Serge Lang's text presents the topics that he feels students should understand before commencing their study of college mathematics. As such, working through this text is a good way for you to supplement what you learned in high school with material that will aid you in studying mathematics in college. Therefore, I particularly recommend it for prospective mathematics majors.
The material in the text is well motivated and clearly presented. While Lang explains how to perform routine calculations, he focuses on the underlying structure of the mathematics. The material is developed logically and results are proved throughout the text. However, the presentation of the material is marred by numerous errors, most, but not all, of which are typographical.
The problems range from routine calculations to proofs. Many of the problems are challenging and some require considerable ingenuity to solve. Answers to some of the exercises are presented in the back of the text. I should warn you that if you are used to artificial textbook problems in which the correct solution is a "nice" number, you will find that is not the case here. Also, it is useful to read through the problem sets before you begin solving them so that you can do related problems at the same time.
The first section of the book covers algebra. Properties of the integers, rational numbers, and real numbers are examined and compared. There is also more routine material on linear equations, systems of linear equations, powers and roots, inequalities, and quadratic equations.
A brief discussion of logic precedes a section on geometry. Basic assumptions about distance, angles, and right triangles are used as a starting point rather than Euclid's postulates. This leads to a discussion of isometries, including reflections, translations, and rotations. Area is discussed in terms of dilations. The treatment here is different from that in the high school text Geometry which Lang wrote with Gene Murrow. I found the material on isometries quite interesting. Be aware that the notation and some of the terminology in this section is not standard.
The third section of the book covers coordinate geometry. Distance is interpreted in terms of coordinates. This leads to a discussion of circles. Transformations are reinterpreted using coordinates. Segments, rays, and lines are presented using parametric equations. A chapter on trigonometry covers standard topics, but also includes a section on rotations. The section concludes with a chapter on conic sections. Of particular interest is a proof that all Pythagorean triples can be generated from points on the unit circle with rational coordinates.
The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text concludes by demonstrating how determinants can be used to solve systems of linear equations.
The eminent mathematicians I. M. Gelfand and Kunihiko Kodaira have also contributed to books intended for high school students. Those of you planning to study mathematics in college would benefit from working through their texts as well.