Division by zero in mathematics and computation

Division by Zero

Mathematics is unique in that it is an objective subject. When answering a question, something is either empirically true or it is completely false; there is little if any subjectivity about it in most situations. One of the most interesting aspects about math is that there are certain laws which are irrefutable and must be accepted in order for the correct answer to be discovered. A particularly intriguing and potentially frustrating aspect of math has to do with the division of zero. According to the laws of math, division by zero is impossible. When someone solves a problem and finds themselves with a fraction where zero is in the denominator, the answer is always undefined because in mathematics, there is no such number. Mathematically, it is not possible to divide a numerator by zero and have either a real or imaginary number for an answer.

There are four basic functions of mathematics: addition, subtraction, multiplication, and of course division. Division is the process of separating a larger number into an equal number of pieces (Fosnot and Dolk 2001). For example, the number four divided by the number two creates two pieces of equal size. We say that two goes into four two times because two multiplied by two is equal to the number four. For a mathematical statement to be true, its reverse must be proven to be true as well. Division is the opposite mathematical concept of multiplication, just as subtraction is the mathematical opposite of addition. If two numbers multiplied is equal to an answer, then that answer divided by one of the numbers in the original problem must provide the other multiplier. If the reverse operation cannot be performed, then this poses a serious problem.

Mathematician Charles Seife writes:

Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and the framework of logic -- and threatens to undermine the very basis of science (2000,-page 214).

In the most basic terms, dividing by zero is a case wherein the divisor, or denominator, is zero. Any number placed in the numerator and then divided by zero is undefined. The reason for this is simple. If the numerator is 2 and the denominator is zero, then the problem concerns a fraction of 2/0. Naive people would give the answer as zero, but this is incorrect. In order for 2/0 to equal zero, then zero times zero would need to be two. According to Charles Seife (2000), "Multiplication by zero should undo division by zero, so 1/0 x 0 should equal 1. However, we say that anything multiplied by zero equals zero!" (page 23). No number times zero will equal the number that is in the numerator. Therefore, the division is impossible and the answer undefined because there is no defined answer for an impossible problem.

Those who understand the aforementioned circumstances question whether 0/0 should be an undefined number or if 0/0 should be either zero or one (Watson 2010,-page 373). The latter suggestion is more easily dealt with. Any number divided by its self is equal to one. So, a potentially logical conclusion would be that since zero is being divided by its self, the answer should be one. However, that assumption is erroneous because zero is not actually a number, but the absence of a number (Kaplan 1999,-page 68). It is a placeholder on the number line separating positive and negative real numbers. If zero divided by itself were indeed equal to one, then problems arise when trying to multiplying that solution by another number, such as zero divided by itself and the quotient then multiplied by another number. If it were one, then the multiplication property of identity would insist that the answer equal the new number and not zero (Stewart 2008). The second hypothesis is a bit more complicated. While it is true that zero times zero equal zero, it is not drive that zero divided by zero equals zero. Also, since any number times zero will equal zero, there is no one unique solution for the answer. Example problems such as 2x0 = 0 and 1450 x 0 = 0 show that a number times zero equals zero. However, at the same time, zero times itself does not equal the number multiplied by zero. The answers cannot be duplicated when the reverse operation is performed (Knifong 1980,-page 179). Even 0 x 0 = 0 is problematic in that zero divided can be divided by itself and multiplied by itself an infinite number of times.

In higher mathematics, such as calculus, the question of division by zero becomes even more complicated. Asymptotes, for example, are lines which correspond to the zeroes of the denominator of a rational function (Kuptsov 2001). This is necessary in determining geometric functions because, since zero can never be in the denominator, the person solving the equation knows that the graph of their equation cannot include the numbers which would allow x to equal zero. For example, if the graph of a line were y = 2/x, x could never be zero because then 2/0 would be an undefined number. The physical presence of x unequal to zero is shown on the graph, but still the solution to a number divided by x is not visible.

Limits are another component of calculus which complicates the division of a number by zero, but still does not change the fact that it is impossibility. Even in calculus, the actual arithmetic value of zero divided by itself cannot be determined. Instead, the function of the limit is to determine a pattern of mathematical quotients to make a best estimate at what such an answer might be if it were to exist in the real world (Weisstein 2012). The established rule for calculus and limits is that the limit of division by zero can be either plus or minus infinity, or that it can have no limit. This is written as either ? Or -?. With limits, the mathematician can get close to approaching the answer to division by zero; however this is only ever an approximation and is never able to fully solve the problem.

There have been advances in mathematics, such as hypothetical and theoretical math topics, such as fractal Cantonian space time. It is an "operational extension to algebraic groups" which poses that there is a place within quantum physics which would allow for a division by zero (Czajko 2004,-page 261). Researchers in this field have postulated that this division will allow for better understanding and utilizing of "mutually dual line vector spaces" (Czajko 2004,-page 262). Scientific inquiry also poses potential situations where there may in fact be ways to divide by zero. However, it must be noted that all these propositions are theoretical and none have been empirically proven.