Division by zero
In mathematics, division by zero, division where the divisor is zero, is a problematic special case. Using fraction notation, the general example can be written as, where is the dividend.
The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor. That is, is equivalent to. By this definition, the quotient is nonsensical, as the product is always rather than some other number. 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 0/0| is also 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. For example, the reciprocal function,, tends to infinity as tends to. When both the numerator and the denominator tend to zero at the same input, the expression is said to take an indeterminate form, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.
As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient can be defined to equal zero; it can be defined to equal a new explicit point at infinity, sometimes denoted by the infinity symbol ; or it can be defined to result in signed infinity, with positive or negative sign depending on the sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior.
In computing, an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate to positive or negative infinity, return a special not-a-number value, or crash the program, among other possibilities.
Elementary arithmetic
The meaning of division
The division can be conceptually interpreted in several ways.In quotitive division, the dividend is imagined to be split up into parts of size , and the quotient is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made. Now imagine instead that zero slices of bread are required per sandwich. Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant.
The quotitive concept of division lends itself to calculation by repeated subtraction: dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this way never terminates. Such an interminable division-by-zero algorithm is physically exhibited by some mechanical calculators.
In partitive division, the dividend is imagined to be split into parts, and the quotient is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies. Now imagine instead that the ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity.
In another interpretation, the quotient represents the ratio. For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of or, proportionally,. To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar. Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio, or proportionally, is perfectly sensible: it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer.
A geometrical appearance of the division-as-ratio interpretation is the slope of a straight line in the Cartesian plane. The slope is defined to be the "rise" divided by the "run" along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope and a vertical line has slope. However, if the slope is taken to be a single real number then a horizontal line has slope while a vertical line has an undefined slope, since in real-number arithmetic the quotient is undefined. The real-valued slope of a line through the origin is the vertical coordinate of the intersection between the line and a vertical line at horizontal coordinate, dashed black in the figure. The vertical red and dashed black lines are parallel, so they have no intersection in the plane. Sometimes they are said to intersect at a point at infinity, and the ratio is represented by a new number ; see below. Vertical lines are sometimes said to have an "infinitely steep" slope.
Inverse of multiplication
Division is the inverse of multiplication, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example. Thus a division problem such as "what is 6 divided by 3?" can be solved by rewriting it as an equivalent equation involving multiplication, "what number times 3 is equal to 6?", and then finding the value for which the statement is true; symbolically, these might be written as and equivalently, where represents the same unknown quantity in each equation; in this case the unknown quantity is, because, so therefore.An analogous problem involving division by zero,, requires determining an unknown quantity satisfying. However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for to make a true statement.
When the problem is changed to, the equivalent multiplicative statement is ; in this case any value can be substituted for the unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient.
Because of these difficulties, quotients where the divisor is zero are traditionally taken to be undefined, and division by zero is not allowed.
Fallacies
A compelling reason for not allowing division by zero is that allowing it leads to fallacies.When working with numbers, it is easy to identify an illegal division by zero. For example:
The fallacy here arises from the assumption that it is legitimate to cancel like any other number, whereas, in fact, doing so is a form of division by.
Using algebra, it is possible to disguise a division by zero to obtain an invalid proof. For example:
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote as.
Early attempts
The Brāhmasphuṭasiddhānta of Brahmagupta is the earliest text to treat zero as a number in its own right and to define operations involving zero. According to Brahmagupta,A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha: "A number remains unchanged when divided by zero."
Bhāskara II's Līlāvatī proposed that division by zero results in an infinite quantity,
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst.
Calculus
studies the behavior of functions using the concept of a limit, the value to which a function's output tends as its input tends to some specific value. The notation means that the value of the function can be made arbitrarily close to by choosing sufficiently close to.In the case where the limit of the real function increases without bound as tends to, the function is not defined at, a type of mathematical singularity. Instead, the function is said to "tend to infinity", denoted, and its graph has the line as a vertical asymptote. While such a function is not formally defined for, and the infinity symbol in this case does not represent any specific real number, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity",. In some cases a function tends to two different values when tends to from above and below ; such a function has two distinct one-sided limits.
A basic example of an infinite singularity is the reciprocal function,, which tends to positive or negative infinity as tends to :
In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,
However, when a function is constructed by dividing two functions whose separate limits are both equal to, then the limit of the result cannot be determined from the separate limits, so is said to take an indeterminate form, informally written. Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in
the separate limits of the numerator and denominator are, so we have the indeterminate form, but simplifying the quotient first shows that the limit exists: