Division by Zero 0/0

Division by zero in mathematics is division in which the divisor equals zero. Such division can be formally written as , where a is the dividend.

In arithmetic with real numbers this expression is meaningless, since:

• for a ≠ 0 there is no number that, multiplied by 0, gives a, so no number can be taken as the quotient a⁄0;
• for a = 0 division by zero is likewise undefined, since any number multiplied by 0 gives 0 and could be taken as the quotient 0⁄0.

Historically, one of the earliest references to the mathematical impossibility of assigning a value to a⁄0 is found in George Berkeley's critique of the calculus of infinitesimals.

Division by zero is not permitted in many algebraic structures (for example, in fields and rings). However, the notion of a ring can be extended so that division by zero becomes possible. The resulting structure is called a wheel.

At the same time, according to the definition of divisibility of integers, if for some integer a and integer b there exists an integer q such that a=qb, then a is said to be exactly divisible by b (denoted ), or that b divides a (denoted b∣a).

According to this definition, the following properties hold:

• Any integer is a divisor of zero:

0⋮a,

and the quotient (for a≠0) equals zero.

• Only zero is divisible by zero:

a⋮0 ⇒ a=0,

and the quotient in this case is undefined.

Since multiplying any number by zero always gives zero, dividing both sides of the equation x × 0 = y × 0, which holds regardless of the values of x and y, by 0 yields the incorrect expression x = y for arbitrarily chosen variables. Since zero may be given not explicitly but in the form of a sufficiently complex mathematical expression, for example as the difference of two values that can be reduced to one another by algebraic transformations, such a division can be a fairly inconspicuous error. Covertly introducing such a division into the course of a proof in order to show the identity of values that are in fact different, thereby proving any absurd statement, is one kind of mathematical sophism .

In programming, depending on the programming language, the data type, and the value of the dividend, an attempt to divide by zero can lead to various consequences. The consequences of division by zero in integer and floating-point arithmetic are fundamentally different:

• An attempt at integer division by zero is always a critical error that makes further execution of the program impossible. It leads either to the generation of an exception (which the program can handle itself, thereby avoiding a crash), or to the immediate termination of the program with an unrecoverable-error message and, possibly, the contents of the call stack. In some programming languages, for example in Go, integer division by a zero constant is considered a syntax error and causes compilation of the program to abort.
• In floating-point arithmetic the consequences can differ from language to language:

• generation of an exception or termination of the program, as with integer division;
• obtaining, as the result of the operation, a special non-numeric value. The computation is not interrupted, and its result may subsequently be interpreted by the program itself or by the user as a meaningful value or as evidence that the computation is incorrect. A widely used convention is that for division of the form a⁄0, where a ≠ 0 — a floating-point number, the result turns out to be positive or negative (depending on the sign of the dividend) infinity — +INF or −INF, while for a = 0 the result is a special value NaN (an abbreviation of the English not a number — «not a number»). This approach is adopted in the IEEE 754 standard, which is supported by many modern programming languages.

An accidental division by zero in a computer program sometimes becomes the cause of expensive or dangerous failures in equipment controlled by the program. For example, on 21 September 1997, a division by zero in the computerized control system of the US Navy cruiser USS Yorktown (CG-48) caused all electronic equipment in the system to shut down, as a result of which the ship's propulsion system stopped working.

• Division (mathematics)
• Zero divisor
• Asymptote
• [[b12607]]
• [[b12606]]
• [[b34]]

See also

• Division by Zero 0/0
• Zero to the power of zero
• Indeterminate form