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  "events": [ {
    "tStartMs": 7785,
    "dDurationMs": 1519,
    "segs": [ {
      "utf8": "In the world of math,"
    } ]
  }, {
    "tStartMs": 9304,
    "dDurationMs": 4080,
    "segs": [ {
      "utf8": "many strange results are possible \nwhen we change the rules."
    } ]
  }, {
    "tStartMs": 13384,
    "dDurationMs": 3842,
    "segs": [ {
      "utf8": "But there’s one rule that most of us \nhave been warned not to break:"
    } ]
  }, {
    "tStartMs": 17226,
    "dDurationMs": 2379,
    "segs": [ {
      "utf8": "don’t divide by zero."
    } ]
  }, {
    "tStartMs": 19605,
    "dDurationMs": 2979,
    "segs": [ {
      "utf8": "How can the simple combination \nof an everyday number"
    } ]
  }, {
    "tStartMs": 22584,
    "dDurationMs": 3879,
    "segs": [ {
      "utf8": "and a basic operation \ncause such problems?"
    } ]
  }, {
    "tStartMs": 26463,
    "dDurationMs": 3230,
    "segs": [ {
      "utf8": "Normally, dividing by smaller \nand smaller numbers"
    } ]
  }, {
    "tStartMs": 29693,
    "dDurationMs": 2628,
    "segs": [ {
      "utf8": "gives you bigger and bigger answers."
    } ]
  }, {
    "tStartMs": 32321,
    "dDurationMs": 2523,
    "segs": [ {
      "utf8": "Ten divided by two is five,"
    } ]
  }, {
    "tStartMs": 34844,
    "dDurationMs": 1599,
    "segs": [ {
      "utf8": "by one is ten,"
    } ]
  }, {
    "tStartMs": 36443,
    "dDurationMs": 2579,
    "segs": [ {
      "utf8": "by one-millionth is 10 million,"
    } ]
  }, {
    "tStartMs": 39022,
    "dDurationMs": 958,
    "segs": [ {
      "utf8": "and so on."
    } ]
  }, {
    "tStartMs": 39980,
    "dDurationMs": 2311,
    "segs": [ {
      "utf8": "So it seems like if you divide by numbers"
    } ]
  }, {
    "tStartMs": 42291,
    "dDurationMs": 2609,
    "segs": [ {
      "utf8": "that keep shrinking \nall the way down to zero,"
    } ]
  }, {
    "tStartMs": 44900,
    "dDurationMs": 3161,
    "segs": [ {
      "utf8": "the answer will grow \nto the largest thing possible."
    } ]
  }, {
    "tStartMs": 48061,
    "dDurationMs": 4662,
    "segs": [ {
      "utf8": "Then, isn’t the answer to 10 \ndivided by zero actually infinity?"
    } ]
  }, {
    "tStartMs": 52723,
    "dDurationMs": 1960,
    "segs": [ {
      "utf8": "That may sound plausible."
    } ]
  }, {
    "tStartMs": 54683,
    "dDurationMs": 3050,
    "segs": [ {
      "utf8": "But all we really know is \nthat if we divide 10"
    } ]
  }, {
    "tStartMs": 57733,
    "dDurationMs": 2941,
    "segs": [ {
      "utf8": "by a number that tends towards zero,"
    } ]
  }, {
    "tStartMs": 60674,
    "dDurationMs": 3060,
    "segs": [ {
      "utf8": "the answer tends towards infinity."
    } ]
  }, {
    "tStartMs": 63734,
    "dDurationMs": 4130,
    "segs": [ {
      "utf8": "And that’s not the same thing as\nsaying that 10 divided by zero"
    } ]
  }, {
    "tStartMs": 67864,
    "dDurationMs": 2650,
    "segs": [ {
      "utf8": "is equal to infinity."
    } ]
  }, {
    "tStartMs": 70514,
    "dDurationMs": 1319,
    "segs": [ {
      "utf8": "Why not?"
    } ]
  }, {
    "tStartMs": 71833,
    "dDurationMs": 4389,
    "segs": [ {
      "utf8": "Well, let’s take a closer look \nat what division really means."
    } ]
  }, {
    "tStartMs": 76222,
    "dDurationMs": 2352,
    "segs": [ {
      "utf8": "Ten divided by two could mean,"
    } ]
  }, {
    "tStartMs": 78574,
    "dDurationMs": 4097,
    "segs": [ {
      "utf8": "\"How many times must \nwe add two together to make 10,”"
    } ]
  }, {
    "tStartMs": 82671,
    "dDurationMs": 3442,
    "segs": [ {
      "utf8": "or, “two times what equals 10?”"
