Q: Can you fix the “1/0 problem” by defining 1/0 as a new number?

The original question was: I’ve been relearning some things about imaginary numbers and the concept behind the number i got me thinking about something else. Is there a reason that the quantity 1/0 could not be defined in a similar way to i, so that functions could a real component, imaginary component, and an undefined component? If so, what implications would this have to mathematics? I can’t see it being very useful but I was curious to see what you think.


Physicist: If that worked it would be extremely useful. However, it just doesn’t jive with the axioms of arithmetic.
People in the math biz are always making up place holders for things that aren’t known or, in some cases, can’t exist. Euler (pronounced “Oiler”, as in “one who oils”) wanted to come up with a number system that included a solution to x2+1=0. He called that solution “i “, for “Imaginary number” or possibly “Incredibly awesome number” (I think it’s the latter).

As it happens, there are no problems involved with defining i.  In fact, it really cleans up a lot of math.  1/0 on the other hand is kind of a train wreck.

Define Q (for “Quite a bit more awesome than i“) as the solution to 0x=1. That is, define Q as “1/0″, so it’s the “multiplicative inverse” of 0.  Right off the bat there’s a problem:
\begin{array}{ll}1=Q\cdot 0\\\Leftrightarrow 1=Q\cdot (1-1)\\\Leftrightarrow 1=Q-Q\\\Leftrightarrow 1=0 \end{array}

This is because zero is defined as “x-x” for any number or variable x.  So by assuming 1=Q\cdot 0 (and the laws of arithmetic) you reach an impossible conclusion!

You could try to patch this problem, for example by declaring that Q-Q\ne 0.  Even so:

\begin{array}{ll}1=Q\cdot 0\\\Rightarrow 1\cdot 0=Q\cdot 0^2\\\Rightarrow 0=Q\cdot 0\\\Rightarrow 0=1 \end{array}

Again, by defining 1=Q\cdot 0 to be true, you’re led to a contradiction.  Mo’ logic, mo’ problems.  You could fix this problem by declaring that associativity doesn’t apply to Q.  That is, (Q\cdot 0)\cdot 0 \ne Q\cdot (0\cdot 0).  Losing associativity is a big deal though.  Without it you can barely do anything.

You can keep going, finding more problems and declaring more “fixes”, but in short order you’ll find that by the time you’re done patching things you’ll have more problems, exceptions, and caveats than just “1/0 is undefined”.

Best to just leave 1/0 undefined.

This entry was posted in -- By the Physicist, Conventions, Logic, Math. Bookmark the permalink.

23 Responses to Q: Can you fix the “1/0 problem” by defining 1/0 as a new number?

  1. Boreeas says:

    1 = Q*0
    => 1*0 = Q*0²
    => 0 = Q*0
    => 0 = 1

    How did you get from step 3 to step 4? If Q*0 = 1, then the first step is already true, and the other steps unnecessary, or am I wrong?

  2. The Physicist The Physicist says:

    This is a “proof by contradiction”.
    The first step is defined to be “true”, but it can’t be because it leads to a false statement (0=1).
    I’ll go back and clear that up.

  3. Neal says:

    Another way of phrasing this problem: for all (rational) numbers but 0, the map “multiplication by *number*” is bijective, so it can be inverted. For 0, the map “multiplication by 0″ is very horrendously not even injective. So it can’t be undone.

  4. Alex says:

    Actually it’s been done

    http://en.wikipedia.org/wiki/Wheel_theory

    “Wheels are a kind of algebra where division is always defined. In particular, division by zero is meaningful.”

    In essence the reciprocal of zero is defined, but the “normal” rules of arithmetic such as “x-x=0″ may not necessarily hold.

  5. The Physicist The Physicist says:

    Awesome!
    Still, given the choice between that set of rules and “don’t divide by zero”, I’d go with the second choice.

  6. Neal says:

    Modification of earlier comment: In any ring R, (right) multiplication maps R into its self-maps. Non-zero-divisors go to non-invertible maps. In particular, 0 is not a zero divisor, so (right) multiplication by 0 is not invertible: “x/0″ makes no sense.

  7. Paul says:

    The wheels construction looks cool. Though we can still say that one can’t “fix” the 1/0 problem — and stay within the framework of (commutative) rings / fields (i.e. the usual “axioms of arithmetic”). In contrast with the inclusion of i to go from reals to complex, which is the simplest of field extensions.

    Neal I believe in your description, “units” should take the place of “zero divisors.”

  8. Neal says:

    Yes, thank you.

  9. marshal says:

    here you try to manipulate Q=1/0 to Q*0=1 this would be done by multiplying both sides by zero which means Q*0=1 or 1*0/0=1 or simply 1=0/0, which is a contradiction . in order for your proof by contradiction to work you have to accept your unintentional corollary, 1=0/0 which would produce a very bizarre set of numbers. now accepting this new invention there is the conclusion that 1=0 or 0=1=0/0 and Q=1/0.

  10. The Physicist The Physicist says:

    I should have been more clear.
    “Q*0=1″ is the definition, and “Q=1/0″ is merely a convenient way of writing Q, since that’s the way every other multiplicative inverse is written. But be careful not to put any weight on it (it doesn’t hold up).

  11. tonyf says:

    As Boreeas pointed out, for your second example, even in ordinary arithmetics 0=Q*0 does not imply 0=1, 0=Q*0 does not imply anything.

