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April 2009Division By Zero In Physics - To do it or not?If there is a single topic in basic mathematics that could be considered "controversial", it is division by zero. Ask three different experts about division by zero and you'll get at least four different answers including any or all of: infinity, undefined, impossible, should be avoided, not a number, zero. You would think that Mathematics, Physics and Software Development would share common rules for simple arithmetic, but they differ in one area: division by zero. In mathematics, Gottfried Leibnitz said it best: "We do not divide by zero" (although this is not unanimous). Similarly, in commercial software development, division by zero is considered "out of scope" and is not performed. Only in physics is division by zero considered to be a valid result. You will find this in predictions of singularities, predictions of regions of infinite gravity or field strength, in the use of the Lorentz Factor, in Special Relativity and other areas. So which approach is correct? Does Leibnitz' method just arbitrarily avoid infinities, or is there a deeper reason to it? Is physics correct in interpretations of regions of infinite density? Is 1/0 a number?
Infinite Gravity or Field Strength?Many of the instances of division by zero in physics follow the general pattern of density calculations like this:
In these cases the volume is zero. In other words the result is referring to a region of space that has zero dimensions. I.e. it does not exist. A region of 0 x 0 x 0 is not a volume, it is a point. A point is no more a volume than a line or a plane is. This seems to escape the notice of those who make claims about infinite density - the volume they are referring to is not a 3-dimensional value and has no physical existence. In software development this would be regarded as out of scope of the equation as it returned no useful information. It also makes Leibnitz' advice look correct.
What's The Problem?We should not lay the blame for our misunderstanding on zero. It is simply that there is a better view of zero in physics: Nothing. If you're like most people, you believe that 1 / 0 = infinity. Why do you believe it? Because someone told you it was so. Perhaps you wondered for a while about "something divided by nothing" but were quickly told to get with the program... In fact 1500 years ago at least some Indian mathematicians believed that division by zero returned zero. It was only during the dark ages that the use of zero as a number and of 1/0=infinity became wide-spread. Over time, following the "invention" of zero we lost our concept of nothing as distinct from a "number zero". The Latin word ciphra for zero actually comes from the Arabic word "sifr" for empty or nothing. It has been forgotten that zero was introduced to represent the absence of a number, not an actual number. In fact, we have taken to using it as an actual number, such as in the time 00:00 (midnight) or as the zeroth element in an array. This confusion about "man-made" zeros versus "natural" zeros such as zero volume has lead to numerous errors in physics. This distinction between "nothing" and "number zero" has largely fallen out of use in mathematics and physics, but is alive and well in software engineering. Why? Because you can't model the real world without it. It is sometimes called "null" and sometimes referred to as "nothing", it is equivalent to an empty set { } - the absence of any value. There is, in fact, a distinction between "nothing" and zero as a number. Nothing is a zero-dimensional value, the empty set { }, but in mathematics it is common to treat zero as a one-dimensional value, like an integer half-way between -1 and 1, a set of a number { 0 }. The single most important difference between these is that the result when dividing by nothing is nothing, as there is nothing to divide by.
What's The Difference?Following the "invention" of zero we lost our concept of nothing as distinct from a "number zero". It has been forgotten that zero was introduced to represent the absence of a number, not an actual number. Using the conventional view of zero { 0 }:
Viewing zero as null, nothing or the empty set { } these may be thought of as:
Which is equivalent to:
Taking the null view, if we interpret zero as null then we also interpret the null results of multiplication and division as zero, i.e. 1 * 0 = 0 and 1 / 0 = 0. We use null and zero interchangeably. Note the variation in the results between the views above: division by zero. Interestingly, the old axiom that you "should not divide by zero" still holds true, not because it arbitrarily avoids infinities but because there is nothing to divide by and it is the right thing to do.
Regards, AJ Corcoran
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