In mathematics, infinity is a concept describing something without any limit, boundless. It is an abstract concept rather than a number that can be manipulated and calculated with. While some ideas of infinity are used in the real number system, infinity itself is not considered a real number.
What is a real number?
Real numbers are numbers that can be plotted on the number line. They include all rational numbers, such as integers and fractions, as well as irrational numbers like π and √2. Real numbers must satisfy the important properties of:
- Closure under addition and multiplication
- Existence of an additive identity (0) and multiplicative identity (1)
- Existence of additive inverses and multiplicative inverses, except for 0
- The commutative, associative, and distributive properties
These properties allow real numbers to be manipulated and calculated with meaningfully. Operations like addition, subtraction, multiplication, and division are well-defined for real numbers.
Properties of infinity
Infinity does not behave like a real number. Some key properties of infinity include:
- Infinity is not part of the real number line. You cannot plot it at some “end point”.
- Arithmetic operations like addition and multiplication are not defined with infinity. Expressions like “infinity + 1” or “infinity * 2” have no meaning.
- Infinity is not reached by successively adding numbers, no matter how large. You can always take a number and add 1 to reach a new, larger number.
- Comparisons like greater than (>) or less than (
These behaviors distinguish infinity from real numbers. While infinity may be used as a concept in limits, series, and set theory, it does not follow the rules required to be a real number.
Infinity in calculus and analysis
In calculus and mathematical analysis, the concept of infinity shows up in a few key ways:
- Limits involving infinity – The limit of a function or sequence may converge towards positive or negative infinity.
- Infinite series – An infinite series sums an infinite number of terms, and may converge towards infinity or a finite limit.
- Infinite integrals – An improper integral may have an infinite interval as its boundary.
While infinity itself is still not treated as a real number here, some meanings are given to equations involving infinity in limits and series. For example, a limit equal to infinity indicates the function grows without bound.
Infinity in set theory
In set theory, infinity shows up through infinite sets – sets that contain an unlimited number of members. Some examples include:
- The set of natural numbers {1, 2, 3, …}
- The set of integers {…, -2, -1, 0, 1, 2, …}
- The set of prime numbers {2, 3, 5, 7, 11, …}
These sets have an infinite number of elements. We cannot list out all the elements due to the unlimited nature. Yet infinite sets can still have mathematical meaning. For example, the set of even numbers has “half” as many members as the set of integers, even though both are infinite.
Georg Cantor’s work developing set theory was key in understanding levels of infinity. Different infinite sets can have different cardinalities, or sizes. The cardinality of the natural numbers is smaller than the cardinality of the real numbers, even though both are infinite.
Infinity in real analysis
In real analysis, ideas related to infinity are used but infinity itself is still not a real number:
- The extended real number line adds two elements, positive infinity (+∞) and negative infinity (-∞), to the real numbers. However, they only represent directions and do not obey normal arithmetic rules.
- The affinely extended real number system adds two new elements, +∞ and −∞. Again, they represent unbounded directions, not actual numbers that can be manipulated.
- The projectively extended real number line adds a single point at infinity. This compactifies the line into a circle, but infinity is not a numerical element.
These constructions allow notions like positive/negative infinity and points at infinity to be used in concepts like limits and continuity. But the “infinities” themselves are not treated as values that can be calculated with.
Uses of infinity in physics and engineering
In physics and engineering contexts, infinity is used as a conceptual tool to understand extreme limiting behavior:
- Limits and approximations – When a physical quantity becomes very large, its behavior may be modeled as approaching infinity. This allows estimation based on limiting properties.
- Infinite series – In techniques like Fourier analysis, a function is expressed as the sum of infinite series, which may converge to the function.
- Asymptotic analysis – The asymptotic behavior as something approaches infinity is studied. This allows simplification of very complex systems.
Again, infinity itself is not treated as a real, manipulable number here. It represents an unbounded conceptual direction for different mathematical and physical concepts. The simplifications and approximations provided by using notions of infinity, however, are extremely useful.
Philosophical perspectives on infinity
From a philosophical perspective, there are different views on the nature of infinity:
- Potential infinity – Infinity arises as a conceptual direction from continuing mathematical processes forever. Numbers don’t stop at some “infinite number”.
- Actual infinity – Infinity exists as an actual, completed totality, not just as a direction. Some philosophers argue this is incoherent.
- Infinity as indefinite – Infinity represents open-endedness and indefiniteness. It is not a fixed state or number.
- Infinity as contradictory – Traditional logic runs into contradictions when dealing with infinity. This illustrates the limits of human knowledge and language.
These perspectives inform debates about whether it is coherent to treat infinity as if it were a number that could be manipulated or mathematically formalized. The contradictions that arise hint that infinity transcends traditional mathematics and logic.
Conclusion
In conclusion, infinity is an abstract concept rather than a real number that can be used in calculations and arithmetic. While infinity shows up in various mathematical fields, it does not follow the rules and properties that define real numbers.
Infinity represents an unbounded limit and direction in many contexts, allowing approximations, simplifications, and philosophical contemplation. But attempting to treat infinity as an actual number that can be added, multiplied, or compared logically leads to contradictions and incoherence. Infinite processes can be studied, but infinity itself lies beyond the finite, structured system of real numbers.