.76, "text": "That"}, {"end": 237.76, "start": 237.04, "text": "means"}, {"end": 238.04, "start": 237.76, "text": "if"}, {"end": 238.32, "start": 238.04, "text": "I"}, {"end": 238.56, "start": 238.32, "text": "take"}, {"end": 238.84, "start": 238.56, "text": "this"}, {"end": 239.24, "start": 238.84, "text": "formula"}, {"end": 239.96, "start": 239.24, "text": "here,"}], "text": " 52 if you do it every week. If you did it every day, it'd be one pound multiplied by one over one over 365 to the power of 365. And that's equal to two pounds 71. And so it would get better if you did it every second or every nanosecond. What if I could do it continuously? Every instant I'm earning interest, continuous interest. What does that look like? That means if I take this formula here,"}, {"chunks": [{"end": 240.0, "start": 240.0, "text": "1"}, {"end": 241.4, "start": 240.0, "text": "plus"}, {"end": 241.68, "start": 241.4, "text": "1"}, {"end": 242.16, "start": 241.68, "text": "over"}, {"end": 242.36, "start": 242.16, "text": "n"}, {"end": 242.4, "start": 242.36, "text": "to"}, {"end": 243.04, "start": 242.4, "text": "the"}, {"end": 243.32, "start": 243.04, "text": "n."}, {"end": 243.68, "start": 243.32, "text": "I'm"}, {"end": 244.0, "start": 243.68, "text": "going"}, {"end": 245.44, "start": 244.0, "text": "to"}, {"end": 245.76, "start": 245.44, "text": "n"}, {"end": 246.32, "start": 245.76, "text": "10"}, {"end": 246.64, "start": 246.32, "text": "to"}, {"end": 246.76, "start": 246.64, "text": "infinity."}, {"end": 247.4, "start": 246.76, "text": "That"}, {"end": 247.72, "start": 247.4, "text": "would"}, {"end": 247.88, "start": 247.72, "text": "be"}, {"end": 248.92, "start": 247.88, "text": "continuous"}, {"end": 250.16, "start": 248.92, "text": "interest."}, {"end": 250.4, "start": 250.16, "text": "Now"}, {"end": 250.96, "start": 250.4, "text": "what"}, {"end": 251.16, "start": 250.96, "text": "is"}, {"end": 251.44, "start": 251.16, "text": "that?"}, {"end": 251.48, "start": 251.44, "text": "What"}, {"end": 251.84, "start": 251.48, "text": "is"}, {"end": 252.12, "start": 251.84, "text": "that"}, {"end": 252.36, "start": 252.12, "text": "value?"}, {"end": 252.88, "start": 252.36, "text": "And"}, {"end": 253.04, "start": 252.88, "text": "that's"}, {"end": 253.2, "start": 253.04, "text": "what"}, {"end": 253.48, "start": 253.2, "text": "Bernoulli"}, {"end": 253.76, "start": 253.48, "text": "wanted"}, {"end": 253.8, "start": 253.76, "text": "to"}, {"end": 254.24, "start": 253.8, "text": "know."}, {"end": 254.44, "start": 254.24, "text": "He"}, {"end": 254.44, "start": 254.44, "text": "didn't"}, {"end": 255.0, "start": 254.44, "text": "work"}, {"end": 255.28, "start": 255.0, "text": "it"}, {"end": 255.48, "start": 255.28, "text": "out."}, {"end": 255.64, "start": 255.48, "text": "He"}, {"end": 255.64, "start": 255.64, "text": "knew"}, {"end": 255.8, "start": 255.64, "text": "it"}, {"end": 255.88, "start": 255.8, "text": "was"}, {"end": 256.52, "start": 255.88, "text": "between"}, {"end": 256.8, "start": 256.52, "text": "2"}, {"end": 257.04, "start": 256.8, "text": "and"}, {"end": 257.36, "start": 257.04, "text": "3."}, {"end": 257.64, "start": 257.36, "text": "So"}, {"end": 257.92, "start": 257.64, "text": "50"}, {"end": 258.4, "start": 257.92, "text": "years"}, {"end": 259.44, "start": 258.4, "text": "later,"}, {"end": 259.96, "start": 259.44, "text": "Euler"}, {"end": 260.24, "start": 259.96, "text": "worked"}, {"end": 260.28, "start": 260.24, "text": "it"}, {"end": 261.16, "start": 260.28, "text": "out."}, {"end": 261.6, "start": 261.16, "text": "Euler,"}, {"end": 261.68, "start": 261.6, "text": "he"}, {"end": 262.0, "start": 261.68, "text": "works"}, {"end": 262.4, "start": 262.0, "text": "everything"}, {"end": 263.0, "start": 262.4, "text": "out."}, {"end": 263.08, "start": 263.0, "text": "He"}, {"end": 263.84, "start": 263.08, "text": "more"}, {"end": 264.48, "start": 263.84, "text": "Gauss."}, {"end": 264.88, "start": 264.48, "text": "It's"}, {"end": 265.56, "start": 264.88, "text": "either"}, {"end": 265.88, "start": 265.56, "text": "Euler"}, {"end": 265.92, "start": 265.88, "text": "or"}, {"end": 266.28, "start": 265.92, "text": "Gauss."}, {"end": 266.68, "start": 266.28, "text": "Say"}, {"end": 267.0, "start": 266.68, "text": "Euler"}, {"end": 267.04, "start": 267.0, "text": "or"}, {"end": 267.32, "start": 267.04, "text": "Gauss,"}, {"end": 267.44, "start": 267.32, "text": "you're"}, {"end": 267.72, "start": 267.44, "text": "probably"}, {"end": 267.88, "start": 267.72, "text": "going"}, {"end": 267.92, "start": 267.88, "text": "to"}, {"end": 268.04, "start": 267.92, "text": "be"}, {"end": 268.36, "start": 268.04, "text": "right."}, {"end": 268.6, "start": 268.36, "text": "And"}, {"end": 268.6, "start": 268.6, "text": "the"}, {"end": 269.0, "start": 268.6, "text": "value"}, {"end": 269.44, "start": 269.0, "text": "was"}, {"end": 269.96, "start": 269.44, "text": "2."