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Concept
Division by zero
Summary
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming ); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").
There are mathematical structures in which is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms).
In computing, a program error may result from an attempt to divide by zero. Depending on the programming environment and the type of number (e.g., floating point, integer) being divided by zero, it may generate positive or negative infinity by the IEEE 754 floating-point standard, generate an exception, generate an error message, cause the program to terminate, result in a special not-a-number value, or crash.
When division is explained at the elementary arithmetic level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table. Each person would receive cookies. Similarly, if there are ten cookies, and only one person at the table, that person would receive cookies.
So, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed among 0 people at a table? Certain words can be pinpointed in the question to highlight the problem.
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