Why can't you divide by zero? - TED-Ed
In the world of math, 00:09 many strange results are possible when we change the rules. 00:13 But there's one rule that most of us have been warned not to break: 00:17 don't divide by zero. 00:19 How can the simple combination of an everyday number 00:22 and a basic operation cause such problems? 00:26 Normally, dividing by smaller and smaller numbers 00:29 gives you bigger and bigger answers. 00:32 Ten divided by two is five, 00:34 by one is ten, 00:36 by one-millionth is 10 million, 00:39 and so on. 00:39 So it seems like if you divide by numbers 00:42 that keep shrinking all the way down to zero, 00:44 the answer will grow to the largest thing possible. 00:48 Then, isn't the answer to 10 divided by zero actually infinity? 00:52 That may sound plausible. 00:54 But all we really know is that if we divide 10 00:57 by a number that tends towards zero, 01:00 the answer tends towards infinity. 01:03 And that's not the same thing as saying that 10 divided by zero 01:07 is equal to infinity. 01:10 Why not? 01:11 Well, let's take a closer look at what division really means. 01:16 Ten divided by two could mean, 01:18 "How many times must we add two together to make 10,” 01:22 or, “two times what equals 10?” 01:26 Dividing by a number is essentially the reverse of multiplying by it, 01:30 in the following way: 01:32 if we multiply any number by a given number x, 01:35 we can ask if there's a new number we can multiply by afterwards 01:39 to get back to where we started. 01:42 If there is, the new number is called the multiplicative inverse of x. 01:47 For example, if you multiply three by two to get six, 01:51 you can then multiply by one-half to get back to three. 01:55 So the multiplicative inverse of two is one-half, 01:59 and the multiplicative inverse of 10 is one-tenth. 02:03 As you might notice, the product of any number and its multiplicative inverse 02:09 is always one. 02:11 If we want to divide by zero, 02:13 we need to find its multiplicative inverse, 02:15 which should be one over zero. 02:19 This would have to be such a number that multiplying it by zero would give one. 02:24 But because anything multiplied by zero is still zero, 02:29 such a number is impossible, 02:31 so zero has no multiplicative inverse. 02:34 Does that really settle things, though? 02:37 After all, mathematicians have broken rules before. 02:40 For example, for a long time, 02:42 there was no such thing as taking the square root of negative numbers. 02:46 But then mathematicians defined the square root of negative one 02:50 as a new number called i, 02:53 opening up a whole new mathematical world of complex numbers. 02:57 So if they can do that, 02:59 couldn't we just make up a new rule, 03:01 say, that the symbol infinity means one over zero, 03:05 and see what happens? 03:07 Let's try it, 03:08 imagining we don't know anything about infinity already. 03:11 Based on the definition of a multiplicative inverse, 03:14 zero times infinity must be equal to one. 03:18 That means zero times infinity plus zero times infinity should equal two. 03:24 Now, by the distributive property, 03:26 the left side of the equation can be rearranged 03:29 to zero plus zero times infinity. 03:32 And since zero plus zero is definitely zero, 03:36 that reduces down to zero times infinity. 03:40 Unfortunately, we've already defined this as equal to one, 03:43 while the other side of the equation is still telling us it's equal to two. 03:48 So, one equals two. 03:51 Oddly enough, that's not necessarily wrong; 03:54 it's just not true in our normal world of numbers. 03:58 There's still a way it could be mathematically valid, 04:00 if one, two, and every other number were equal to zero. 04:05 But having infinity equal to zero 04:07 is ultimately not all that useful to mathematicians, or anyone else. 04:12 There actually is something called the Riemann sphere 04:16 that involves dividing by zero by a different method, 04:19 but that's a story for another day. 04:21 In the meantime, dividing by zero in the most obvious way 04:25 doesn't work out so great. 04:27 But that shouldn't stop us from living dangerously 04:30 and experimenting with breaking mathematical rules 04:33 to see if we can invent fun, new worlds to explore.
