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Section 3.1 Infinitesimals

You are likely already familiar with several types of numbers, including integers (such as \(\pm2\)), rational numbers (such as \(\frac23\)), irrational numbers (such as \(\sqrt2\)), and transcendental numbers (such as \(\pi\)). Putting these number systems together, we get the real numbers, denoted \(\RR\text{.}\) Although the real numbers are widely used as the basis for calculus, there are other number systems. We will extend the real numbers by adding both infinitesimal and infinite numbers; the resulting number system is called the hyperreal numbers, denoted \(\HR\text{.}\)  1 Historically, the formal definition of the real numbers postdates the invention of calculus by 200 years. The hyperreal numbers are arguably a better language for expressing the original description of calculus, but they were not shown to be consistent until another 100 years after the formalization of the real numbers. An infinitesimal number \(\alpha\) satisfies

\begin{equation} -r \lt \alpha \lt r \qquad (\forall r\in\RR)\tag{3.1.1} \end{equation}

and is therefore smaller than any positive real number, and bigger than any negative real number. We divert from standard usage, however, in that we do not consider zero to be infinitesimal; for us, all infinitesimal numbers are either positive or negative. The hyperreals also contain the reciprocals of infinitesimals, which are infinite numbers.

Most of the properties of real numbers extend to the hyperreal numbers; this is in fact a deep theorem in nonstandard analysis, called the transfer principle. In particular, you can add, subtract, multiply, and divide (nonzero) hyperreal numbers as usual.