}], "text": " 1 plus 1 over n to the n. I'm going to n 10 to infinity. That would be continuous interest. Now what is that? What is that value? And that's what Bernoulli wanted to know. He didn't work it out. He knew it was between 2 and 3. So 50 years later, Euler worked it out. Euler, he works everything out. He more Gauss. It's either Euler or Gauss. Say Euler or Gauss, you're probably going to be right. And the value was 2."}, {"chunks": [{"end": 270.84, "start": 270.0, "text": "7,"}, {"end": 271.68, "start": 270.84, "text": "1,"}, {"end": 272.04, "start": 271.68, "text": "8,"}, {"end": 272.4, "start": 272.04, "text": "2,"}, {"end": 272.92, "start": 272.4, "text": "8,"}, {"end": 273.28, "start": 272.92, "text": "1,"}, {"end": 273.52, "start": 273.28, "text": "8,"}, {"end": 274.0, "start": 273.52, "text": "2,"}, {"end": 274.32, "start": 274.0, "text": "8,"}, {"end": 274.96, "start": 274.32, "text": "4,"}, {"end": 275.48, "start": 274.96, "text": "5,"}, {"end": 276.16, "start": 275.48, "text": "9,"}, {"end": 276.28, "start": 276.16, "text": "and"}, {"end": 276.48, "start": 276.28, "text": "so"}, {"end": 276.6, "start": 276.48, "text": "on."}, {"end": 276.76, "start": 276.6, "text": "We"}, {"end": 277.12, "start": 276.76, "text": "were"}, {"end": 277.36, "start": 277.12, "text": "pretty"}, {"end": 277.8, "start": 277.36, "text": "close"}, {"end": 278.0, "start": 277.8, "text": "when"}, {"end": 278.12, "start": 278.0, "text": "we"}, {"end": 278.36, "start": 278.12, "text": "were"}, {"end": 278.72, "start": 278.36, "text": "doing"}, {"end": 278.96, "start": 278.72, "text": "it"}, {"end": 279.32, "start": 278.96, "text": "daily,"}, {"end": 279.52, "start": 279.32, "text": "weren't"}, {"end": 279.52, "start": 279.52, "text": "we?"}, {"end": 279.64, "start": 279.52, "text": "It"}, {"end": 279.84, "start": 279.64, "text": "was"}, {"end": 280.24, "start": 279.84, "text": "already"}, {"end": 280.88, "start": 280.24, "text": "271"}, {"end": 281.08, "start": 280.88, "text": "at"}, {"end": 281.16, "start": 281.08, "text": "daily."}, {"end": 281.44, "start": 281.16, "text": "You're"}, {"end": 281.84, "start": 281.44, "text": "right,"}, {"end": 282.12, "start": 281.84, "text": "you're"}, {"end": 282.56, "start": 282.12, "text": "right."}, {"end": 282.72, "start": 282.56, "text": "We're"}, {"end": 282.88, "start": 282.72, "text": "getting"}, {"end": 283.32, "start": 282.88, "text": "closer,"}, {"end": 283.64, "start": 283.32, "text": "weren't"}, {"end": 283.92, "start": 283.64, "text": "we?"}, {"end": 284.36, "start": 283.92, "text": "We're"}, {"end": 284.44, "start": 284.36, "text": "getting"}, {"end": 285.0, "start": 284.44, "text": "closer"}, {"end": 285.0, "start": 285.0, "text": "and"}, {"end": 285.4, "start": 285.0, "text": "closer"}, {"end": 285.4, "start": 285.4, "text": "to"}, {"end": 285.76, "start": 285.4, "text": "this"}, {"end": 286.56, "start": 285.76, "text": "value."}, {"end": 286.72, "start": 286.56, "text": "So"}, {"end": 287.32, "start": 286.72, "text": "already"}, {"end": 287.48, "start": 287.32, "text": "we're"}, {"end": 287.88, "start": 287.48, "text": "quite"}, {"end": 288.28, "start": 287.88, "text": "close"}, {"end": 288.6, "start": 288.28, "text": "to"}, {"end": 288.96, "start": 288.6, "text": "it."}, {"end": 289.32, "start": 288.96, "text": "If"}, {"end": 289.72, "start": 289.32, "text": "you"}, {"end": 290.0, "start": 289.72, "text": "did"}, {"end": 290.2, "start": 290.0, "text": "it"}, {"end": 290.72, "start": 290.2, "text": "forever,"}, {"end": 290.96, "start": 290.72, "text": "though,"}, {"end": 291.0, "start": 290.96, "text": "of"}, {"end": 291.52, "start": 291.0, "text": "course,"}, {"end": 291.68, "start": 291.52, "text": "you"}, {"end": 291.72, "start": 291.68, "text": "would"}, {"end": 291.72, "start": 291.72, "text": "have"}, {"end": 291.96, "start": 291.72, "text": "this"}, {"end": 292.72, "start": 291.96, "text": "irrational"}, {"end": 293.0, "start": 292.72, "text": "number."}, {"end": 293.2, "start": 293.0, "text": "Now,"}, {"end": 293.56, "start": 293.2, "text": "Euler"}, {"end": 293.88, "start": 293.56, "text": "called"}, {"end": 294.36, "start": 293.88, "text": "this"}, {"end": 294.52, "start": 294.36, "text": "E."}, {"end": 294.84, "start": 294.52, "text": "He"}, {"end": 295.32, "start": 294.84, "text": "didn't"}, {"end": 295.68, "start": 295.32, "text": "name"}, {"end": 295.68, "start": 295.68, "text": "it"}, {"end": 295.68, "start": 295.68, "text": "after"}, {"end": 296.32, "start": 295.68, "text": "himself,"}, {"end": 296.92, "start": 296.32, "text": "although"}, {"end": 297.04, "start": 296.92, "text": "it"}, {"end": 297.28, "start": 297.04, "text": "is"}, {"end": 297.48, "start": 297.28, "text": "now"}, {"end": 297.64, "start": 297.48, "text": "known"}, {"end": 297.76, "start": 297.64, "text": "as"}, {"end": 297.92, "start": 297.76, "text": "the"}, {"end": 298.24, "start": 297.92, "text": "Euler"}, {"end": 298.6, "start": 298.24, "text": "constant."}, {"end": 298.8, "start": 298.6, "text": "Why"}, {"end": 299.04, "start": 298.8, "text": "did"}, {"end": 299.28, "start": 299.04, "text": "he"}, {"end": 299.64, "start": 299.28, "text": "call"}, {"end": 299.64, "start": 299.64, "text": "it"}, {"end": 299.96, "start": 299.64, "text": "E?"