Why can’t you divide by zero? – TED-Ed


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? Normally, dividing by smaller
and smaller numbers gives you bigger and bigger answers. Ten divided by two is five, by one is ten, by one-millionth is 10 million, and so on. So it seems like if you divide by numbers that keep shrinking
all the way down to zero, the answer will grow
to the largest thing possible. Then, isn’t the answer to 10
divided by zero actually infinity? That may sound plausible. But all we really know is
that if we divide 10 by a number that tends towards zero, the answer tends towards infinity. And that’s not the same thing as
saying that 10 divided by zero is equal to infinity. Why not? Well, let’s take a closer look
at what division really means. Ten divided by two could mean, “How many times must
we add two together to make 10,” or, “two times what equals 10?” Dividing by a number is essentially
the reverse of multiplying by it, in the following way: if we multiply any number
by a given number x, we can ask if there’s a new number
we can multiply by afterwards to get back to where we started. If there is, the new number is called
the multiplicative inverse of x. For example, if you multiply
three by two to get six, you can then multiply
by one-half to get back to three. So the multiplicative inverse
of two is one-half, and the multiplicative inverse
of 10 is one-tenth. As you might notice, the product of any
number and its multiplicative inverse is always one. If we want to divide by zero, we need to find
its multiplicative inverse, which should be one over zero. This would have to be such a number that
multiplying it by zero would give one. But because anything multiplied
by zero is still zero, such a number is impossible, so zero has no multiplicative inverse. Does that really settle things, though? After all, mathematicians
have broken rules before. For example, for a long time, there was no such thing as taking
the square root of negative numbers. But then mathematicians defined
the square root of negative one as a new number called i, opening up a whole new
mathematical world of complex numbers. So if they can do that, couldn’t we just make up a new rule, say, that the symbol infinity
means one over zero, and see what happens? Let’s try it, imagining we don’t know
anything about infinity already. Based on the definition
of a multiplicative inverse, zero times infinity must be equal to one. That means zero times infinity plus
zero times infinity should equal two. Now, by the distributive property, the left side of the equation
can be rearranged to zero plus zero times infinity. And since zero plus zero
is definitely zero, that reduces down to zero times infinity. Unfortunately, we’ve already defined
this as equal to one, while the other side of the equation
is still telling us it’s equal to two. So, one equals two. Oddly enough,
that’s not necessarily wrong; it’s just not true
in our normal world of numbers. There’s still a way it could
be mathematically valid, if one, two, and every other number
were equal to zero. But having infinity equal to zero is ultimately not all that useful
to mathematicians, or anyone else. There actually is something called
the Riemann sphere that involves dividing by zero
by a different method, but that’s a story for another day. In the meantime, dividing by zero
in the most obvious way doesn’t work out so great. But that shouldn’t stop us
from living dangerously and experimenting
with breaking mathematical rules to see if we can invent
fun, new worlds to explore.

100 thoughts on “Why can’t you divide by zero? – TED-Ed”

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  2. Thank you for this video and easy explanation. On a different note, Indians invinted the zero, so I hear. Then Arabs were the first to implement it (Not sure of the exact history).

  3. Oooh so you're saying that if we divide the cake pieces by zero, then we'll get infinite number of cake pieces?

  4. 1546 Maths Problem Of The Year: division by zero | Mathematicians: nope

    1547 Maths Problem Of The Year: 1/0 | Mathematicians: Don’t care

    1548 Maths Problem Of The Year: 1 over zero | Mathematicians: No

    …some time later…

    1572 Maths Problem Of The Year: square root of -1 | Mathematicians: IMAGINARY NUMBERS TIME

    1573 Maths Problem Of The Year: Divide by zero | Mathematicians: Big no

  5. When you ask Siri what is 0 divided by 0 and the only thing you remember is “you have no friends”

  6. I always thought that it could be that anything divided by zero, would equal zero.
    If you look at it in the sense that “divide” means “split into groups” then:
    10 divided by 2 equals 2 groups of 5.
    10 divided by 1 equals 1 group of 10.
    10 divided by 0 equals 0 groups of 10 (which ultimately equals zero).

    It’s the reverse of saying:
    2 x 5 = 10
    1 x 10 = 10
    0 x 10 = 10

  7. That's funny. There's a video explaining why you can't divide by zero. I always thought that was, you know, SELF-EXPLANATORY.

  8. Sir (0*♾ )+ (0* ♾)=0*2♾=2, 2♾ may be greater than real ♾. Sorry if I was wrong ,I am seventh class student…….

  9. Zero isn’t a number and so the division can never work. Can’t see how this calculation is even being discussed. Hardly a brain teaser

  10. You can DEFINITELY divide any number by zero. Your only problem is that any number divided by 0 is infinity.

    Let X be any number from R excluding 0, as it's a special case.
    Limit as i->0 (X/i) = lim as i->0+ (X/i) = lim as i-> 0- (X/i) = Infinity
    Therefore, any X real number divided by 0 is infinity. QED

  11. 0/0=infinity
    a/b=c
    c*b=a so,
    10/2=5
    5*2=10
    so if you change it up put zero in the equation,
    x*0=0
    0/0=x
    x can be any number(including 0) in this- x*0=0 equation – 5*0=0, 827*0=0, etc then those turn into 0/0=5, 0/0=827
    the fact that you can plug any number of infinity in for x makes 0/0 infinity

    and if u find a flaw in this, please comment it because i'm very curious if this actually works

    EDIT: instead of infinity, it might be more correct to call the answer "all real numbers"

  12. Me before even clicking the video: IT'S NOT GOING TO BE ANYTHING, BECAUSE IT'S NOTHING!!!
    my 14-year-old brain: (┛✧Д✧))┛彡┻━┻

  13. I know the reason!!!!

    YOU CAN'T DIVIDE BY ZERO JUST BECAUSE NO NUMBER MULTIPLIED BY ZERO EQUALS A NUMBER THAT IS MORE THAN ZERO!!!
    Note: Zero ÷ zero = every number because if a number that is more than zero will be mltiplied by zero, it always will be zero

  14. What you did wrong is that when you multiplied infinity by itself it didn't become infinity^2, you dimwit.

    I need to point out that this is a joke in the case that self-proclaimed mathematicians get angry.

  15. Simply put, and what he's saying long windedly is, 10 divided by 0 = 10……….0 divided by 10 = 0……….0 divided by 0 = 0.

  16. One more interesting fact: if you divide a number by some value which approaches 0 from the right (positive numbers) then the answer approaches positive infinity; but if the value approaches 0 from the left (negative numbers) then the answer approaches negative infinity!
    You can see this by ploting the graph of (1/x). There is a complete discontinuity in the function at x = 0

  17. Just wondering, aint (0x infinity)(0x infinity)= 0x 2infinty because infinity ain't no number, it's undefined like x. Because adding them together seems wrong.

  18. If you had (insert variable) of (insert any matter) and you make them in groups of zero
    You can get infinite groups of nothing
    You will get infinite groups
    But of nothing
    Unless nothing equals a random number but 0 then
    You either don’t try this
    Or you get or take away your (insert matter) in (insert a random number)
    Ok

  19. If you could divide by zero then the rule "something can't come from nothing" would be broken since if you divide a number with itself then you get 1 so 0÷0=1 but…

  20. if there are an infinite amount of numbers between 0 and 1 …( 1/2, 1/4, 1/8, 1/16…) and an infinite number of numbers between 1 and two, then infinity plus infinity should be two.

  21. If you have one cookie and divide it between your zero friends, cookie monster gets it all. Nom nom nom.
    So:
    1 / 0 = greedy cookie monster

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