Undefined (mathematics) in the context of Contradiction

0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently dividing by 0 is generally considered to be undefined in arithmetic.

As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya.

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 ⁠${\displaystyle {\tfrac {a}{0}}}$⁠, where ⁠${\displaystyle a}$⁠ 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, ⁠${\displaystyle c={\tfrac {a}{b}}}$⁠ is equivalent to ⁠${\displaystyle c\times b=a}$⁠. By this definition, the quotient ⁠${\displaystyle q={\tfrac {a}{0}}}$⁠ is nonsensical, as the product ⁠${\displaystyle q\times 0}$⁠ is always ⁠${\displaystyle 0}$⁠ rather than some other number ⁠${\displaystyle a}$⁠. 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 zero is zero, the expression ⁠${\displaystyle {\tfrac {0}{0}}}$⁠ is also undefined.

In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if ${\displaystyle f}$ takes real numbers as input, and if ${\displaystyle f(0.5)}$ does not equal ${\displaystyle f(1/2)}$ then ${\displaystyle f}$ is not well defined (and thus not a function). The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.

A function that is not well defined is not the same as a function that is undefined. For example, if ${\displaystyle f(x)={\frac {1}{x}}}$, then even though ${\displaystyle f(0)}$ is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of ${\displaystyle f}$.