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In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible.  It can't be done in general.

In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible.  It can't be done in general.

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Another place where infinity is considered acceptable is in [[Floating-point]] arithmetic in computers.  The IEEE-754 standard (which all modern computers support) allows the values <math>\infty\,</math> and <math>- \infty\,</math>. But their treatment is very different from other numbers. To represent <math>\infty\</math>, the exponent bits are all set to 1, and the mantissa bits are all set to 0, with the sign bit being set to 0 or 1 for <math>+\infty\</math> and <math>-\infty\</math> respectively.

Also, in some non-standard models of Peano Arithmetic, <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.

Also, in some non-standard models of Peano Arithmetic, <math>\infty\,</math> is treated as an actual number. But, once again, this is not standard mathematics.