What is value of Zero divided by Zero (0/0) – One, Zero, Infinite, Indeterminate, or Undefined?

It is not uncommon to see students getting confused on asking, “What is the value of zero divided by zero?” This is often because they have forgotten the actual definition and therefore apply their ‘common sense’ to deduce the result based on other facts they have learned or are aware of.

If they happen to think that any number divided by itself is one, they may conclude that zero divided to zero should also be one. In other words, one may tend to think that since, for example, \(7/7=1\), \(4/4=1\), \(1/1=1\), so \(0/0=1\).

If they happen to think that zero divided by any number yields zero, they may guess that zero divided by zero should also be zero. In other words, one may tend to think that since, for example, \(0/7=0\), \(0/4=0\), \(0/1=0\), so \(0/0=0\).

Students who are aware with limits may get mistaken by thinking that zero divided to zero is infinite or indeterminate. They may think that when a positive number is divided by another positive number, the result becomes larger and larger as the number in the denominator is made smaller and smaller; and the result approaches to infinite as the number in denominator approaches zero. For example, \(7/14=0.5\), \(7/7=1\), \(7/3.5=2\), \(7/0.35=20\), \(7/0.05=140\), and so on. One may mistakenly extrapolate this and think that when the number in denominator becomes zero, the result becomes infinite.

Another mistake by many students aware with limits is that they think that \(0/0\) is indeterminate. Certain forms of limits (like \(0/0\) and \(0^0\)) are said to be indeterminate because merely knowing the limiting behavior of individual parts of the expression is not sufficient to actually determine the overall limit. For example, consider the expression \((x^2-4)/(x-2)\) as value of \(x\) tends to 2. As value of \(x\) tends to 2, both the numerator and denominator of the given expression approach to zero but the limit of overall expression is \(4\). This is because the given expression is same as \((x-2)(x+2)/(x-2)\) which simplifies to \((x+2)\) and which becomes equal to \(4\) as \(x\) tends to 2. Since \(0/0\) form in limits is indeterminate, many students start believing that division of zero by zero is indeterminate in arithmetic too.

But all of above conclusions about the value of \(0/0\) are wrong. Value of \(0/0\) is simply undefined because division by zero to any number (including zero) is undefined. Division and multiplication are inverse operations – if we divide real number \(a\) by real number \(b\) to get real number \(c\), then it means we will get number \(a\) if we multiply \(c\) by \(b\) provided that \(b\) is not \(0\). If \(b\) is \(0\), then for every real value of \(c\) we get \(c \times b=0\). We, therefore, say that division by zero is undefined.

In special case where \(a\) and \(b\) both are \(0\), one can argue that expression \(a/b=c\) is equivalent to \(c \times b=a\), that is, we recover \(a\) by multiplying \(b\) and \(c\). But this can not be accepted as the valid argument because multiplying \(c\) with \(b\) (which is \(0\)) will yield \(a\) (which is also \(0\)) for each and every real value of \(c\) and hence there is no definite value of \(c\). Thus, \(0/0\) can have no definite value if not undefined.

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