Undefined is a term used when a mathematical result has no meaning.
More precisely, undefined "values" occur when an expression is evaluated for input values outside of its domain.
- (if no complex numbers)
- (if no complex numbers)
- (units in radians, no complex infinity)
- (if no complex infinity)
Upon extending the functions definition to include a domain and range containing the complex numbers, we find that undefined only occurs upon a division by zero - or zero to the power of zero - because a logical and mathematical contradiction results. It must necessarily be excluded from possibility. See division by zero for more information and examples on this. No contradiction results if one allows for complex values, thus much of the domain and range we excluded earlier in simpler algebra can now be re-included.
Allowing for complex values, we find that the above example expressions equal the complex values or and .
Thus, until complex numbers are introduced and functions extended to include them (in higher levels of algebra), most functions are restricted to their real number domains which yield a real number range, and any number outside of that restriction is summarily regarded as "undefined".
In truth, however, all mathematical concepts included, division by zero is one of the few cases where a true undefined occurs. In such instances, the input value which results in a division by zero should have been left out of the domain. Thus, even division by zero is a special case of the evaluation of an expression outside of its domain.
If x=0 then . This is undefined in the truest sense of the word because, even if you allow for complex values, the expression still has no meaning if one doesn't allow complex infinity.
Unlike undefined values, indeterminates occur when we have a division of zero by zero, or division of infinities.
In the case in which we have a zero divided by another zero, we call it indeterminate. In all levels of math we exclude this situation from the allowed domain. However, in lower level algebras we still refer to it as undefined for simplicity sake. Whereas this is a simpler for young minds, it is ambiguous, technically inexact, and is generally a misnomer for the mathematical truth which is revealed in higher level algebras and calculus.
In the most general sense, it is still a division by zero and thus still an undefined value. There is, however, a technical distinction between undefined and indeterminate which takes a bit of explaining. A decent approach is a demonstration:
Suppose the value a was a non-zero number and . In this case, b is undefined. Notice that upon algebraic rearranging, we can show that . We know from the zero identity of multiplication that zero times any number is zero. Thus, 0b is equal to zero and the expression simplifies to a=0. Since a was defined ahead of time to be a non-zero number, we have arrived at a contradiction. The simple fact of the matter is that no value of b is possible for which multiplication by zero yields a non-zero value. Thus, b is undefined amongst all numbers.
Now suppose that a was zero. We now have . Algebra will now show that . The same zero identity guarantees that 0b equals zero regardless of what b is. The equation is not a contradiction any longer, its a theorem and is true for all values of b. Thus, no value of b is determinable from the equation as all values of b hold true.
Calculus and the Evaluation of Limits
Though the arithmetic evaluation of divisions by zero are mathematically impossible, producing either an indeterminate or undefined value, the study of calculus has necessitated their treatment with the use of limits.
Limits allow us to study the properties and trends of expressions at infinite and infinitesimal values. Whereas we cannot evaluate an expression that divides by zero, we may analyze the expression for how it operates "near" a division by zero.
Division by zero is always undefined or indeterminate, such as . However, upon introducing the limit, we find that is evaluable and in fact the expression "approaches" infinity.
Limits do not say what is at a given value. Limits only describe how things are trending near a given value.
Thus, even in calculus and the implementation of limits, division by zero is still not permitted. The function may be described, however, near certain values. Limits never technically equal anything. They tend toward.
A common misconception in amateurs is that the simple arithmetic operation is evaluable as "infinity", or otherwise explainable with limits. They fail to realize that introducing limits changes the question entirely.
How does this section pertain to indeterminate and undefined expressions?
In the case of undefined expressions, limits that evaluate them will tend to infinity or negative infinity. In the case of indeterminate expressions, however, the limits that evaluate them will tend toward a finite real value.
- An undefined a/0 form:
- An indeterminate 0/0 form:
Graphical Properties of Undefined and Indeterminate
The graphs of functions which have undefined and indeterminate values equally illustrate the distinctions between the two.
As stated in the previous section, undefined values tend toward infinity. Consequently, a graphical illustration of undefined shows a vertical asymptote, on either one or both sides of which the function increases in magnitude without bound. The function never crosses or touches the asymptote (by definition) and thus there is no value - not even infinite - at the asymptote.
Indeterminate values are a bit different. On a graph they will not have a vertical asymptote at all as the indeterminate value. Instead, there will be a point-gap: a single point which is excluded from the domain and has no range- or output-value. The function on either side of the point-gap will approach one or two distinct finite values.
In some cases, a function may be asymptotic and thus undefined when approaching a domain-value from one side, but when approached from the opposite side it is indeterminate and thus a finite real value is approached. These functions tend to be piece-wise