}], "text": " 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, and so on. We were pretty close when we were doing it daily, weren't we? It was already 271 at daily. You're right, you're right. We're getting closer, weren't we? We're getting closer and closer to this value. So already we're quite close to it. If you did it forever, though, of course, you would have this irrational number. Now, Euler called this E. He didn't name it after himself, although it is now known as the Euler constant. Why did he call it E?"}, {"chunks": [{"end": 300.16, "start": 300.0, "text": "It"}, {"end": 300.4, "start": 300.16, "text": "was"}, {"end": 300.44, "start": 300.4, "text": "just"}, {"end": 300.6, "start": 300.44, "text": "a"}, {"end": 301.2, "start": 300.6, "text": "letter."}, {"end": 301.28, "start": 301.2, "text": "He"}, {"end": 301.56, "start": 301.28, "text": "might"}, {"end": 301.76, "start": 301.56, "text": "have"}, {"end": 302.16, "start": 301.76, "text": "used"}, {"end": 302.36, "start": 302.16, "text": "A,"}, {"end": 302.4, "start": 302.36, "text": "B,"}, {"end": 302.68, "start": 302.4, "text": "C"}, {"end": 302.88, "start": 302.68, "text": "and"}, {"end": 302.92, "start": 302.88, "text": "D"}, {"end": 303.12, "start": 302.92, "text": "already"}, {"end": 303.4, "start": 303.12, "text": "for"}, {"end": 304.16, "start": 303.4, "text": "something"}, {"end": 304.44, "start": 304.16, "text": "else,"}, {"end": 304.8, "start": 304.44, "text": "right?"}, {"end": 305.08, "start": 304.8, "text": "So"}, {"end": 305.36, "start": 305.08, "text": "you"}, {"end": 305.88, "start": 305.36, "text": "use"}, {"end": 306.4, "start": 305.88, "text": "the"}, {"end": 306.64, "start": 306.4, "text": "next"}, {"end": 307.0, "start": 306.64, "text": "one."}, {"end": 307.24, "start": 307.0, "text": "Bit"}, {"end": 307.56, "start": 307.24, "text": "of"}, {"end": 307.8, "start": 307.56, "text": "a"}, {"end": 308.76, "start": 307.8, "text": "coincidence."}, {"end": 309.0, "start": 308.76, "text": "It's"}, {"end": 309.24, "start": 309.0, "text": "a"}, {"end": 309.64, "start": 309.24, "text": "lovely"}, {"end": 310.2, "start": 309.64, "text": "coincidence."}, {"end": 310.56, "start": 310.2, "text": "I"}, {"end": 310.88, "start": 310.56, "text": "fully"}, {"end": 311.6, "start": 310.88, "text": "believe"}, {"end": 311.6, "start": 311.6, "text": "that"}, {"end": 312.24, "start": 311.6, "text": "he's"}, {"end": 312.48, "start": 312.24, "text": "not"}, {"end": 312.76, "start": 312.48, "text": "being"}, {"end": 312.8, "start": 312.76, "text": "a"}, {"end": 313.16, "start": 312.8, "text": "jerk"}, {"end": 313.52, "start": 313.16, "text": "here,"}, {"end": 313.84, "start": 313.52, "text": "naming"}, {"end": 314.08, "start": 313.84, "text": "it"}, {"end": 314.28, "start": 314.08, "text": "after"}, {"end": 314.8, "start": 314.28, "text": "himself,"}, {"end": 315.68, "start": 314.8, "text": "but"}, {"end": 316.2, "start": 315.68, "text": "it's"}, {"end": 316.44, "start": 316.2, "text": "a"}, {"end": 316.56, "start": 316.44, "text": "lovely"}, {"end": 317.88, "start": 316.56, "text": "coincidence"}, {"end": 318.04, "start": 317.88, "text": "that"}, {"end": 318.36, "start": 318.04, "text": "it's"}, {"end": 318.36, "start": 318.36, "text": "E"}, {"end": 318.84, "start": 318.36, "text": "for"}, {"end": 319.4, "start": 318.84, "text": "Euler's"}, {"end": 319.92, "start": 319.4, "text": "number."}, {"end": 320.04, "start": 319.92, "text": "Would"}, {"end": 320.28, "start": 320.04, "text": "you"}, {"end": 320.28, "start": 320.28, "text": "have"}, {"end": 320.6, "start": 320.28, "text": "called"}, {"end": 320.68, "start": 320.6, "text": "it"}, {"end": 321.12, "start": 320.68, "text": "G"}, {"end": 321.2, "start": 321.12, "text": "if"}, {"end": 321.2, "start": 321.2, "text": "you"}, {"end": 321.76, "start": 321.2, "text": "discovered"}, {"end": 321.8, "start": 321.76, "text": "it?"}, {"end": 322.24, "start": 321.8, "text": "I"}, {"end": 322.52, "start": 322.24, "text": "would"}, {"end": 322.64, "start": 322.52, "text": "not"}, {"end": 323.04, "start": 322.64, "text": "have"}, {"end": 323.4, "start": 323.04, "text": "called"}, {"end": 323.44, "start": 323.4, "text": "it"}, {"end": 323.6, "start": 323.44, "text": "G."}, {"end": 324.16, "start": 323.6, "text": "No."}, {"end": 324.28, "start": 324.16, "text": "I"}, {"end": 324.28, "start": 324.28, "text": "would"}, {"end": 324.52, "start": 324.28, "text": "have"}, {"end": 324.8, "start": 324.52, "text": "hoped"}, {"end": 325.24, "start": 324.8, "text": "somebody"}, {"end": 325.68, "start": 325.24, "text": "else"}, {"end": 325.84, "start": 325.68, "text": "would"}, {"end": 325.84, "start": 325.84, "text": "have"}, {"end": 326.24, "start": 325.84, "text": "called"}, {"end": 326.28, "start": 326.24, "text": "it"}, {"end": 326.88, "start": 326.28, "text": "G,"}, {"end": 327.48, "start": 326.88, "text": "and"}, {"end": 327.76, "start": 327.48, "text": "then"}, {"end": 327.8, "start": 327.76, "text": "I"}, {"end": 327.92, "start": 327.8, "text": "would"}, {"end": 328.16, "start": 327.92, "text": "have"}, {"end": 328.36, "start": 328.16, "text": "accepted"}, {"end": 328.36, "start": 328.36, "text": "that."