    } ]
  }, {
    "tStartMs": 86113,
    "dDurationMs": 4340,
    "segs": [ {
      "utf8": "Dividing by a number is essentially \nthe reverse of multiplying by it,"
    } ]
  }, {
    "tStartMs": 90453,
    "dDurationMs": 1948,
    "segs": [ {
      "utf8": "in the following way:"
    } ]
  }, {
    "tStartMs": 92401,
    "dDurationMs": 3022,
    "segs": [ {
      "utf8": "if we multiply any number \nby a given number x,"
    } ]
  }, {
    "tStartMs": 95423,
    "dDurationMs": 4230,
    "segs": [ {
      "utf8": "we can ask if there’s a new number \nwe can multiply by afterwards"
    } ]
  }, {
    "tStartMs": 99653,
    "dDurationMs": 2633,
    "segs": [ {
      "utf8": "to get back to where we started."
    } ]
  }, {
    "tStartMs": 102286,
    "dDurationMs": 5058,
    "segs": [ {
      "utf8": "If there is, the new number is called \nthe multiplicative inverse of x."
    } ]
  }, {
    "tStartMs": 107344,
    "dDurationMs": 4101,
    "segs": [ {
      "utf8": "For example, if you multiply \nthree by two to get six,"
    } ]
  }, {
    "tStartMs": 111445,
    "dDurationMs": 4119,
    "segs": [ {
      "utf8": "you can then multiply \nby one-half to get back to three."
    } ]
  }, {
    "tStartMs": 115564,
    "dDurationMs": 3830,
    "segs": [ {
      "utf8": "So the multiplicative inverse \nof two is one-half,"
    } ]
  }, {
    "tStartMs": 119394,
    "dDurationMs": 4570,
    "segs": [ {
      "utf8": "and the multiplicative inverse \nof 10 is one-tenth."
    } ]
  }, {
    "tStartMs": 123964,
    "dDurationMs": 5270,
    "segs": [ {
      "utf8": "As you might notice, the product of any\nnumber and its multiplicative inverse"
    } ]
  }, {
    "tStartMs": 129234,
    "dDurationMs": 2030,
    "segs": [ {
      "utf8": "is always one."
    } ]
  }, {
    "tStartMs": 131264,
    "dDurationMs": 2199,
    "segs": [ {
      "utf8": "If we want to divide by zero,"
    } ]
  }, {
    "tStartMs": 133463,
    "dDurationMs": 2380,
    "segs": [ {
      "utf8": "we need to find \nits multiplicative inverse,"
    } ]
  }, {
    "tStartMs": 135843,
    "dDurationMs": 3301,
    "segs": [ {
      "utf8": "which should be one over zero."
    } ]
  }, {
    "tStartMs": 139144,
    "dDurationMs": 5828,
    "segs": [ {
      "utf8": "This would have to be such a number that\nmultiplying it by zero would give one."
    } ]
  }, {
    "tStartMs": 144972,
    "dDurationMs": 4171,
    "segs": [ {
      "utf8": "But because anything multiplied\nby zero is still zero,"
    } ]
  }, {
    "tStartMs": 149143,
    "dDurationMs": 2413,
    "segs": [ {
      "utf8": "such a number is impossible,"
    } ]
  }, {
    "tStartMs": 151556,
    "dDurationMs": 3286,
    "segs": [ {
      "utf8": "so zero has no multiplicative inverse."
    } ]
  }, {
    "tStartMs": 154842,
    "dDurationMs": 2501,
    "segs": [ {
      "utf8": "Does that really settle things, though?"
    } ]
  }, {
    "tStartMs": 157343,
    "dDurationMs": 3640,
    "segs": [ {
      "utf8": "After all, mathematicians \nhave broken rules before."
    } ]
  }, {
    "tStartMs": 160983,
    "dDurationMs": 1730,
    "segs": [ {
      "utf8": "For example, for a long time,"
    } ]
  }, {
    "tStartMs": 162713,
    "dDurationMs": 4061,
    "segs": [ {
      "utf8": "there was no such thing as taking \nthe square root of negative numbers."
    } ]
  }, {
    "tStartMs": 166774,
    "dDurationMs": 4087,
    "segs": [ {
      "utf8": "But then mathematicians defined \nthe square root of negative one"
    } ]
  }, {
    "tStartMs": 170861,
    "dDurationMs": 2423,
    "segs": [ {
      "utf8": "as a new number called i,"
    } ]
  }, {
    "tStartMs": 173284,
    "dDurationMs": 4519,
    "segs": [ {
      "utf8": "opening up a whole new\nmathematical world of complex numbers."
    } ]
  }, {
    "tStartMs": 177803,
    "dDurationMs": 1442,
    "segs": [ {
      "utf8": "So if they can do that,"
    } ]
  }, {
    "tStartMs": 179245,
    "dDurationMs": 1958,
    "segs": [ {
      "utf8": "couldn’t we just make up a new rule,"
    } ]
  }, {
    "tStartMs": 181203,
    "dDurationMs": 4061,
    "segs": [ {
      "utf8": "say, that the symbol infinity \nmeans one over zero,"
    } ]
  }, {
    "tStartMs": 185264,
    "dDurationMs": 2279,
    "segs": [ {
      "utf8": "and see what happens?"
