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Division by Zero


Division by zero

Consider the expression which shows dividing a number by zero. This division is considered undefined or not possible. This is because, there exists no number real or otherwise, such that

Division indicates how many times the denominator "fits" into the numerator. If you calculate to determine how many times will fit in , you will never reach no matter how many zeros you try. Therefore, certain conditions must be specified when s denominator contains a variable. For example,


An illustration

Another way to illustrate why division by is undefined is to study the graph of By using a table of values some points on the curve can be found.

Graphing the points on a coordinate plane allows the behavior of the function to be shown. The red points show what happens as approaches from the left, and the green points show what happens as approaches from the right.

The closer comes to on both sides, the more extreme the values of the functions become. The quotient approaches when approaches from the right, and when approaches from the left. Because is a function, can only produce one value when Therefore, division by is undefined.


The consequences of dividing by zero
Suppose dividing by zero was defined. Then, the "logic" below would be accepted as true. Let and be any real numbers/

Since both sides of the equation are divided by and the expression was validly manipulated to create a false statement.