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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