In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as {\textstyle {\tfrac {a}{0}}}{\textstyle {\tfrac {a}{0}}} where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives a (assuming {\textstyle a\neq 0}{\textstyle a\neq 0}), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression {\displaystyle {\tfrac {0}{0}}}{\displaystyle {\tfrac {0}{0}}} 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 {\textstyle {\tfrac {a}{0}}}{\textstyle {\tfrac {a}{0}}} is contained in Anglo-Irish philosopher George Berkeley’s criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").[1]