Division by zero

In mathematics, division by zero, division where the divisor (denominator) is zero, is a problematic special case. Using fraction notation, the general example can be written as ⁠

a

0

{\displaystyle {\tfrac {a}{0}}}

⁠, where ⁠

a

{\displaystyle a}

⁠ is the dividend (numerator).

The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor. That is, ⁠

c

=

a

b

{\displaystyle c={\tfrac {a}{b}}}

⁠ is equivalent to ⁠

c

×

b

=

a

{\displaystyle c\times b=a}

⁠. By this definition, the quotient ⁠

q

=

a

0

{\displaystyle q={\tfrac {a}{0}}}

⁠ is nonsensical, as the product ⁠

q

×

0

{\displaystyle q\times 0}

⁠ is always ⁠

0

{\displaystyle 0}

⁠ rather than some other number ⁠

a

{\displaystyle a}

⁠. Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called fields leaves division by zero undefined, and situations where division by zero might occur must be treated with care. Since any number multiplied by 0 is 0, the expression ⁠

0

0

{\displaystyle {\tfrac {0}{0}}}

⁠ is also left undefined.

Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type of mathematical singularity.