Published: Mathematical Education, Journal of the Foundation for Mathematical Education and Enlightenment. — Moscow, 1997. — No.1-3; 1998. — No1-4, — 234 s. Publication of the outstanding mathematician Academician of the Russian Academy of Sciences . R. Shafarevich is designed for senior schoolchildren and teachers working on an in-depth program of studying mathematics . Chapters I, II, III were published in the 1997 issues of the journal, Chapters IV, V, VI and VII were published in the 1998 issues. Number. Irrational numbers. Irrationality of other square roots. Decomposition into simple multipliers. polynomial. Roots and divisibility of a polynomial. Multiple Roots and Derivatives. Binoma Formula . Members of Bernoulli. Many. Sets and subsets. Combinatorics. Algebra of sets . Probability Language . Chebyshev's Inequality . Simple numbers. The infinity of prime numbers. Proof of the infinity of prime numbers by Euler. Function PI(n). Chebyshev inequalities for PI(n). Number and polynomial. Real numbers and polynomials. Axioms of real numbers. Limits and infinite amounts. Set real numbers by decimals. Actual roots of polynomials. Sturm's Theorem . Endless sets of . Equilibrium . Continuum. Subtle set. Normal numbers. Graded rows. Multi-members as producing functions. Graded rows. Partitio Numerorum. Euler's Pentagonal Theorem . Producing function for Bernoulli numbers.