The development of automatic digital computers has made it possible to carry out computations involving a very large number of arithmetic operations and this has stimulated a study of the cumulative effect of rounding errors. In this book I have given an elementary introduction to this subject which is based on the work which has been done in the Mathematics Division of N.P.L. in the past few years. Some of the material presented here has already appeared in published papers, but much of it has hitherto been available only in the form of rough notes for lectures given in this country and the United States.