Why can't you divide by zero? - TED-Ed

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  • Published: 23 April 2018
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    View full lesson: https://ed.ted.com/lessons/why-can-t-you-divide-by-zero

    In the world of math, many strange results are possible when we change the rules. But there’s one rule that most of us have been warned not to break: don’t divide by zero. How can the simple combination of an everyday number and a basic operation cause such problems?

    Lesson by TED-Ed, animation by Nick Hilditch.

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Comments • 6 997

  • TED-Ed
    TED-Ed   1 years back

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    • Vincent RedDragon
      Vincent RedDragon  1 months back

      TED-Ed I just want to ask,when I tried 0 divided 0 in calculator,the result is ERROR. Why?

    • Talote1983
      Talote1983  2 months back

      The voice is super annoying

    • Andrew Long
      Andrew Long  2 months back

      Why do they use anti matter for i its till has postive mass if it was negative we would have no matter on contacted of each other

    • Alejandro Cardone
      Alejandro Cardone  3 months back

      At minute 1:56 the video shows a triple equality that isn't true. I know what you wanted to say, but you have to be careful with notation.

  • Alan
    Alan  1 days back

    When God divided O, boobs were created.

    • GlowstoneLove Pad
      GlowstoneLove Pad  1 days back

      1 does equal 2.

      We have two variables, a and b. As long as they're the same and not zero, it's fine.
      They could be both 3, both 7, or both 1371742, but they can't be both zero.

      1. a = b
      2. a^2 = a*b(multiply by a)
      3. a^2 - b^2 = a*b - b^2(subtract b^2)
      4. (a+b)*(a-b) = b*(a-b)(factor left side as a difference of squares: a^2 - b^2 = (a+b)*(a-b) and right side as a distributive property, a*b - b*c = b*(a-c))
      5. a+b = b(divide by a-b)
      6. b+b = b(since a = b, you can substitute a for b in an equation, and vice versa)
      7. 2b = b(for every a, 2a = a + a)
      8. 2 = 1(divide by b)

      B is not zero, defined by the given a and b are not zero.

      Now that we have that 2 = 1, we can create an infinite list of axioms, each inferred from the last one:

      1. 2 = 1
      2. 1 = 0(subtract 1)
      3. 2 = 0(combine 1 and 2: if a = b and b = c, a = c)
      4. 3 = 1(add 1)
      5. 3 = 0(combine 4 and 2)
      6. 4 = 1(add 1)
      7. 4 = 0(combine 6 and 2)
      ... .... .....
      z. m = 1(add 1)
      z+1. m = 0(combine z and 2)
      z+2. m+1 = 1(add 1)
      z+3. m+1 = 0(combine z+2 and 2)
      ... and so on.

      Now we have an infinite list of things, if we continue this on infinitely.
      Extract line 1 from the list, and all of the even numbered lines that are after 3(4, 6, 8... )
      Now we have the list:

      2. 1 = 0
      3. 2 = 0
      5. 3 = 0
      7. 4 = 0
      9. 5 = 0
      ... and so on.

      Now renumber the lines.

      1. 1 = 0
      2. 2 = 0
      3. 3 = 0
      ... and so on.

      Now we have an infinite list saying that every positive integer = 0. Let's call this list l, and the items in it 1, 2, 3, 4...

      By the rule if a = b, -a = -b(multiplying both sides by -1), since -0 = 0, we can create another list, called l'(l prime):

      1'. -1 = 0
      2'. -2 = 0
      3'. -3 = 0
      ... and so on.

      Now we have an infinite list saying that every negative integer = 0.

      Also, if we have a fraction, we can substitute the top for zero, making that every fraction is zero too, call this l''(l double-prime):

      1''. 1/2 = 0
      2''. 1/3 = 0
      3''. 2/3 = 0
      ... and so on, continuing with 1/4, 2/4, 3/4, 1/5, 2/5, 3/5... and so on...

      This list, l''(l double prime) says that every fraction of two positive integers inbetween 0 and 1 is equal to zero too.

      If you add 0 to 0 you get 0 also, and so if we make all of the mixed numbers = 0, in a list l'''(l triple prime).
      Let's call infinity "t" for short:

      1'''. 1 1/2 = 0
      2'''. 2 1/2 = 0
      ... ... ...
      t'''. t 1/2 = 0
      t + 1'''. 1 1/3 = 0
      ... ... ... and so on... ... ...

