The question of whether one can add to infinity touches on a profound distinction in mathematics. The symbol $\infty$ is not a number like 5 or 10, which is why standard arithmetic rules often break down when applied to it. Instead, infinity represents a concept of unboundedness, a limit, or the size of an endless set. Understanding this conceptual nature is the first step in exploring the rules of infinite arithmetic.
Defining Infinity: Not Just a Big Number
Infinity is not merely a very large number; it is a limit that signifies a quantity without bound. A large number, like a googol ($10^{100}$), is finite, meaning you can always point to it on a number line. Infinity, by contrast, is a state of endlessness.
Mathematicians differentiate between two types of infinity: potential and actual. Potential infinity describes a process that has no end, such as continuously counting $1, 2, 3, \ldots$. Actual infinity treats the completed result of that endless process as a single, fixed size. The set of all whole numbers is an example of an actual infinity.
The infinity symbol ($\infty$) is most often used to represent potential infinity in calculus, describing a variable that increases without limit. In set theory, actual infinity allows mathematicians to compare the sizes of completed infinite collections. This difference explains why treating infinity like a regular number causes confusion in simple arithmetic.
The Rules of Infinite Addition
When you add a finite quantity to infinity, the result remains infinity. This is formally expressed as $\infty + N = \infty$, where $N$ is any finite number. Imagine an infinitely long line of objects; adding one more object does not change the fact that the line is still infinitely long.
Similarly, adding one infinity to another infinity yields the same infinite result: $\infty + \infty = \infty$. This rule holds because combining two unbounded quantities does not create a quantity that is more unbounded. For instance, the set of all positive integers is infinite, and the set of all negative integers is also infinite. Combining them results in the set of all integers, which is still simply one size of infinity.
This principle is consistent when considering infinite processes, such as adding the terms of a divergent series like $1 + 2 + 3 + 4 + \dots$. Since the sum of this sequence grows without limit, its value is $\infty$. Adding another infinite series that also diverges to $\infty$ will still result in an unbounded sum.
When Arithmetic Fails: Indeterminate Forms
The rules of infinite addition collapse in certain subtraction and division scenarios, leading to “indeterminate forms.” An indeterminate form is an expression where the final value cannot be determined from the limits of the individual parts alone. The most relevant indeterminate form is $\infty – \infty$.
The result of $\infty – \infty$ is not zero, because the two infinities might be growing at different rates. For example, consider the limit of $(x^2 + 5) – x^2$ as $x$ approaches infinity; the expression is $\infty – \infty$, but the result is 5. If you consider $x^3 – x^2$, the term $x^3$ grows much faster, making the limit $\infty$.
The indeterminate nature means the context of the problem dictates the true answer, which could be any finite number, zero, or infinity itself. Similarly, $\frac{\infty}{\infty}$ is indeterminate, and its outcome depends entirely on the relative growth rates of the functions involved. This ambiguity highlights that $\infty$ is a placeholder for a process, and the specific functions must be analyzed to find the true limiting value.
Beyond Simple Addition: Countable and Uncountable Sets
A sophisticated understanding of actual infinity, developed by Georg Cantor, reveals that not all infinite sets are the same size. These “sizes” are called cardinalities, introducing a hierarchy of infinities. The smallest infinite size is countable infinity, which is the cardinality of the set of natural numbers (1, 2, 3, 4, $\dots$).
A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers, meaning you could theoretically list them. This includes the set of all integers and the set of all fractions, both of which are countably infinite. Even though it seems like there are twice as many integers as natural numbers, they share the same cardinality.
Cantor proved that the set of all real numbers, which includes all the numbers on the number line, is an uncountable infinity, a larger size. The real numbers are so densely packed that no list could ever contain all of them. This demonstrates that infinite addition can lead to a result of a different magnitude.