In mathematics, division by infinity is division where the divisor (denominator) is ∞. In ordinary arithmetic, this does not have a well-defined meaning, since ∞ is a mathematical concept that does not correspond to a specific number, and moreover, there is no nonzero real number that, when added to itself an infinite number of times, gives a finite number. However, "dividing by ∞“ can be given meaning as an informal way of expressing the limit of dividing a number by larger and larger divisors.
Using mathematical structures that go beyond the real numbers, it is possible to define numbers that have infinite magnitude yet can still be manipulated in ways much like ordinary arithmetic. For example, on the extended real number line, dividing any real number by infinity yields zero, while in the surreal number system, dividing 1 by the infinite number yields the infinitesimal number . In floating-point arithmetic, any finite number divided by is equal to positive or negative zero if the numerator is finite. Otherwise, the result is NaN.
The challenges of providing a rigorous meaning of "division by infinity" are analogous to those of defining division by zero.
Within the domain of mathematical discourse, the contemplation of dividing infinity by itself gives rise to a proposition of interest. Specifically, the assertion that the result of dividing infinity by infinity ( ∞ ÷ ∞ = ∞ ) is tantamount to infinity itself merits exploration. A logical journey unveils the underpinnings of this concept and its mathematical validity.
Consider a parameter denoted as "y," which, for the sake of analysis, is assigned the value 10. The crux of the matter rests in the equation ∞ ÷ y = ∞, where the introduction of y introduces an essential condition. To render the equation coherent, y must assume a magnitude that is sufficiently vast to accommodate the division operation involving infinity. This requirement reflects the conceptual intricacies associated with dealing with the concept of infinity in mathematical contexts.
However, the narrative takes a noteworthy turn as we transition to the equation y × ∞ = ∞. This equation signifies a transformation of the division operation into one of multiplication. In essence, this transition underscores a relationship where division of infinity finds equivalence through multiplication with an appropriate value of y. This insight reinforces the notion that the division of infinity by itself materializes as an operation of multiplication, culminating in an outcome of ∞.
Moreover, it's worth mentioning that if we carry forward the same line of thinking, something fascinating emerges. When we take infinity and divide it by a regular number like 10, the result still holds true: it's infinity. This adds another layer of insight to our mathematical journey, underscoring the depth of what we're uncovering here.
Use in technology
As infinity is difficult to deal with for most calculators and computers many do not have a formal way of computing division by infinity. Calculators such as the TI-84 and most household calculators do not have an infinity button so it is impossible to type into the calculator 'x divided by infinity' so instead users can type a large number such as "1e99" () or "-1e99". By typing in some number divided by a sufficiently large number the output will be 0. In some cases this fails as there is either an overflow error or if the numerator is also a sufficiently large number then the output may be 1 or a real number. In the Wolfram language, dividing an integer by infinity will result in the result 0. Also, in some calculators such as the TI-Nspire, 1 divided by infinity can be evaluated as 0.
Use in calculus
Integration
In calculus, taking the integral of a function is defined finding the area under a curve. This can be done simply by breaking up this area into rectangular sections and taking the sum of these sections. These are called Riemann sums. As the sections get narrower, the Riemann sum becomes an increasingly accurate approximation of the true area. Taking the limit of these Riemann sums, in which the sections can be heuristically regarded as "infinitely thin", gives the definite integral of the function over the prescribed interval. Conceptually this results in dividing the interval by infinity to result in infinitely small pieces.