      And therefore, every fraction is equal to zero. Again, we have l quadruple prime(l'''') and l quintuple prime (l''''') saying that every negative fraction is also equal to zero:

      1''''. -1/2 = 0
      2''''. -1/3 = 0
      3''''. -2/3 = 0
      ... ... ... and so on... ... ...

      1'''''. -1 1/2 = 0
      2'''''. -2 1/2 = 0
      ... ... ... and so on... ... ...

      l'''' is just a negative of l'', and l''''' is a negative of l'''.

      Since every fraction above 1 is just a mixed number, all of the fractions above 1 = 0 list is the same as l''', and the fractions below -1 are in l'''''.

      Now we can merge all of these l lists together, in a superlist called q.

      We have a list saying that every integer and fraction, positive and negative, is equal to zero now. Now that, since if a = b and b = c, a = c, every integer and fraction, positive and negative, is equal to eachother too!

      Since therefore the numbers picked as a and b are zero too, therefore this could never exist since then a and b would violate the zero-rule. But then this could never have been proved, and therefore the rules of math are back to regular. Therefore the numbers picked as a and b would not be zero anymore, this could be proved.

      So this could be proved, which means that a and b are zero, which means this could not be proved, which means that a and b are no longer zero, which means this could be proved again. It's a paradox.

      • 1000 subscribers without a video

        Because your teacher keeps telling you so, it's simple

        • Mannem Nageswarao
          Mannem Nageswarao  3 days back

          What is 0÷0

          • Yinit Belloraaa
            Yinit Belloraaa  3 days back

            Any number that is devided by zero is zero

            Example 10/0= zero not infinity

            • GlowstoneLove Pad
              GlowstoneLove Pad  1 days back

              if a / 0 = 0, 0 * 0 = a
              So what you're saying is that the product of 0 and 0 is any number, therefore the square root of any number is 0?

          • 321 Cubing
            321 Cubing  3 days back

            Is this not just advanced algebra and what they try to find out

            • ra s.
              ra s.  4 days back

              I broke a math rule, but failed the exam.

              • ifigenio buenrostro
                ifigenio buenrostro  4 days back

                Maybe current mathematics can't divide by zero but surely future mathematics will

                • PraWin Shidore
                  PraWin Shidore  4 days back

                  0 is amazing by the way 🤔😁😜

                  • Morning Star
                    Morning Star  4 days back

                    After watching this ,I clicked Like but and app said "Removed from liked videos".

                    I was like wt!!! I already watched that many times ago and literally forgot.😅

                    • rianma 123
                      rianma 123  5 days back

                      You have a 20 cm ruler
                      You divided it by 3 doesn't mean you cut the ruler into 3 pieces of ruler that each piece is 20:3 cm long.
                      It means you cut the ruler into 3 pieces of ruler and you can just keep 1 piece.
                      So a 20cm ruler divide by 3 =a 20:3cm ruler
                      Not = 3 20:3 rulers
                      So when your mum tells you to divide a bread by 3, you cut the bread into 3 pieces and throw 2 pieces away.

                      • Mr. Formal
                        Mr. Formal  5 days back

                        infinity ÷ 0 is less than or equal to infinity.

                        • Aditya Kant Kushwaha
                          Aditya Kant Kushwaha  6 days back

                          Please add Hindi language in captions

                          • Dude Trust Me
                            Dude Trust Me  6 days back

                            As an English education major this finally put me to sleep. Thanks math!

                            • KT7rocks Jr
                              KT7rocks Jr  6 days back

                              Infinity divided by 0

                              Thanos:DID YOU SAY INFINITY?

                              • Eunhyouk Shin
                                Eunhyouk Shin  7 days back

                                1:56 This notation can be quite misleading

                                • Sub for no good reason H

                                  Remember infinity is a SYMBOL for a number we can't count to/it's an infinite amount of zeros. So it doesn't =infinity (saying infinity is a number)it is infinity

                                  • Francesco Sorce
                                    Francesco Sorce  7 days back

                                    3:30
                                    there's the fallacy

                                    you have said "let the symbol infty be a number so that infty *0 =1" and you also said "assuming we know nothing about infinity"

                                    then why should 2 infty =infty?

                                    that is a preconceived notion of how infinity works

                                    • Francesco Sorce
                                      Francesco Sorce  7 days back

                                      there are other ways to show this method isn't valid, just that this one in particular was not it


                                      for example, let y so that 0y=1

                                      1+1=2
                                      0y+0y=2

                                      2 0 y = 2

                                      (2 0) y = 2

                                      0 y = 2

                                      but 0y =1



                                      however note, this disproved EITHER associativity OR the existence of y, not necessarily the latter.

