The theory of hypernumbers and extrafunctions emanated from physically directed thinking and was derived by a natural extension of the classical approach to the real number universe construction. Namely, an important class of problems that appear in contemporary physics and involve infinite values inspired this theory. As it is known, many mathematical models, which are used in modern theories of elementary particles (such as gauge theories), imply divergence of analytically calculated properties of physical systems. The simplest example is the case of a free electron when its interaction with photons changes the energy of the electron so that the energy becomes infinite (in a model). Mathematical investigation of many physical problems gives rise to divergent integrals and series that are such mathematical constructions that have, in some sense, infinite values. However, physical measurements give, as the result, only finite values. That is why, many methods of divergence elimination (regularization), i.e., of elimination of infinity, have been elaborated. Nevertheless the majority of them were not well grounded mathematically because they utilized operations with formal expressions that had neither mathematical nor physical meaning. Moreover, there are such models in physics that contain infinities that cannot be eliminated by these methods based on existing mathematical theories. Only in the theory of hyperintegration, based on the theory of hypernumbers, all divergent integrals and series that appear in the calculations with physical quantities become correctly grounded as strict mathematical objects.

In addition to this, theory of hyperintegration suggests a new approach to functional (path) integrals. An important peculiarity of this approach is that functional integrals are treated as ordinary integrals in which hypermeasures are used instead of ordinary measures. Moreover, it is possible to apply this approach to develop an integral calculus for arbitrary functional spaces.

The new theory provides also new facilities for mathematics. For example, there was time when mathematicians (such as L.Euler) manipulated with divergent series (that have infinite values) in the same way as they treated convergent series (that have finite values). But it was demonstrated (in the context of real and complex numbers) that such manipulations were not mathematically correct and led to contradictions. Transition to hypernumbers provides correct mathematical means to deal with such constructions in a proper way.

Other mathematical problems that are facilitated by application of the theory of hypernumbers and extrafunctions are connected with definitions of norms and distances for unbounded functions and operators.

In addition, the theory of hypernumbers provides possibilities for a solution of another important scientific problem. Mathematical modeling of many phenomena in physics (as well as in chemistry and biology) often requires a physicist to operate with such functions that are not differentiable or even not continuous (or are discontinuous) at any point. For example, it may be important to find a derivative of such a function. This causes certain difficulties because methods of the classical mathematical analysis do not provide appropriate means for dealing with such situations. As a consequence, new branches of mathematics appeared: the distribution theory and nonstandard. Thus, to make possible to utilize classical methods of analysis, in the distribution theory there are means of applying the operation of differentiation to arbitrary continuous functions. The nonstandard analysis does not have similar means but provides possibilities to deal with infinite numbers. In spite of all these new abilities, these theories unfortunately do not give complete solutions to many problems that arise in physics. While in the distribution theory any continuous function has the derivative, this generalized derivative may be undefined at many points. Moreover, the problems of the divergence that appear in theoretical physics remain unsolved in the distribution theory. In order to do the same thing in the theory of extrafunctions, i.e., the functions defined for hypernumbers and having them as possible values, the differential calculus for extrafunctions has been constructed. It makes it possible to differentiate any continuous function and to determine values of the generalized derivatives (which are called extraderivatives) at any point. If we take an ordinary function f which has the classical derivative f ', then the, so-called, extraderivative also coincides with f '. In this partial case it coincides with the generalized derivative either in the sense of the distribution theory (as a functional) or in the sense of the non-smooth analysis (as a function). It demonstrates that the calculus of extrafunctions is a natural extension of the classical calculus (as well as the theory of distributions). If the classical derivative f ' of an ordinary function f at some point x does not exist, then the complete generalized derivative at the same point is not a single number but some set of such numbers. It means that a generalized derivative is a relation on the set of real or complex numbers (or, more generally, hypernumbers).

The problem of differentiation is even harder in the nonstandard analysis than in the theory of distributions. Really, in the nonstandard analysis derivatives of ordinary functions coincide with the derivatives in the classical sense. Consequently, there exist continuous functions that are not differentiable in the nonstandard analysis, and the nonstandard analysis does not extend the class of functions that may be investigated by means of differential calculus. This peculiarity demonstrates the essential differences between the nonstandard analysis and the theory of extrafunctions because methods of the latter theory extend the universe of differentiable functions to a great extent. Consequently, these methods are much more powerful in solution of various physical problems.