    For your first example, as you noticed yourself, the problem is defining Q-Q=0. Actually, the question of giving 0/1 a value has already been solved, in two somewhat related ways: so-called non-standard analysis, and many computer implementations of arithmetics. In the latter we introduce two numbers in addition to the standard ones: NaN and Inf, informally we can think of them as “not a number” and “infinity”. Here 1/0=Inf, Inf-Inf=NaN. Your first example would work because Inf is not the multiplicative inverse of 0, instead any positive number divided by zero is directly defined as Inf (and as -Inf for negative), and 0*Inf=NaN. So the trick to get all arithmetic operations well defined (i.e., also division by zero when only ordinary numbers are concerned) is that it is not enough to introduce only one additional non-conventional number (“Q”), you need two new numbers (“Q=Inf” _and_ “NaN”) to get both ordinary number divided by zero well defined, and all the operations concerning the new non-conventional numbers well defined in a consistent way.

  12. Tom says:

    Well im just throwing this out there… 1/0 doesnt work in general as zero times any number will not equal one, even if you define an arbitrary symbol to equal this impossible number, that symbol really represents an either negative or positive infinite limit and cannot be defined as a single value, and as for defining Q as NaN and Inf is counterproductive for the original question, as operations using two non-existant numbers cannot be arithmetically calculated not to mention you just defined one symbol as two different values, which cannot exist in real functions, so then when does this become useful?

  13. The Physicist The Physicist says:

    You’re certainly right, there is no number that when multiplied by zero is one. Which is why “Q” in the post isn’t an actual number, it’s just some mathematical object with the correct property, in very much the same way that “i” is just an “impossible” mathematical object.
    I think what tonyf wanted to do was abandon Q altogether and replace it with two new symbols. But, there are plenty of examples of systems involving more than one made-up mathematical objects. For example, “quaternions” include the usual “i”, but also include “j” and “k”, which have identical properties to i, but interact weirdly with each other.

  14. doronshadmi says:

    x > 0
    x/0 = y
    x = y*0
    x > AND = 0, which is a contradiction.

  15. doronshadmi says:

    A better one:
    x ≠ 0
    x/0 = y
    x = y*0
    x ≠ AND = 0, which is a contradiction.

  16. John Morgan says:

    This thread on 1/0 puts me in mind of something I used to tell my kids when they were very young and fascinated by the idea of infinity (this was before the public school system managed to beat all interest whatever in mathematics out of one of them-to my instant, white-hot, and abiding fury.) I’m no great shakes at explaining mathematics to the very young, but the Riemann construction showed me what I think is a good way to think about infinity and numbers:

    “Infinity is NOT a number, but you can sometimes think of it as a place”

    Having said that, the person who posed the original question was close to the idea of an extension to the reals (AKA the “point at infinity”) which Riemann so memorably visualized.

  17. pi says:

    If I define a function called f(x) = x^2 – 6x + 9 / x^2 – 9
    then, if x = 3, then f(3) = 0/0, which is undefined.
    But, if I simplify the fraction by factorization, it would equal, f(x) = (x-3) / (x+3). If you evaluate for f(3) now, it will now equal to 0/6 = 0.
    Another example is g(x)= x-1/x-1
    If I put in g(1)=0/0, which is undefined.
    But g(x) can be simplified to 1/1 =1 . And g(1) is 1.
    How is this possible? Doesn’t it violate mathematics? I mean if one expression can give you an answer, then simplifying it should also give you the answer because you have only simplified the approach to the solution. How can it have two solutions?

  18. The Physicist The Physicist says:

    Simplifying doesn’t quite leave the equation alone. In the examples you gave the new simplified functions keep their undefined points. So, (x-1)/(x-1) = 1, for x≠1. The “for x≠1″ is important there, but it’s subtle.

  19. Gavin Grubb says:

    If you divide any number that’s not zero by zero you will get negative or positive infinity because zero tries to turn the number into zero by turning it into X groups of zero. (Hence why division makes a number into something smaller.) Now if you’re going to say that the answer should be zero because it makes it into zero groups, you’re wrong!

    Dividing any number by zero is basically breaking it down by any means possible until you reach nothing. Now multiplying by zero is like dividing by any other number; it gets smaller. When dividing, it’s the exact opposite; it gets [U]Value[/U].

    What about 0/0? It suppose to be every number that can be multiplied by zero, or 1, -500, 0, or infinity… This means the theory of something coming from nothing can be true, but it’s not exactly clear…

    What if, 0/0 was suppose to happen, that would mean the equation was suppose to give value to anything. This could be the base of our number line…

  20. Pingback: Q: Is there a number set that is “above” complex numbers? | Ask a Mathematician / Ask a Physicist

  21. The Wonderer says:

    I have often thought of the same thing as this, but you have proved me wrong that this cant be done unless you make some special rules to go along with it. Alephs have their own special rules why not this?

  22. I think the problem deals with SO MUCH imaginary numbers, which in essence isn’t real. To mean something; a number MUST be in front of a [unit], but zero cancels all units.!!! so if you reach definition for 1/0 then i can turn my lead into gold… a*[lead]=b*[gold],

    In essence 1/0 is the philosopher’s stone which in essence is again OUT OF THIS universe [units]… because one apple IS EVEN AN ESSENTIALLY DIFFERENT to another apple, just in math world can be treated as equal….

  23. Pingback: Q: Where do the weird rules for rational numbers come from? (Dealing with fractions) | Ask a Mathematician / Ask a Physicist

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>