                                  • Thijmen Willems
                                    Thijmen Willems  1 weeks back

                                    John has four apples, he distributes them fairly with Mike, how many apples do they each have? Two.

                                    John has zero apples, he distributes them fairly with Mike, how many apples do they each have? Zero because there are no apples to distribute.

                                    there are four apples on the table, they are not divided because there are zero people to distribute them, how many apples does everyone have? zero because there are zero people to divide them. 4 ÷ 0 = 0.


                                    SOLVED
                                    (Google calculator says that 4 ÷ 0 equals infinity)

                                    • Naff YT
                                      Naff YT  1 weeks back

                                      my head.....

                                      • Hdtomo
                                        Hdtomo  1 weeks back

                                        They is an answer to any ÷0. That is an infinity. And if it's 1÷0 it's infinity because it's not how much 0 is in 1. It's just the opposite of times. Look it's hard to explain but it's still infinity. Because reasons

                                        • Maxwell Baron
                                          Maxwell Baron  1 weeks back

                                          Easy answer to the title question without watching the video is because you can’t turn nothing into something.

                                          • Imagination TM
                                            Imagination TM  1 weeks back

                                            You can't divide 0 by 0 because that's just like saying I have 0 dollars and I divide it by 0 people. Check on our calculator

                                            • David Alejandro Moreno Perea

                                              actually
                                              once i asked my math teacher in fifth grade: What is 1 divided by 0? (said it before that became a math dilemma)
                                              the next day he told me this:
                                              "Any number times 0 would be 1"
                                              Edit: acctually happened

                                              • Yash Rai
                                                Yash Rai  1 weeks back

                                                Its equal to 0 dude

                                                • Fake Person
                                                  Fake Person  1 weeks back

                                                  1:30 . 10÷2 means if you divide ten twice, how much is a single piece's value. Not how many times 2 can be subtracted from 10. I know, I'm being a snob. But there's a difference.

                                                  • solus 22
                                                    solus 22  2 weeks back

                                                    *number comparaison*
                                                    1 one
                                                    10 ten
                                                    100 hundred
                                                    10^3 thousand
                                                    10^6 million
                                                    10^100 googol
                                                    10^10^100 googolplex
                                                    g:64 graham number
                                                    aleph 0 infinity
                                                    aleph 1 infinity of infinity
                                                    *1 divided by 0*

                                                    • Roni Shaheen
                                                      Roni Shaheen  2 weeks back

                                                      3:30 ∞ + ∞ =2∞ so 0x2∞ =2

                                                    • Bry Bry
                                                      Bry Bry  2 weeks back

                                                      How about 5 ÷ 0 =
                                                      $5 ÷ 0 friends = $0
                                                      $0 given to your friends XD

                                                      • Kowi Kowi
                                                        Kowi Kowi  2 weeks back

                                                        but somehow you must explain that the 5 dollar suddenly stop existing

                                                    • Bry Bry
                                                      Bry Bry  2 weeks back

                                                      If you cant divide by 0
                                                      Why can we answer multiplication
                                                      5 × 0 = 0

                                                      But how about this way
                                                      0 ÷ 0 = 5

                                                      • Gnome Ann
                                                        Gnome Ann  2 weeks back

                                                        Bry Bry, you are right, 0/0 can be 5. But also...

                                                        1 * 0 = 0
                                                        0/0 = 1

                                                        It can be 1, too. In fact, it can give any number in existence. Since it can be any number, and we don’t know which one, we call it indeterminate. We need more information to find its limit.

                                                    • Richard Hough
                                                      Richard Hough  2 weeks back

                                                      10 divided by 2 means that 10 items are divided into 2 groups of 5 of these things. This would indicate that 10 divided by 0 means there are 0 groups, thus dividing by 0 equals 0?

                                                      • Eggnachohead_rblx
                                                        Eggnachohead_rblx  2 weeks back

                                                        i thought the answer was always ERROR

                                                        • Nobody
                                                          Nobody  2 weeks back

                                                          my math teacher make me divide by zero im not confused

                                                          • _ShadowShot _
                                                            _ShadowShot _  2 weeks back

                                                            Anything to the power of zero is one.
                                                            But something cannot come of nothing.
                                                            So my question is...



                                                            What is zero to the power of zero?