}, {"end": 328.96, "start": 328.36, "text": "Euler"}, {"end": 329.8, "start": 328.96, "text": "proved"}, {"end": 329.96, "start": 329.8, "text": "that"}], "text": " It was just a letter. He might have used A, B, C and D already for something else, right? So you use the next one. Bit of a coincidence. It's a lovely coincidence. I fully believe that he's not being a jerk here, naming it after himself, but it's a lovely coincidence that it's E for Euler's number. Would you have called it G if you discovered it? I would not have called it G. No. I would have hoped somebody else would have called it G, and then I would have accepted that. Euler proved that"}, {"chunks": [{"end": 330.16, "start": 330.0, "text": "This"}, {"end": 330.32, "start": 330.16, "text": "was"}, {"end": 330.72, "start": 330.32, "text": "irrational."}, {"end": 331.64, "start": 330.72, "text": "He"}, {"end": 332.36, "start": 331.64, "text": "found"}, {"end": 332.44, "start": 332.36, "text": "a"}, {"end": 332.8, "start": 332.44, "text": "formula"}, {"end": 333.36, "start": 332.8, "text": "for"}, {"end": 333.52, "start": 333.36, "text": "e,"}, {"end": 334.52, "start": 333.52, "text": "which"}, {"end": 335.12, "start": 334.52, "text": "was"}, {"end": 335.52, "start": 335.12, "text": "a"}, {"end": 335.72, "start": 335.52, "text": "new"}, {"end": 336.08, "start": 335.72, "text": "formula,"}, {"end": 336.28, "start": 336.08, "text": "not"}, {"end": 336.56, "start": 336.28, "text": "this"}, {"end": 336.92, "start": 336.56, "text": "one"}, {"end": 337.84, "start": 336.92, "text": "here,"}, {"end": 337.96, "start": 337.84, "text": "a"}, {"end": 338.24, "start": 337.96, "text": "different"}, {"end": 338.56, "start": 338.24, "text": "formula,"}, {"end": 339.08, "start": 338.56, "text": "and"}, {"end": 339.48, "start": 339.08, "text": "it"}, {"end": 339.88, "start": 339.48, "text": "showed"}, {"end": 340.4, "start": 339.88, "text": "that"}, {"end": 340.6, "start": 340.4, "text": "it"}, {"end": 341.04, "start": 340.6, "text": "was"}, {"end": 341.52, "start": 341.04, "text": "irrational."}, {"end": 341.88, "start": 341.52, "text": "I'll"}, {"end": 342.48, "start": 341.88, "text": "quickly"}, {"end": 342.92, "start": 342.48, "text": "show"}, {"end": 343.56, "start": 342.92, "text": "you"}, {"end": 343.76, "start": 343.56, "text": "that."}, {"end": 344.76, "start": 343.76, "text": "He"}, {"end": 345.28, "start": 344.76, "text": "found"}, {"end": 345.48, "start": 345.28, "text": "that"}, {"end": 345.76, "start": 345.48, "text": "e"}, {"end": 346.0, "start": 345.76, "text": "was"}, {"end": 346.0, "start": 346.0, "text": "equal"}, {"end": 346.0, "start": 346.0, "text": "to"}, {"end": 346.08, "start": 346.0, "text": "2"}, {"end": 346.92, "start": 346.08, "text": "plus"}, {"end": 347.4, "start": 346.92, "text": "1"}, {"end": 347.92, "start": 347.4, "text": "over"}, {"end": 348.16, "start": 347.92, "text": "1"}, {"end": 349.04, "start": 348.16, "text": "plus"}, {"end": 350.0, "start": 349.04, "text": "1"}, {"end": 350.28, "start": 350.0, "text": "over"}, {"end": 350.4, "start": 350.28, "text": "2"}, {"end": 351.12, "start": 350.4, "text": "plus"}, {"end": 351.6, "start": 351.12, "text": "1"}, {"end": 351.84, "start": 351.6, "text": "over"}, {"end": 352.24, "start": 351.84, "text": "1"}, {"end": 353.12, "start": 352.24, "text": "plus"}, {"end": 353.32, "start": 353.12, "text": "1"}, {"end": 353.6, "start": 353.32, "text": "over"}, {"end": 354.24, "start": 353.6, "text": "4"}, {"end": 355.08, "start": 354.24, "text": "plus"}, {"end": 355.48, "start": 355.08, "text": "1"}, {"end": 355.92, "start": 355.48, "text": "over"}, {"end": 356.32, "start": 355.92, "text": "1"}, {"end": 357.16, "start": 356.32, "text": "plus"}, {"end": 357.36, "start": 357.16, "text": "1"}, {"end": 357.92, "start": 357.36, "text": "over"}, {"end": 358.32, "start": 357.92, "text": "1"}, {"end": 359.12, "start": 358.32, "text": "plus"}, {"end": 359.52, "start": 359.12, "text": "1"}, {"end": 359.68, "start": 359.52, "text": "over"}, {"end": 359.96, "start": 359.68, "text": "6."}], "text": " This was irrational. He found a formula for e, which was a new formula, not this one here, a different formula, and it showed that it was irrational. I'll quickly show you that. He found that e was equal to 2 plus 1 over 1 plus 1 over 2 plus 1 over 1 plus 1 over 4 plus 1 over 1 plus 1 over 1 plus 1 over 6."}, {"chunks": [{"end": 360.12, "start": 360.0, "text": "And"}, {"end": 360.36, "start": 360.12, "text": "this"}, {"end": 360.76, "start": 360.36, "text": "goes"}, {"end": 360.96, "start": 360.76, "text": "on"}, {"end": 361.44, "start": 360.96, "text": "forever."}, {"end": 361.8, "start": 361.44, "text": "This"}, {"end": 362.12, "start": 361.8, "text": "is"}, {"end": 362.36, "start": 362.12, "text": "a"}, {"end": 362.72, "start": 362.36, "text": "fraction"}, {"end": 362.88, "start": 362.72, "text": "that"}, {"end": 363.32, "start": 362.88, "text": "goes"}, {"end": 363.36, "start": 363.32, "text": "on"}, {"end": 364.04, "start": 363.36, "text": "forever,"}, {"end": 364.68, "start": 364.04, "text": "continuous"}, {"end": 365.2, "start": 364.68, "text": "fraction."