Division by zero

From Exampleproblems

In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as $\frac{a}{0}$ , where a is the dividend. Whether this expression can be assigned a meaningful (well-defined) value depends upon how the expression is interpreted.

1 Algebraic interpretation
- 1.1 Fallacies based on division by zero
- 1.2 Abstract algebra
2 Limits and division by zero
3 In mathematical analysis
4 Other number systems
5 Division by zero in computer arithmetic
6 See also

Algebraic interpretation

It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, the value of a division by zero is undefined, as it is in any field. The reason is that division is defined to be the inverse operation of multiplication. This means that the value of

${a \over b}$

is the solution x of the equation

$b x = a \quad$

whenever such a value exists and is unique. Otherwise the expression ${a \over b}$ is undefined.

For b = 0, the equation bx = a can be rewritten as 0x = a or simply 0 = a. Thus, in this case, the equation bx = a has no solution if a is not equal to 0, and has any x as a solution if a equals 0. In either case, ${a \over b}$ is undefined. Conversely, for the number systems mentioned above, the expression ${a \over b}$ is always defined if b is not equal to zero.

Fallacies based on division by zero

It is possible to disguise a special case of division by zero in an algebraic argument, leading to spurious proofs that 2 = 1 such as the following:

1) For any real number $x$ :

$x^2 - x^2 = x^2 - x^2 \quad$

2) Factoring both sides in two different ways:

$(x - x)(x + x) = x(x - x)\quad$

3) Dividing both sides by $x - x$ , giving $(0 / 0)$ :

$(0/0)(x + x) = x(0/0) \quad$

4) Simplified, yields:

$(1)(x + x) = x(1) \quad$

5) Which is:

$2x = x \quad$

6) Since this is valid for any value of $x$ , we can plug in $x = 1$ .

$2 = 1 \quad$

The fallacy is the assumption in step 4 that $(x - x) / (x - x)$ -- which is $(0 / 0)$ -- simplifies to $1$ . This proof is for the special case of dividing by zero when the numerator is zero. The fallacy results from the assumption that $0 / 0 = 1$ -- an assumption that generates the absurdity that $2 = 1$ .

This special case proof is instructive in that it is commonly held to be a general proof on how division by zero is destructive to math (a virtual Weapon of Math Destruction). In reality it only shows that $0 / 0 = 1$ is bad for algebra. This proof, strictly speaking, is not a general case against division by zero.

If one were to rewrite the proof and assume that $0 / 0 = 0$ , the absurdity would be erased. This flexibility on assuming the value of $0 / 0$ gets to the real issue: $0 / 0$ is indeterminate on the Reals. In the above proof it was determined to be $1$ -- possibly because of the rule that $a / a = 1$ . But when $a = 0$ we have an indeterminate meaning, and we are also free to work as if $0 / 0 = 0$ .

Still, in practice, division by a term in any algebraic argument will require either an explicit assumption that the term is not zero, or a separate justification showing that the term can never be zero.

Abstract algebra

Similar statements are true in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication, so as above, division poses problems only when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression

${2 \over 2}$

This should be the solution x of the equation

$2x = 2 \quad$

But this equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. The problem occurs because 2 is not invertible under multiplication.

Limits and division by zero

Image:Hyperbola one over x.jpg

At first glance it seems possible to define ${a \over 0}$ by considering the limit of ${a \over b}$ as b approaches 0. For any nonzero a, it is known that

$\lim_{b \to 0{+}} {a \over b} = {+}\infty$

and

$\lim_{b \to 0{-}} {a \over b} = {-}\infty$

Therefore, we might consider defining ${a \over 0}$ as $+\infty$ for positive a, and $-\infty$ for negative a. However, this definition is not generally useful, because positive and negative infinity are not real numbers, and the equation

$0 \, x = a$

still has no solution for any finite a. Furthermore, there is no obvious definition of ${0 \over 0}$ that can be derived from considering the limit of a ratio. The limit

$\lim_{(a,b) \to (0,0)} {a \over b}$

does not exist. Limits of the form

$\lim_{x \to 0} {f(x) \over g(x)}$

in which both f(x) and g(x) approach 0 as x approaches 0, may converge to any value or may not converge at all. See l'Hopital's rule for discussion and examples of limits of ratios.

In mathematical analysis

In distribution theory one can extend the function

${1 \over x}$

to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at $x = 0$ ; a sophisticated answer refers to the singular support of the distribution.

Other number systems

Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other mathematical structures, for instance on the Riemann sphere (see also poles in complex analysis). In hyperreal numbers and surreal numbers, division by non-zero infinitesimals is possible. If a number system forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, it can be extended to a wheel in which division by zero is always possible, but division has then a slightly different meaning.

Division by zero in computer arithmetic

The IEEE floating-point standard, supported by almost all modern processors, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, a/0 is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0. These definitions are derived from the properties of limits of ratios, as discussed above.

Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and simply generate an incorrect result for the division (often 0). If an exception is raised, the usual result is aborting whatever program it occurred in, although some programs (especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities.

Division by zero

From Exampleproblems

Contents

Algebraic interpretation

Fallacies based on division by zero

Abstract algebra

Limits and division by zero

In mathematical analysis

Other number systems

Division by zero in computer arithmetic

See also

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