                                                            • Gnome Ann
                                                              Gnome Ann  2 weeks back

                                                              _ShadowShot _, 0^0 is indeterminate/undefined. By itself it has no definable value.

                                                          • SayNuthinTillYaSeeClaude

                                                            Disappointed to see you break the rule of equals. 2x3=6x0.5=3 isn't correct; not all of them are the same.

                                                            Poor form Ted talks, poor form

                                                            • Austin Liu
                                                              Austin Liu  2 weeks back

                                                              Infinity times infinity is infinity squared 3:20

                                                              • Gnome Ann
                                                                Gnome Ann  2 weeks back

                                                                Austin Liu, but infinity to the power of any positive number is infinity.

                                                            • Agata Malecka
                                                              Agata Malecka  2 weeks back

                                                              Divide $10by 0 u still got $10 You don't have more money also divide 10 $ by 0.00001 it can not be logically more than $10

                                                              • Anonymoose
                                                                Anonymoose  2 weeks back

                                                                That's... that's not how division works.

                                                            • Mr. Anonymous
                                                              Mr. Anonymous  2 weeks back

                                                              3:28 Wouldn't ∞ plus ∞ equal 2∞?

                                                              • Mr. Anonymous
                                                                Mr. Anonymous  2 weeks back

                                                                @Gnome Ann I understand that you mean that ∞ + ∞ would still algebraically equal ∞. But 2∞ doesn't only mean that its value is ∞. It more importantly denotes that ∞ can be written twice, like ∞ + ∞ which actually does matter in this case because it would then set the equation true, and 0 + 2∞ would = 2.

                                                              • Gnome Ann
                                                                Gnome Ann  2 weeks back

                                                                Mr. Anonymous, it does, but writing out 2∞ is unnecessary, since infinity times any number except zero is equal to infinity, so:

                                                                ∞ + ∞ = ∞(1 + 1) = 2∞ = ∞

                                                                (Note that it is not really formal math to do operations on infinity like this, if we wanted to be more technical, we would use aleph-0 or limits).

                                                            • TheProgrammingJedi
                                                              TheProgrammingJedi  2 weeks back

                                                              Excellent explanation. Thank you.

                                                              • Deva Preethi
                                                                Deva Preethi  2 weeks back

                                                                I've wondered about this so many times. Here's what I think:
                                                                Where do we use division in basic day to day life?? Like dividing things between people. If you have 10 candies and want to divide it equally to 2 people, you will divide 10 by 2, so you will give 5 to each. If you have 3 apple and want to give equal pieces to 2 people, you will give 1 each (3/2, Quotient= 1 reminder=1), 1 will be reminder but if you choose to absolute you further divide that remaining apple into 2 and give half each (3/2=1.5, 2*1.5 = 3), so the answer will be 1 and half each . So if you are having 20 KG mango, you give equal kg of fruits to 0 people, how will you divide 20kg with 0 people? You don't divide because there is no one to give it to, reminder is 20kg.(20/0=?) So what must be multiplied to 0 to get reminder 20 itself. Answer is 0. But 0*0 is not equal to 20. ughhhhh.

                                                                • Gopal Roy
                                                                  Gopal Roy  2 weeks back

                                                                  0 discovered in India and explored with Einstein in 20th century, breaking rules of Newton.

                                                                  So breaking rules will continue in future also..

                                                                  • Gopal Roy
                                                                    Gopal Roy  2 weeks back

                                                                    Alot better explained.. The rules applies in limit and calculas

                                                                    • Sebastian Gruber-Kersting

                                                                      Yes, fun worlds to explore, while completly pointless by any means, apart from the fun about nonsense.
                                                                      Really?!
                                                                      Mathematicians hate to be unemployed ... that might be TRUE!

                                                                      • BangGamer 1338 YT
                                                                        BangGamer 1338 YT  3 weeks back

                                                                        0:0=1?

                                                                        • seHun eXo
                                                                          seHun eXo  3 weeks back

                                                                          well all i knew is that MITOCHONDRION is the powerhouse of the cell.

                                                                          • Moepowerplant
                                                                            Moepowerplant  3 weeks back

                                                                            3:40 looks like heavy quantum stuff... Is it possible to divide by zero in a quantum world?

                                                                            • Ethan Ton
                                                                              Ethan Ton  3 weeks back

                                                                              10 ÷ 0 = 0

                                                                              There is no multiplication
                                                                              0 × 0 = 0
                                                                              So 0 equals nothing and if 10 × 0 would be 0 because there would be nothing there
                                                                              Just a theory