A new direction of mathematics is presented. Sub-interval analysis is simple and effective tool for practical and theoretical calculations and estimations. Feasible applications of sub-interval analysis are: Economics, including accounting, audit, microeconomics, macroeconomics, banking, insurance, utility theory, econometrics, etc. Internet, including databases, E-stores, E-shops, forums, blogs, search, etc. Statistics. Fourier analysis. Large databases. Preliminary recognition. Sociology. Elections. Military science. Engineering. Technics. …