Division by zero

Division by zero is an attempted division whose denominator is zero. In ordinary arithmetic it is undefined because no number satisfies the required quotient equation.

Division: Division is the arithmetic operation that determines how many times one quantity is contained in another. A quotient is defined through multiplication by the denominator, which cannot recover a unique value when that denominator is zero.
Zero: Zero is the additive identity: adding it to any number leaves that number unchanged. The denominator at issue is the number that has no multiplicative inverse.
Zero divided by zero: Zero divided by zero is an undefined quotient that can yield different limiting values in different contexts. Unlike a nonzero numerator over zero, its quotient equation is satisfied by every number, so it is indeterminate.
IEEE 754: IEEE 754 is the standard for binary and decimal floating-point arithmetic. It defines how floating-point division by zero is represented, including signed infinities and invalid results.
Multiplicative inverse: The multiplicative inverse of a nonzero number is the number that multiplies with it to produce one. Division by a number requires its multiplicative inverse; zero has none.
Field (mathematics): A field is a set with addition and multiplication in which every nonzero element has a multiplicative inverse. Field axioms explicitly exclude zero from the elements that may be divided by.
Indeterminate form: An indeterminate form is an expression whose limit cannot be determined from its displayed form alone. The form zero over zero often appears in limits, where additional analysis is needed.
Computer algebra system: A computer algebra system manipulates mathematical expressions symbolically and performs exact calculations. Such systems must preserve domain restrictions when simplifying expressions involving a zero denominator.
Limits: A limit describes the value a function approaches as its input approaches a specified point. Limits can describe behavior near zero without assigning a value to division by zero.
Multiplication: Multiplication is an arithmetic operation that combines quantities, often interpreted as repeated scaling. Division by zero would require multiplication by zero to recover a nonzero dividend, which is impossible.

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