}, {"end": 365.48, "start": 365.2, "text": "But"}, {"end": 365.64, "start": 365.48, "text": "you"}, 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"start": 386.96, "text": "it"}, {"end": 387.4, "start": 387.08, "text": "up"}, {"end": 387.64, "start": 387.4, "text": "to"}, {"end": 388.12, "start": 387.64, "text": "18"}, {"end": 388.48, "start": 388.12, "text": "decimal"}, {"end": 389.04, "start": 388.48, "text": "places."}, {"end": 389.48, "start": 389.04, "text": "To"}, {"end": 389.72, "start": 389.48, "text": "do"}, {"end": 389.96, "start": 389.72, "text": "that,"}], "text": " And this goes on forever. This is a fraction that goes on forever, continuous fraction. But you can see it goes on forever because there's a pattern, and that pattern does hold. You've got 2, 1, 1, 4, 1, 1, 6, 1, 1, 8. So you can see that pattern goes on forever. And if the fraction goes on forever, it means it's an irrational number. If it didn't go on forever, it would terminate. And if it terminates, you can write it as a fraction. And he also worked out the value for e. He did it up to 18 decimal places. To do that,"}, {"chunks": [{"end": 390.04, "start": 390.0, "text": "He"}, {"end": 390.44, "start": 390.04, "text": "had"}, {"end": 390.64, "start": 390.44, "text": "a"}, {"end": 390.88, "start": 390.64, "text": "different"}, {"end": 391.36, "start": 390.88, "text": "formula"}, {"end": 391.44, "start": 391.36, "text": "to"}, {"end": 391.6, "start": 391.44, "text": "do"}, {"end": 391.8, "start": 391.6, "text": "that."}, {"end": 391.8, "start": 391.8, "text": "I'll"}, {"end": 391.8, "start": 391.8, "text": "show"}, {"end": 391.84, "start": 391.8, "text": "you"}, {"end": 392.12, "start": 391.84, "text": "that"}, {"end": 392.4, "start": 392.12, "text": "one."}, {"end": 392.68, "start": 392.4, "text": "So"}, {"end": 392.84, "start": 392.68, "text": "this"}, {"end": 393.64, "start": 392.84, "text": "time,"}, {"end": 393.68, "start": 393.64, "text": "he"}, {"end": 393.92, "start": 393.68, "text": "worked"}, {"end": 394.12, "start": 393.92, "text": "out"}, {"end": 394.12, "start": 394.12, "text": "that"}, {"end": 394.12, "start": 394.12, "text": "e"}, {"end": 394.2, "start": 394.12, "text": "was"}, {"end": 394.44, "start": 394.2, "text": "equal"}, {"end": 394.84, "start": 394.44, "text": "to"}, {"end": 395.24, "start": 394.84, "text": "1"}, {"end": 396.64, "start": 395.24, "text": "plus"}, {"end": 397.0, "start": 396.64, "text": "1"}, {"end": 397.16, "start": 397.0, "text": "over"}, {"end": 397.4, "start": 397.16, "text": "1"}, {"end": 397.92, "start": 397.4, "text": "factorial"}, {"end": 398.56, "start": 397.92, "text": "plus"}, {"end": 398.72, "start": 398.56, "text": "1"}, {"end": 399.0, "start": 398.72, "text": "over"}, {"end": 399.24, "start": 399.0, "text": "2"}, {"end": 400.28, "start": 399.24, "text": "factorial"}, {"end": 400.44, "start": 400.28, "text": "plus"}, {"end": 400.84, "start": 400.44, "text": "1"}, {"end": 401.28, "start": 400.84, "text": "over"}, {"end": 401.52, "start": 401.28, "text": "3"}, {"end": 402.2, "start": 401.52, "text": "factorial"}, {"end": 402.64, "start": 402.2, "text": "plus"}, {"end": 402.96, "start": 402.64, "text": "1"}, {"end": 403.24, "start": 402.96, "text": "over"}, {"end": 403.48, "start": 403.24, "text": "4"}, {"end": 404.04, "start": 403.48, "text": "factorial."}, {"end": 404.28, "start": 404.04, "text": "And"}, {"end": 404.52, "start": 404.28, "text": "this"}, {"end": 404.52, "start": 404.52, "text": "is"}, {"end": 404.56, "start": 404.52, "text": "something"}, {"end": 404.8, "start": 404.56, "text": "that's"}, {"end": 405.08, "start": 404.8, "text": "going"}, {"end": 405.68, "start": 405.08, "text": "on"}, {"end": 406.36, "start": 405.68, "text": "forever."}, {"end": 406.76, "start": 406.36, "text": "It's"}, {"end": 406.76, "start": 406.76, "text": "a"}, {"end": 407.56, "start": 406.76, "text": "nice"}, {"end": 408.16, "start": 407.56, "text": "formula."}, {"end": 408.24, "start": 408.16, "text": "If"}, {"end": 408.84, "start": 408.24, "text": "you're"}, {"end": 409.04, "start": 408.84, "text": "happy"}, {"end": 409.08, "start": 409.04, "text": "with"}, {"end": 409.56, "start": 409.08, "text": "factorials,"}, {"end": 410.16, "start": 409.56, "text": "factorials"}, {"end": 410.44, "start": 410.16, "text": "means"}, {"end": 410.6, "start": 410.44, "text": "you're"}, {"end": 411.72, "start": 410.6, "text": "multiplying"}, {"end": 411.92, "start": 411.72, "text": "all"}, {"end": 412.28, "start": 411.92, "text": "the"}, {"end": 412.96, "start": 412.28, "text": "numbers"}, {"end": 413.2, "start": 412.96, "text": "up"}, {"end": 413.36, "start": 413.2, "text": "to"}, {"end": 413.56, "start": 413.36, "text": "that"}, {"end": 414.08, "start": 413.56, "text": "value."}, {"end": 414.32, "start": 414.08, "text": "So"}, {"end": 414.56, "start": 414.32, "text": "if"}, {"end": 414.56, "start": 414.56, "text": "it"}, {"end": 414.6, "start": 414.56, "text": "was"}, {"end": 415.0, "start": 414.6, "text": "4"}, {"end": 415.48, "start": 415.0, "text": "factorial,"}, {"end": 415.52, "start": 415.48, "text": "it'd"}, {"end": 416.0, "start": 415.52, "text": "be"}, {"end": 416.24, "start": 416.0, "text": "4"}, {"end": 416.52, "start": 416.24, "text": "times"}, {"end": 416.76, "start": 416.52, "text": "3"}, {"end": 417.0, "start": 416.76, "text": "times"}, {"end": 417.16, "start": 417.0, "text": "2"}, {"end": 417.4, "start": 417.16, "text": "times"}, {"end": 417.72, "start": 417.4, "text": "1."