### Something About Infinity in Mathematics

### Infinity symbol

The infinity symbol (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.

^{}

### Calculus

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.^{}

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#### Real analysis

In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that*x*grows without bound, and means the value of

*x*is decreasing without bound. If

*f*(

*t*) ≥ 0 for every

*t*, then

- means that
*f*(*t*) does not bound a finite area from to - means that the area under
*f*(*t*) is infinite. - means that the total area under
*f*(*t*) is finite, and equals

- means that the sum of the infinite series converges to some real value .
- means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

#### Complex analysis

As in real analysis, in complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. means that the magnitude of*x*grows beyond any assigned value. A point labeled 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 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 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.

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