Dividing by Zero!

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Division by zero is one of the baffling mathematical relationships that leave many students confused. Although many of higher level students come to terms with the answer “Infinity!” many a times the mathematical reasoning behind the relationship is not quite understood by many. The reason is not far to see; when a student is introduced to basics of mathematical operations like addition, multiplication, division and fractions, he often gets confirmation of truth of these relations when he applies them to day to day transactions like when he buys candies or shares them among his friends. But nowhere, he encounters the need or the likelihood of encountering the reality of ‘infinity’!

To understand the implication of division of zero, one has to understand the operation division very clearly and be able to relate the mathematical concept to the real world. Without this, the confusion around the division by zero is not cleared. The same goes for the concept of Infinity. First of all, we must understand that Infinity is not a number! Because if you put it in the form of a number, no matter how large, still you can add one number to it to get still higher valued number. Infinity refers to a concept expressing a very large number or refers to a quantity that has no bound or end. It is said that the concept of mathematical infinity is first formulated by Indian mathematician Bhaskaracharya with the relation n/0=  though the concept of infinity was stated in Upanishads (part of Yajurveda) by the statement, “If you remove something from Infinity, what remains still is Infinity”.  The symbol for infinity was introduced by John Wallis in 1655. The symbol represents snake symbol which depicts the snake eating its own tail, symbolizing the endless nature of the infinity.

Division of a number by another involves finding how many parts of the second number (divisor) are contained in the first number (dividend). The answer is referred as quotient. For example 12/3 = 4. In this case, you have to find how many times the quantities of “3” are found in 12? Or how many times can you take out 3 out of 12. The answer is 4. You can take 3’s 4 times to completely empty the quantity 12. Also, you can observe that division is the reverse process of multiplication. The difference between the two operations is that the division is always found intuitively where as the multiplication is a straightforward application of mathematical tables.

Coming to the question of division by zero, we have to apply the same logic to determine the value obtained by dividing any number by zero.

How many times we can take out ‘0’s from a given quantity to completely empty it? Suppose the jar of candies contains 12 candies and you are asked to remove 0 candies at a time and find out after how many operations the candies in the jar will be emptied? The obvious answer is that it will never be emptied even if you remove billions of billion times!

Another way of understanding the division by zero is to divide a number from smaller numbers to find the quotient getting higher in value. For example 1/0.01= 100 while 1/0.00001 = 10000. By the same logic, if you divide by smallest quantity imaginable (=0) the quotient will approach infinity!

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