Hyperreal number

In mathematics, the hyperreal numbers are the elements of one of several possible field extensions ∗ R {\displaystyle \ast \mathbb {R} } of the field of real numbers R {\displaystyle \mathbb {R} } , extensions that include certain classes of infinite and infinitesimal numbers. A hyperreal number x {\displaystyle x} is said to be finite when | x | < n {\displaystyle {|x|}<n} for some integer n {\displaystyle n} . Similarly, x {\displaystyle x} is said to be infinitesimal when | x | < 1 / n {\displaystyle {|x|}<1/n} for all positive integers n {\displaystyle n} . The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R {\displaystyle \mathbb {R} } are also valid in ∗ R {\displaystyle *\mathbb {R} } . For example, the commutative law of addition, x + y = y + x {\displaystyle x+y=y+x} , holds for the hyperreals just as it does for the reals; since R {\displaystyle \mathbb {R} } is a real closed field, so is ∗ R {\displaystyle *\mathbb {R} } . Similarly, since sin ⁡ ( π n ) = 0 {\displaystyle \sin({\pi n})=0} for all integers n {\displaystyle n} , one also has sin ⁡ ( π h ) = 0 {\displaystyle \sin({\pi h})=0} for all hyperintegers h {\displaystyle h} . The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This...