None of the numbers are "real world" - they are human-made representations of either something in the real world, or something else in some culture. Bridges between cultures and real world are built both ways. Imaginary numbers used to be purely cultural for a while, until bridges Cherlin mentions allowed people to see them as models of something in physical reality. Same goes for quaternions and many exotic geometries. Relationships with reality is one of the main distinctions between most mathematical and most scientific frameworks and methodologies, which is a huge topic. Cheers, Maria Droujkova <a href="http://www.naturalmath.com">http://www.naturalmath.com</a> Make math your own, to make your own math. <div class="gmail_quote">On Sun, Aug 16, 2009 at 7:29 AM, Edward Cherlin <<a href="mailto:echerlin@gmail.com">echerlin@gmail.com</a>> wrote: <blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> On Sun, Aug 16, 2009 at 12:25 AM, <<a href="mailto:forster@ozonline.com.au">forster@ozonline.com.au</a>> wrote: > Alan > > We study complex numbers and transfinite numbers even though they aren't real world. Root(-1) isn't real world but its a useful abstraction to study. This turns out not to be the case. Complex numbers are required in classical electricity and in all parts of Quantum Mechanics, where the imaginary part of a wave is necessary to represent its phase, where probabilities of states are represented by the product of the wave function with its complex conjugate, and where state-change operators (such as _i_ times partial derivative with respect to spatial coordinates) routinely involve complex values. Infinities and infinitesimals appear as real-world values in the theory of games of perfect information. See Conway, On Numbers and Games, and Berlekamp, Conway, and Guy, Winning Ways for Your Mathematical Plays. <a href="http://lists.sugarlabs.org/listinfo/iaep" target="_blank"></a></blockquote></div>