What Is Infinity?

Infinity (symbol: ) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, “infinity” is often treated as if it were a number (i.e., it counts or measures things: “an infinite number of terms”) but it is not the same sort of number as natural or real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number.
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In real analysis, the symbol \infty, called “infinity”, is used to denote an unbounded limit. x \rightarrow \infty means that x grows without bound, and x \to -\infty means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then

  • \int_{a}^{b} \, f(t)\ dt \ = \infty means that f(t) does not bound a finite area from a to b
  • \int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty means that the area under f(t) is infinite.
  • \int_{-\infty}^{\infty} \, f(t)\ dt \ = a means that the total area under f(t) is finite, and equals a

Infinity is also used to describe infinite series:

  • \sum_{i=0}^{\infty} \, f(i) = a means that the sum of the infinite series converges to some real value a.
  • \sum_{i=0}^{\infty} \, f(i) = \infty means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

Infinity can be used not only to define a limit but as a value in the extended real number system. Points labeled +\infty and -\infty can be added to the topological space of the real numbers, producing the two-pointcompactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat +\infty and -\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and so forth for higher dimensions.


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