}, {"end": 418.04, "start": 417.72, "text": "OK,"}, {"end": 418.4, "start": 418.04, "text": "why"}, {"end": 418.8, "start": 418.4, "text": "is"}, {"end": 418.84, "start": 418.8, "text": "e"}, {"end": 419.08, "start": 418.84, "text": "a"}, {"end": 419.64, "start": 419.08, "text": "big"}, {"end": 419.96, "start": 419.64, "text": "deal?"}], "text": " He had a different formula to do that. I'll show you that one. So this time, he worked out that e was equal to 1 plus 1 over 1 factorial plus 1 over 2 factorial plus 1 over 3 factorial plus 1 over 4 factorial. And this is something that's going on forever. It's a nice formula. If you're happy with factorials, factorials means you're multiplying all the numbers up to that value. So if it was 4 factorial, it'd be 4 times 3 times 2 times 1. OK, why is e a big deal?"}, {"chunks": [{"end": 420.36, "start": 420.0, "text": "It's"}, {"end": 421.12, "start": 420.36, "text": "because"}, {"end": 421.4, "start": 421.12, "text": "E"}, {"end": 421.52, "start": 421.4, "text": "is"}, {"end": 421.72, "start": 421.52, "text": "the"}, {"end": 422.16, "start": 421.72, "text": "natural"}, {"end": 422.68, "start": 422.16, "text": "language"}, {"end": 422.76, "start": 422.68, "text": "of"}, {"end": 423.24, "start": 422.76, "text": "growth."}, {"end": 423.36, "start": 423.24, "text": "And"}, {"end": 423.6, "start": 423.36, "text": "I'll"}, {"end": 423.92, "start": 423.6, "text": "show"}, {"end": 423.96, "start": 423.92, "text": "you"}, {"end": 424.36, "start": 423.96, "text": "why."}, {"end": 424.96, "start": 424.36, "text": "OK."}, {"end": 425.2, "start": 424.96, "text": "Let's"}, {"end": 425.68, "start": 425.2, "text": "draw"}, {"end": 425.76, "start": 425.68, "text": "a"}, {"end": 426.4, "start": 425.76, "text": "graph."}, {"end": 426.76, "start": 426.4, "text": "y"}, {"end": 427.64, "start": 426.76, "text": "equals"}, {"end": 427.64, "start": 427.64, "text": "e"}, {"end": 428.04, "start": 427.64, "text": "to"}, {"end": 428.12, "start": 428.04, "text": "the"}, {"end": 428.28, "start": 428.12, "text": "x."}, {"end": 428.36, "start": 428.28, "text": "So"}, {"end": 428.52, "start": 428.36, "text": "we're"}, {"end": 429.32, "start": 428.52, "text": "taking"}, {"end": 429.68, "start": 429.32, "text": "powers"}, {"end": 430.0, "start": 429.68, "text": "of"}, {"end": 430.0, "start": 430.0, "text": "e."}, {"end": 430.32, "start": 430.0, "text": "So"}, {"end": 430.84, "start": 430.32, "text": "here"}, {"end": 431.04, "start": 430.84, "text": "at"}, {"end": 431.52, "start": 431.04, "text": "0,"}, {"end": 431.68, "start": 431.52, "text": "this"}, {"end": 431.72, "start": 431.68, "text": "would"}, {"end": 432.24, "start": 431.72, "text": "cross"}, {"end": 432.44, "start": 432.24, "text": "at"}, {"end": 432.76, "start": 432.44, "text": "1."}, {"end": 432.84, "start": 432.76, "text": "So"}, {"end": 432.92, "start": 432.84, "text": "if"}, {"end": 433.04, "start": 432.92, "text": "you"}, {"end": 433.28, "start": 433.04, "text": "took"}, {"end": 433.44, "start": 433.28, "text": "a"}, {"end": 434.2, "start": 433.44, "text": "point"}, {"end": 434.28, "start": 434.2, "text": "on"}, {"end": 434.44, "start": 434.28, "text": "this"}, {"end": 435.36, "start": 434.44, "text": "graph,"}, {"end": 435.96, "start": 435.36, "text": "the"}, {"end": 436.64, "start": 435.96, "text": "value"}, {"end": 436.76, "start": 436.64, "text": "at"}, {"end": 436.96, "start": 436.76, "text": "that"}, {"end": 439.24, "start": 436.96, "text": "point"}, {"end": 439.92, "start": 439.24, "text": "is"}, {"end": 440.0, "start": 439.92, "text": "e"}, {"end": 440.16, "start": 440.0, "text": "to"}, {"end": 440.4, "start": 440.16, "text": "the"}, {"end": 440.64, "start": 440.4, "text": "power"}, {"end": 441.6, "start": 440.64, "text": "x."}, {"end": 441.84, "start": 441.6, "text": "And"}, {"end": 442.0, "start": 441.84, "text": "this"}, {"end": 442.24, "start": 442.0, "text": "is"}, {"end": 442.68, "start": 442.24, "text": "why"}, {"end": 442.96, "start": 442.68, "text": "it's"}, {"end": 443.48, "start": 442.96, "text": "important."}, {"end": 443.64, "start": 443.48, "text": "The"}, {"end": 446.16, "start": 443.64, "text": "gradient"}, {"end": 446.28, "start": 446.16, "text": "at"}, {"end": 446.48, "start": 446.28, "text": "that"}, {"end": 447.04, "start": 446.48, "text": "point,"}, {"end": 447.32, "start": 447.04, "text": "the"}, {"end": 447.72, "start": 447.32, "text": "gradient"}, {"end": 447.8, "start": 447.72, "text": "of"}, {"end": 447.92, "start": 447.8, "text": "the"}, {"end": 448.28, "start": 447.92, "text": "curve"}, {"end": 448.4, "start": 448.28, "text": "at"}, {"end": 448.48, "start": 448.4, "text": "that"}, {"end": 449.28, "start": 448.48, "text": "point,"}, {"end": 449.44, "start": 449.28, "text": "is"}, {"end": 449.52, "start": 449.44, "text": "e"}, {"end": 449.64, "start": 449.52, "text": "to"}, {"end": 449.88, "start": 449.64, "text": "the"}, {"end": 449.96, "start": 449.88, "text": "x."