The question of whether one can add to infinity often arises from our everyday understanding of numbers. Infinity, however, is not a number in the conventional sense, and applying standard arithmetic to it can lead to counterintuitive results. It represents endlessness or boundlessness, transcending countable quantities and measurement.
Understanding Infinity: Not Just a Number
Infinity is a mathematical concept signifying something without end or limit, distinct from a specific numerical value. Unlike finite numbers that can be counted, measured, or reached, infinity cannot be quantified. The symbol for infinity, ∞, introduced by mathematician John Wallis in 1655, represents this boundless idea. It symbolizes a potential infinity, indicating a process that continues indefinitely, such as the sequence of natural numbers (1, 2, 3, …), which never reaches an end.
Think of infinity as a direction on a number line that extends without bound, rather than a fixed point. It does not behave like the numbers we typically use in arithmetic. For instance, you cannot have “infinity apples” or “infinity miles” because these are physical quantities that can always be increased. This conceptual nature is why standard mathematical operations, designed for finite values, often yield different outcomes when applied to infinity.
The Math of “Adding” to Infinity
When adding to infinity, standard arithmetic rules do not apply. If any finite number is added to infinity, the result remains infinity. For example, infinity plus one (∞ + 1) still equals infinity, whether you add 1 or a million.
This principle extends to other basic operations. Subtracting a finite number from infinity also leaves infinity unchanged. Similarly, multiplying infinity by any finite positive number results in infinity. These behaviors highlight that infinity is not a quantity that can be increased or decreased by finite amounts. If infinity were a number, adding 1 to it and getting the same result would imply that 1 equals 0, which contradicts fundamental arithmetic.
The underlying reason is that infinity represents an unbounded state. Adding a finite value to something already without limit does not make it “more” infinite; it simply remains endless. This non-intuitive behavior illustrates why mathematicians treat infinity as a concept or a limit, rather than a fixed numerical value.
Beyond Simple Addition: Different Infinities
While adding a finite number to infinity does not change its “size,” mathematics recognizes that different “sizes” of infinity can exist. This concept emerges from set theory, where infinite sets can have different cardinalities, or measures of their size. For instance, the set of natural numbers (1, 2, 3, …) is considered countably infinite because its elements can be put into a one-to-one correspondence with the counting numbers.
A famous illustration is Hilbert’s Grand Hotel, a thought experiment with an infinite number of occupied rooms. If a new guest arrives, the manager can move each guest from room (n) to (n+1), freeing room 1. Even if an infinite number of new guests arrive, the hotel can accommodate them by moving existing guests to rooms with numbers twice their current room (n to 2n), leaving all odd-numbered rooms vacant. This paradox shows that adding to a countably infinite set does not increase its cardinality.
However, there are also uncountably infinite sets, such as the set of all real numbers (including decimals) between 0 and 1. Georg Cantor demonstrated that these sets are “larger” than countably infinite sets because their elements cannot be matched one-to-one with the natural numbers. Despite these varying “sizes,” adding any finite number to either a countably or uncountably infinite set still results in that same infinity.
Infinity in Everyday Thought and Science
The concept of infinity extends beyond abstract mathematics, appearing in various fields as a conceptual tool or theoretical limit. In cosmology, for example, scientists ponder whether the universe is spatially infinite, meaning it extends without end. While current observations do not definitively confirm an infinite universe, the idea helps frame models of cosmic scale and structure.
Philosophical discussions have long engaged with infinity, from ancient Greek notions of the boundless to contemplations of infinite time or eternal existence. Philosophers like Aristotle distinguished between “potential” infinity, a process that can continue without end, and “actual” infinity, which he generally rejected as a completed state. These explorations examine the implications of endlessness on existence and understanding.
In computer science, infinity is sometimes represented in floating-point number systems to handle calculations that exceed the maximum representable value or involve division by zero. While not a true mathematical infinity, these representations indicate a value too large to be contained within the system’s finite memory. Programmers might also use the concept of an “infinite loop” to describe a sequence of instructions that repeats endlessly without a termination condition.