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Posted on Jan 02, 2017
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When division is explained at the elementary arithmetic level, it is often considered as a description of dividing a set of objects into equal parts. As an example, consider having ten apples, and these apples are to be distributed equally to five people at a table. Each person would receive = 2 apples. Similarly, if there are 10 apples, and only one person at the table, that person would receive = 10 apples.
So for dividing by zero - what is the number of apples that each person receives when 10 apples are evenly distributed amongst 0 people? Certain words can be pinpointed in the question to highlight the problem. The problem with this question is the "when". There is no way to distribute 10 apples amongst 0 people. In mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So , at least in elementary arithmetic, is said to be meaningless, or undefined.
Similar problems occur if we have 0 apples and 0 people, but this time the problem is in the phrase "the number". A partition is possible (of a set with 0 elements into 0 parts), but since the partition has 0 parts, vacuously every set in our partition has a given number of elements, be it 0, 2, 5, or 1000. If there are, say, 5 apples and 2 people, the problem is in "evenly distribute". In any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other.
In all of the above three cases, , and , one is asked to consider an impossible situation before deciding what the answer will be, and that is why the operations are undefined in these cases.
To understand division by zero, we must check it with multiplication: multiply the quotient by the divisor to get the original number. However, no number multiplied by zero will produce a product other than zero. To satisfy division by zero, the quotient must be bigger than all other numbers, i.e., infinity. This connection of division by zero to infinity takes us beyond elementary arithmetic (see below).
A recurring theme even at this elementary stage is that for every undefined arithmetic operation, there is a corresponding question that is not well-defined. "How many apples will each person receive under a fair distribution of ten apples amongst three people?" is a question that is not well-defined because there can be no fair distribution of ten apples amongst three people.
There is another way, however, to explain the division: if we want to find out how many people, who are satisfied with half an apple, can we satisfy by dividing up one apple, we divide 1 by 0.5. The answer is 2. Similarly, if we want to know how many people, who are satisfied with nothing, can we satisfy with 1 apple, we divide 1 by 0. The answer is infinite; we can satisfy infinite people, that are satisfied with nothing, with 1 apple.
Clearly, we cannot extend the operation of division based on the elementary combinatorial considerations that first define division, but must construct new number systems.
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