}], "text": " It's because E is the natural language of growth. And I'll show you why. OK. Let's draw a graph. y equals e to the x. So we're taking powers of e. So here at 0, this would cross at 1. So if you took a point on this graph, the value at that point is e to the power x. And this is why it's important. The gradient at that point, the gradient of the curve at that point, is e to the x."}, {"chunks": [{"end": 451.32, "start": 450.0, "text": "And"}, {"end": 451.76, "start": 451.32, "text": "the"}, {"end": 453.32, "start": 451.76, "text": "area"}, {"end": 453.64, "start": 453.32, "text": "under"}, {"end": 453.64, "start": 453.64, "text": "the"}, {"end": 454.92, "start": 453.64, "text": "curve,"}, {"end": 455.12, "start": 454.92, "text": "which"}, {"end": 455.6, "start": 455.12, "text": "means"}, {"end": 455.92, "start": 455.6, "text": "the"}, {"end": 456.36, "start": 455.92, "text": "area"}, {"end": 456.64, "start": 456.36, "text": "under"}, {"end": 456.64, "start": 456.64, "text": "the"}, {"end": 456.72, "start": 456.64, "text": "curve"}, {"end": 457.12, "start": 456.72, "text": "all"}, {"end": 457.28, "start": 457.12, "text": "the"}, {"end": 457.76, "start": 457.28, "text": "way"}, {"end": 458.76, "start": 457.76, "text": "down"}, {"end": 458.88, "start": 458.76, "text": "to"}, {"end": 459.28, "start": 458.88, "text": "minus"}, {"end": 459.6, "start": 459.28, "text": "infinity,"}, {"end": 459.72, "start": 459.6, "text": "is"}, {"end": 459.72, "start": 459.72, "text": "e"}, {"end": 460.4, "start": 459.72, "text": "to"}, {"end": 460.64, "start": 460.4, "text": "the"}, {"end": 460.92, "start": 460.64, "text": "x."}, {"end": 462.28, "start": 460.92, "text": "And"}, {"end": 462.56, "start": 462.28, "text": "it's"}, {"end": 462.88, "start": 462.56, "text": "the"}, {"end": 463.56, "start": 462.88, "text": "only"}, {"end": 463.92, "start": 463.56, "text": "function"}, {"end": 464.12, "start": 463.92, "text": "that"}, {"end": 464.32, "start": 464.12, "text": "has"}, {"end": 464.48, "start": 464.32, "text": "that"}, {"end": 464.76, "start": 464.48, "text": "property."}, {"end": 464.88, "start": 464.76, "text": "So"}, {"end": 465.52, "start": 464.88, "text": "it"}, {"end": 465.72, "start": 465.52, "text": "has"}, {"end": 466.32, "start": 465.72, "text": "the"}, {"end": 466.68, "start": 466.32, "text": "same"}, {"end": 467.28, "start": 466.68, "text": "value,"}, {"end": 468.28, "start": 467.28, "text": "gradient,"}, {"end": 468.64, "start": 468.28, "text": "and"}, {"end": 469.44, "start": 468.64, "text": "area"}, {"end": 469.72, "start": 469.44, "text": "at"}, {"end": 470.04, "start": 469.72, "text": "every"}, {"end": 470.52, "start": 470.04, "text": "point"}, {"end": 470.8, "start": 470.52, "text": "along"}, {"end": 470.92, "start": 470.8, "text": "the"}, {"end": 471.72, "start": 470.92, "text": "line."}, {"end": 472.04, "start": 471.72, "text": "So"}, {"end": 472.28, "start": 472.04, "text": "at"}, {"end": 472.6, "start": 472.28, "text": "1,"}, {"end": 473.32, "start": 472.6, "text": "the"}, {"end": 473.84, "start": 473.32, "text": "value"}, {"end": 474.12, "start": 473.84, "text": "is"}, {"end": 474.24, "start": 474.12, "text": "e,"}, {"end": 474.64, "start": 474.24, "text": "because"}, {"end": 474.8, "start": 474.64, "text": "it's"}, {"end": 474.92, "start": 474.8, "text": "e"}, {"end": 475.04, "start": 474.92, "text": "to"}, {"end": 475.04, "start": 475.04, "text": "the"}, {"end": 475.32, "start": 475.04, "text": "power"}, {"end": 475.84, "start": 475.32, "text": "1."}, {"end": 475.96, "start": 475.84, "text": "The"}, {"end": 476.24, "start": 475.96, "text": "value"}, {"end": 476.48, "start": 476.24, "text": "is"}, {"end": 477.24, "start": 476.48, "text": "2.718."}, {"end": 477.4, "start": 477.24, "text": "The"}, {"end": 478.04, "start": 477.4, "text": "gradient"}, {"end": 478.16, "start": 478.04, "text": "is"}, {"end": 479.32, "start": 478.16, "text": "2.718."}, {"end": 479.56, "start": 479.32, "text": "And"}, {"end": 479.6, "start": 479.56, "text": "the"}, {"end": 479.96, "start": 479.6, "text": "area"}], "text": " And the area under the curve, which means the area under the curve all the way down to minus infinity, is e to the x. And it's the only function that has that property. So it has the same value, gradient, and area at every point along the line. So at 1, the value is e, because it's e to the power 1. The value is 2.718. The gradient is 2.718. 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The reason this is important then, because it's unique in having this property as well, it becomes the natural language of calculus. And calculus is the maths of rate of change and growth and areas, maths like that. And if you are interested in those things, if you write it in terms of E, then the maths becomes much simpler. Because if you don't write it in terms of E, you get lots"}, {"chunks": [{"end": 510.64, "start": 510.0, "text": "nasty"}, {"end": 511.88, "start": 510.64, "text": "constants"}, {"end": 512.32, "start": 511.88, "text": "and"}, {"end": 512.32, "start": 512.32, "text": "the"}, {"end": 512.44, "start": 512.32, "text": "maths"}, {"end": 512.76, "start": 512.44, "text": "is"}, {"end": 513.24, "start": 512.76, "text": "really"}, {"end": 513.76, "start": 513.24, "text": "messy."