    } ]
  }, {
    "tStartMs": 187543,
    "dDurationMs": 1050,
    "segs": [ {
      "utf8": "Let's try it,"
    } ]
  }, {
    "tStartMs": 188593,
    "dDurationMs": 3130,
    "segs": [ {
      "utf8": "imagining we don’t know \nanything about infinity already."
    } ]
  }, {
    "tStartMs": 191723,
    "dDurationMs": 2569,
    "segs": [ {
      "utf8": "Based on the definition \nof a multiplicative inverse,"
    } ]
  }, {
    "tStartMs": 194292,
    "dDurationMs": 4182,
    "segs": [ {
      "utf8": "zero times infinity must be equal to one."
    } ]
  }, {
    "tStartMs": 198474,
    "dDurationMs": 6056,
    "segs": [ {
      "utf8": "That means zero times infinity plus\nzero times infinity should equal two."
    } ]
  }, {
    "tStartMs": 204530,
    "dDurationMs": 1957,
    "segs": [ {
      "utf8": "Now, by the distributive property,"
    } ]
  }, {
    "tStartMs": 206487,
    "dDurationMs": 2901,
    "segs": [ {
      "utf8": "the left side of the equation \ncan be rearranged"
    } ]
  }, {
    "tStartMs": 209388,
    "dDurationMs": 3231,
    "segs": [ {
      "utf8": "to zero plus zero times infinity."
    } ]
  }, {
    "tStartMs": 212619,
    "dDurationMs": 3569,
    "segs": [ {
      "utf8": "And since zero plus zero \nis definitely zero,"
    } ]
  }, {
    "tStartMs": 216188,
    "dDurationMs": 3861,
    "segs": [ {
      "utf8": "that reduces down to zero times infinity."
    } ]
  }, {
    "tStartMs": 220049,
    "dDurationMs": 3668,
    "segs": [ {
      "utf8": "Unfortunately, we’ve already defined \nthis as equal to one,"
    } ]
  }, {
    "tStartMs": 223717,
    "dDurationMs": 4636,
    "segs": [ {
      "utf8": "while the other side of the equation \nis still telling us it’s equal to two."
    } ]
  }, {
    "tStartMs": 228353,
    "dDurationMs": 2885,
    "segs": [ {
      "utf8": "So, one equals two."
    } ]
  }, {
    "tStartMs": 231238,
    "dDurationMs": 3226,
    "segs": [ {
      "utf8": "Oddly enough, \nthat's not necessarily wrong;"
    } ]
  }, {
    "tStartMs": 234464,
    "dDurationMs": 3668,
    "segs": [ {
      "utf8": "it's just not true \nin our normal world of numbers."
    } ]
  }, {
    "tStartMs": 238132,
    "dDurationMs": 2582,
    "segs": [ {
      "utf8": "There’s still a way it could\nbe mathematically valid,"
    } ]
  }, {
    "tStartMs": 240714,
    "dDurationMs": 4507,
    "segs": [ {
      "utf8": "if one, two, and every other number \nwere equal to zero."
    } ]
  }, {
    "tStartMs": 245221,
    "dDurationMs": 2493,
    "segs": [ {
      "utf8": "But having infinity equal to zero"
    } ]
  }, {
    "tStartMs": 247714,
    "dDurationMs": 5170,
    "segs": [ {
      "utf8": "is ultimately not all that useful \nto mathematicians, or anyone else."
    } ]
  }, {
    "tStartMs": 252884,
    "dDurationMs": 3230,
    "segs": [ {
      "utf8": "There actually is something called \nthe Riemann sphere"
    } ]
  }, {
    "tStartMs": 256114,
    "dDurationMs": 3295,
    "segs": [ {
      "utf8": "that involves dividing by zero \nby a different method,"
    } ]
  }, {
    "tStartMs": 259409,
    "dDurationMs": 2364,
    "segs": [ {
      "utf8": "but that’s a story for another day."
    } ]
  }, {
    "tStartMs": 261773,
    "dDurationMs": 4192,
    "segs": [ {
      "utf8": "In the meantime, dividing by zero \nin the most obvious way"
    } ]
  }, {
    "tStartMs": 265965,
    "dDurationMs": 1768,
    "segs": [ {
      "utf8": "doesn’t work out so great."
    } ]
  }, {
    "tStartMs": 267733,
    "dDurationMs": 2941,
    "segs": [ {
      "utf8": "But that shouldn’t stop us \nfrom living dangerously"
    } ]
  }, {
    "tStartMs": 270674,
    "dDurationMs": 2931,
    "segs": [ {
      "utf8": "and experimenting \nwith breaking mathematical rules"
    } ]
  }, {
    "tStartMs": 273605,
    "dDurationMs": 3878,
    "segs": [ {
      "utf8": "to see if we can invent \nfun, new worlds to explore."
    } ]
  } ]
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