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Every non-infinite hyperreal number $x$ is approximately equal to some real number $r\text{,}$ formally called its standard part, in the sense that $x-r$ is either zero or infinitesimal.  1 A better name for the standard part of a non-infinite hyperreal number would be the real part, but that would conflict with the usage of this term for complex numbers. We will write $x\cong r$ to indicate that $r$ is the standard part of $x\text{.}$  2 The most common notation for standard part is $r=\hbox{st}(x)\text{;}$ other notations are also used.
We will often turn this notation around and write $r\cong x\text{,}$ which should be interpreted as defining $r$ to be the standard part of $x\text{;}$ it should be clear from the context which of these two forms is meant. In this notation, an infinitesimal number $\alpha$ satisfies $\alpha\cong0\text{,}$ and any non-infinite hyperreal number is either real, or can be expressed as $r+\alpha$ for some real number $r$ and infinitesimal number $\alpha\text{.}$
More generally, we often need to compare hyperreal numbers, neither of which is real. Two nonzero hyperreal numbers $x,y$ will be called close, written $x\approx y\text{,}$ if $\frac{y}{x}\cong1\text{.}$ A nonzero hyperreal number $x$ is close to zero, written either as $x\approx0$ or $0\approx x\text{,}$ if $x$ is infinitesimal. Finally, we include the special case $0\approx0\text{,}$ so that every hyperreal number is indeed close to itself.
The sum or difference of infinitesimals is clearly infinitesimal, as is their product. The product of an infinitesimal and a (nonzero) real number is again infinitesimal. However, division of infinitesimals could be infinitesimal, real, or infinite, depending on the relative “order” of the infinitesimals. For example, if $\alpha$ is infinitesimal, $\frac{\alpha^2}{\alpha}=\alpha$ is infinitesimal, $\frac{2\alpha}{\alpha}=2$ is real, and $\frac{\alpha}{\alpha^2}=\frac{1}{\alpha}$ is infinite. Similarly, the product of an infinitesimal and an infinite number could again be infinitesimal, real, or infinite. Analogous statements hold for operations on two infinite numbers.