}, {"end": 513.8, "start": 513.76, "text": "If"}, {"end": 514.08, "start": 513.8, "text": "you're"}, {"end": 514.36, "start": 514.08, "text": "trying"}, {"end": 514.44, "start": 514.36, "text": "to"}, {"end": 515.04, "start": 514.44, "text": "deliberately"}, {"end": 515.48, "start": 515.04, "text": "avoid"}, {"end": 515.76, "start": 515.48, "text": "using"}, {"end": 515.88, "start": 515.76, "text": "E,"}, {"end": 516.2, "start": 515.88, "text": "you're"}, {"end": 516.52, "start": 516.2, "text": "making"}, {"end": 517.08, "start": 516.52, "text": "it"}, {"end": 517.44, "start": 517.08, "text": "hard"}, {"end": 517.6, "start": 517.44, "text": "for"}, {"end": 518.48, "start": 517.6, "text": "yourself."}, {"end": 518.72, "start": 518.48, "text": "It's"}, {"end": 518.96, "start": 518.72, "text": "the"}, {"end": 519.68, "start": 518.96, "text": "natural"}, {"end": 520.4, "start": 519.68, "text": "language"}, {"end": 520.56, "start": 520.4, "text": "of"}, {"end": 520.84, "start": 520.56, "text": "growth."}, {"end": 521.16, "start": 520.84, "text": "And"}, {"end": 521.4, "start": 521.16, "text": "of"}, {"end": 521.96, "start": 521.4, "text": "course"}, {"end": 522.6, "start": 521.96, "text": "E"}, {"end": 522.84, "start": 522.6, "text": "is"}, {"end": 523.76, "start": 522.84, "text": "famous"}, {"end": 524.2, "start": 523.76, "text": "for"}, {"end": 524.52, "start": 524.2, "text": "bringing"}, {"end": 525.64, "start": 524.52, "text": "together"}, {"end": 526.28, "start": 525.64, "text": "all"}, {"end": 526.36, "start": 526.28, "text": "the"}, {"end": 526.72, "start": 526.36, "text": "famous"}, {"end": 527.2, "start": 526.72, "text": "mathematical"}, {"end": 527.76, "start": 527.2, "text": "constants"}, {"end": 528.0, "start": 527.76, "text": "with"}, {"end": 529.16, "start": 528.0, "text": "this"}, {"end": 530.24, "start": 529.16, "text": "formula,"}, {"end": 530.64, "start": 530.24, "text": "Euler's"}, {"end": 531.52, "start": 530.64, "text": "formula,"}, {"end": 531.6, "start": 531.52, "text": "which"}, {"end": 531.8, "start": 531.6, "text": "is"}, {"end": 532.16, "start": 531.8, "text": "E"}, {"end": 532.76, "start": 532.16, "text": "to"}, {"end": 532.84, "start": 532.76, "text": "the"}, {"end": 533.12, "start": 532.84, "text": "i"}, {"end": 533.64, "start": 533.12, "text": "pi"}, {"end": 534.56, "start": 533.64, "text": "plus"}, {"end": 535.6, "start": 534.56, "text": "1"}, {"end": 536.2, "start": 535.6, "text": "equals"}, {"end": 537.44, "start": 536.2, "text": "0."}, {"end": 537.96, "start": 537.44, "text": "So"}, {"end": 538.2, "start": 537.96, "text": "there"}, {"end": 538.52, "start": 538.2, "text": "we"}, {"end": 539.16, "start": 538.52, "text": "have"}, {"end": 539.56, "start": 539.16, "text": "all"}, {"end": 539.6, "start": 539.56, "text": "the"}, {"end": 539.96, "start": 539.6, "text": "big"}], "text": " nasty constants and the maths is really messy. If you're trying to deliberately avoid using E, you're making it hard for yourself. It's the natural language of growth. And of course E is famous for bringing together all the famous mathematical constants with this formula, Euler's formula, which is E to the i pi plus 1 equals 0. 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We've got e, we've got i square root of minus 1, and we've got pi, of course. We've got 1 and 0. And they bring them all together in one formula, which is often voted as the most beautiful formula in mathematics. I've seen it so often that I'm kind of jaded to it. Don't put that in the video. 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They've got an incredible array of lessons something like over 7,000 videos from world experts, top professors around the world on all sorts of subjects from photography to paleontology to prime numbers. No matter what you're into you're gonna find something. What I've been looking at just recently on the site is paleontology and one of my favorite subjects Egyptology."}, {"chunks": [{"end": 600.2, "start": 600.0, "text": "They've"}, {"end": 600.36, "start": 600.2, "text": "got"}, {"end": 600.6, "start": 600.36, "text": "some"}, {"end": 600.8, "start": 600.6, "text": "great"}, {"end": 601.2, "start": 600.8, "text": "videos"}, {"end": 601.28, "start": 601.2, "text": "on"}, {"end": 601.88, "start": 601.28, "text": "those."}, {"end": 602.08, "start": 601.88, "text": "Now"}, {"end": 602.36, "start": 602.08, "text": "plans"}, {"end": 602.48, "start": 602.36, "text": "on"}, {"end": 602.72, "start": 602.48, "text": "The"}, {"end": 602.72, "start": 602.72, "text": "Great"}, {"end": 603.12, "start": 602.72, "text": "Courses"}, {"end": 603.6, "start": 603.12, "text": "Plus"}, {"end": 604.12, "start": 603.6, "text": "start"}, {"end": 604.6, "start": 604.12, "text": "from"}, {"end": 605.96, "start": 604.6, "text": "$14.99"}, {"end": 606.08, "start": 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Our thanks again to them for supporting this video."}]}} |