Infinities

There are two infinities in IEEE-754 floating point, a positive and a negative infinity. For any finite positive \(x\), we have

• Addition/Subtraction rules
  • \(\pm x+\infty = \infty\)
  • \(\pm x-\infty = -\infty\)
  • \(\infty + \infty = \infty - (-\infty) = \infty\)
  • \(-\infty - \infty = -\infty + (-\infty) = -\infty\)
• Multiplication rules
  • \(x\infty = (-x)(-\infty) = \infty\)
  • \((-x)\infty = x(-\infty) = -\infty\)
  • \((-\infty)\infty = \infty(-\infty) = -\infty\)
  • \(\infty\infty = (-\infty)(-\infty) = \infty\)
• Division rules
  • \(\pm x/\infty = \pm x/(-\infty) = 0\)
  • \(\pm x/0 = \pm\infty\)
  • \(\pm x/(-0) = \mp\infty\)
  • \(0/\infty = 0\)

All other operations produce a NaN ("Not-a-Number") special value.

We can treat them like numbers, which gives us the extended real number line. The extended reals have the additional rules for operators when at most 1 operand is infinite. The basic operations are

• \(a+\infty = \infty + a = \infty\) for finite \(a\)
• \(a-\infty = -\infty+ a = -\infty\) for finite \(a\)
• \(a\times(\pm\infty) = \pm\infty\) for \(0\lt a\lt+\infty\)
• \(a\times(\pm\infty) = \mp\infty\) for \(-\infty\lt a\lt 0\)
• \(a/(\pm\infty) = 0\) for finite \(a\)
• \((\pm\infty)/a = \pm\infty\) for \(0\lt a\lt\infty\)
• \((\pm\infty)/a = \mp\infty\) for \(-\infty\lt a\lt 0\)
• \(a^{\infty} = \infty\) and \(a^{-\infty}=0\) for \(1\lt a\lt\infty\)
• \(0^{a} = 0\) and \((+\infty)^{a}=\infty\) for \(0\lt a\lt\infty\)
• \(0^{a} = \infty\) and \((+\infty)^{a}=0\) for \(-\infty\lt a\lt 0\)

When both are infinite in disagreeable ways, like \(\infty - \infty\), the operation is undefined. For agreeable situations, we could define the operation, like \(\infty + \infty = \infty\).

Another approach is to use the projective real number line. Intuitively, look at the plane \(\mathbb{R}^{2}\) and take the $x$-axis as the real number system. Now construct a unit circle centered at the point \((0, 1)\).

We can associate each point on the $x$-axis with a point on the circle by constructing a line from our point \((x,0)\) and connecting it to the North pole of the circle \((0, 2)\). This line will intersect the circle exactly once (besides the North pole), and that intersection point is identified with the point on the number line.

We can then "complete" the number line with an additional infinity, namely, the North pole of this circle.

But this identifies positive infinity with negative infinity: they're both mapped to the North pole.

We could use hyperreals to extend the real number system with infinitesimals and, for closure under division, infinities. This attempts to emulate the use of infinitesimals we find in Leibniz and Newton.

Dual numbers are slightly simpler, and correspond to \(\mathbb{R}[\varepsilon]/(\varepsilon^{2})\). Or if you prefer, adding a number \(\varepsilon\) treated as a variable, but whose square is zero. We can extend smooth functions on the real number system to the dual reals by \(f\colon\mathbb{R}\to\mathbb{R}\) having \(f(a+\varepsilon b) = f(a) + b\varepsilon f'(a)\) where \(f'(x)\) is the derivative of \(f\).

The mathematicians who study hyperreals use insanely sophisticated methods (I'm thinking of smooth infinitesimal analysis, which requires category theory for simple calculus). I'm not sure the juice is worth the squeeze.