Infinity Plus Infinity – A Comprehensive Guide

The exploration of ‘infinity plus infinity’ delves beyond traditional arithmetic, ushering readers into a universe where numbers transform into abstract philosophies and where the infinite stretches even further than one might believe. Join us as we unravel this mathematical enigma and probe the intricate tapestry of infinity.

Defining Infinity Plus Infinity

The “infinity plus infinity,” in the context of standard arithmetic involving real numbers, the result is still considered infinity. Symbolically, it can be represented as:

$\infty +\infty =\infty$

However, it’s crucial to note that operations involving infinity are not arithmetic in the traditional sense.

“Infinity” is not a traditional number in the same way that integers or real numbers are, but rather a concept or representation of an unbounded quantity. In the context of set theory and mathematics, there are different “sizes” or “cardinalities” of infinity, but for most elementary and real number applications, infinity is used to represent an immeasurable or limitless quantity.

This expression simply captures the idea that combining two unbounded quantities yields another unbounded quantity. Proper manipulation and understanding of infinity involve more advanced mathematical concepts and are primarily discussed within the realms of set theory, calculus, and other higher branches of mathematics.

Properties of Infinity Plus Infinity

Dealing with infinity requires an understanding that we’re not handling a standard number but a concept that describes an unbounded quantity. However, infinity can be engaged with some “arithmetic-like” operations, especially in the context of calculus and set theory. Let’s delve into the properties of “infinity plus infinity”:

• Commutativity

  • Just as with standard numbers, the order in which you add infinities does not affect the result.

$\infty +\infty =\infty$

• • Similarly, $\infty +a=\infty$ for any real number $a$.

• Non-Finiteness

  • Adding infinity to infinity doesn’t yield a finite number. It’s still infinite.

• Undefined Operations

  • Certain operations with infinity are undefined. For instance:
    • $\infty -\infty$ is undefined because the difference between two infinite quantities isn’t well-defined in standard arithmetic.
    • ∞ / ∞​ is also undefined for similar reasons.

• Cardinality in Set Theory

  • In set theory, there are different sizes or cardinalities of infinity. The size of the set of natural numbers (0, 1, 2, 3, …) is denoted by ℵ0​ (aleph-null) and is considered countably infinite.
  • When you “add” two countably infinite sets (like natural numbers and integers), the resulting set is still countably infinite. So, in a sense, ℵ0​ + ℵ0​ = ℵ0​.
  • However, the set of real numbers between 0 and 1 has a greater cardinality, often considered “uncountably infinite.”

• Extended Real Number System

  • In the extended real number system, which is often used in calculus and analysis, positive infinity ($\infty$) and negative infinity ($-\infty$) are treated as distinct entities. In this system:
    • $\infty +\infty =\infty$
    • $-\infty +(-\infty )=-\infty$
    • But operations like $\infty -\infty$ or $\infty +(-\infty )$ remain undefined.

• No Multiplicative Identity

  • For standard numbers, multiplying by 1 leaves the number unchanged. But there’s no such identity for infinity. In fact:
    • $\infty \times 0$ is undefined.

Exercise

Example 1

Basic Arithmetic With Infinity: ∞

Solution

By definition, when you add infinity to infinity, the result is still infinity ∞.

Example 2

Limit of Sum of Functions

Suppose f(x) = x and g(x) = 2x. Find: lim x→∞ (f(x) + g(x))

Solution

lim x→∞ (f(x) + g(x))

= lim x→∞ (x + 2x)

= lim x→∞ 3x

= ∞

Example 3

Limit of a Rational Function

Find: lim x→∞ $x^2$ +3x / (x + 5)

Solution

Divide each term by x:

lim x→∞ x + 3 ​/ (1 + 5/x)

The term 5/x goes to 0 as x goes to infinity. Therefore:

lim x→∞ (x + 3) / 1 = ∞

Applications 

The concept of infinity, as well as operations involving it like “infinity plus infinity”, appears in various disciplines and fields, both within and outside of mathematics. Here are some of the notable applications:

• Mathematics

  • Calculus: In calculus, the notion of limits often leads to results where functions approach infinity. The sum of two functions, each going to infinity, typically goes to infinity, e.g., when determining the behavior of functions at asymptotes.
  • Set Theory: The study of different “sizes” of infinity, like countable and uncountable infinities, comes into play. For instance, the cardinality of natural numbers added to itself (i.e., the cardinality of integers) remains countably infinite.

• Physics

  • Cosmology: The study of the universe’s origin and fate may sometimes invoke concepts of infinity, especially when discussing the possible “size” of the universe or its future expansion.
  • Quantum Field Theory: Some calculations might yield “infinities” which are then renormalized to give meaningful physical results.

• Computer Science

  • Algorithms and Complexity: Infinite loops or tasks can sometimes be conceptualized as tasks with infinite runtime. Considering the combination of multiple such tasks might invoke “infinity plus infinity”-like ideas.
  • Theory of Computation: Turing machines, which are a foundational concept in computation theory, work with potentially infinite tapes. Studying their behavior and combinations can sometimes touch upon concepts of combined infinities.

• Economics

  • Utility Theory: Some economic models might consider the idea of “infinite utility” when discussing certain types of decision-making paradigms.

• Philosophy

  • Metaphysics: Philosophical discussions about the nature of the universe, time, and existence might involve ideas about infinity and its various operations.

• Theology

  • Concepts of the “infinite” often appear in religious and theological contexts, such as the infinite nature of God or the universe. The combination of infinite concepts, though less directly analogous to “infinity plus infinity”, might still be relevant in certain discussions or interpretations.

• Art and Literature

  • Infinity is a powerful theme that can symbolize endlessness, eternity, or insurmountable. The idea of combining or stacking such immense concepts can be a tool for emphasizing overwhelming scales or emotions.