The multiplicative inverse of zero

This is silly post, but was inspired while reading something which mentioned that zero has no multiplicative inverse. Since grade school I’ve always been puzzled by how math has weird corner cases like this, another: 1 divided by 3, never ends! Anyway, what if zero did have an inverse? Is there some Abstract Algebra thingy that has such a property?

So, lets call the inverse “j”, not ∞, it already has some meanings. Then we have: j0=1. Or, 0=1/j. This means that 0/0 = 1. I.e., (1/j)/(1/j) = 1*j/j*1 = 1.

Is there a physical meaning? What could 1/0 mean? Since multiplication can be interpreted as a scaling, then what happens when you scale something to zero? Well, using the analogy of a map or engineering drawing, a scaling doesn’t destroy a physical measurement, it allows a smaller or large value represent the other value. In a floor plan we may see something like 1/4 inch = 1 foot. In a full size scale, the multiplier is 1. As the multiplier becomes smaller we can represent larger measures. At zero scale, we represent infinity. Thus, scaling infinity to zero, still scales the whole thing, which is 1.

Yeah, doesn’t really work, unless your a mystic. Oh well, like I said silly. I won’t quit my day job.


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4 thoughts on “The multiplicative inverse of zero”

  1. I was wondering if IEEE had assigned some way for computers to use it, in particular allowing divide by zero to fail gracefully.

    1. From the Wikipedia page we have this: “The IEEE floating-point standard, supported by almost all modern floating-point units, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. The standard supports signed zero, as well as infinity and NaN (not a number). There are two zeroes, +0 (positive zero) and −0 (negative zero) and this removes any ambiguity when dividing. In IEEE 754 arithmetic, a ÷ +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0. The infinity signs change when dividing by −0 instead.”

  2. It’s not that easy. I actually thought of a multiplicative inverse, and I also called it j, and I am currently working on that with a math professor in a math forum. But understand that rather than just making a new number we are making a new system. There are, for example, rules for multiplication that you will have to change for multiplication with j to work. j(0) = 1 = j(0+0) = j(0)+j(0) = 2; 1 = 2. Right there we have a problem. So, there has to be rules for distributivity, associativity, etc. which I have already defined.

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