Something About Infinity in Mathematics

Infinity symbol

The infinity symbol \infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E infinity (HTML: ∞ ∞) and in LaTeX as \infty.
It was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology.


Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties.

Real analysis

In real analysis, the symbol \infty, called "infinity", denotes 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 is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled +\infty and -\infty can be added to the topological space of the real numbers, producing the two-point compactification 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 introduces a line at infinity in plane geometry, and so forth for higher dimensions.

Complex analysis

As in real analysis, in complex analysis the symbol \infty, called "infinity", denotes an unsigned infinite limit. x \rightarrow \infty means that the magnitude |x| of x grows beyond any assigned value. A point labeled \infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = \infty for any nonzero complex number z. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Nonstandard analysis

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Howard Jerome Keisler's book (see below).

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