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Early users of the differential calculus made use of the concept of an infinitesimal; a quantity so small that although it is not zero its square and higher powers are zero.  There are no real numbers with this property and its use was discouraged as nonrigorous.  It was however a very convenient construct and it did not seem to result in incorrect results.

In 1960 Abraham Robinson discovered a way to provide a rigorous basis for infinitesimals.  The webpage explains this foundation for infinitesimals.  It requires a new concept of number different from the real numbers.

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