This unique book provides a new and well-motivated introduction to
calculus and analysis, historically significant fundamental areas of
mathematics that are widely used in many disciplines. It begins with
familiar elementary high school geometry and algebra, and develops
important concepts such as tangents and derivatives without using any
advanced tools based on limits and infinite processes that dominate the
traditional introductions to the subject. This simple algebraic method
is a modern version of an idea that goes back to René Descartes and that
has been largely forgotten. Moving beyond algebra, the need for new
analytic concepts based on completeness, continuity, and limits becomes
clearly visible to the reader while investigating exponential functions.
The author carefully develops the necessary foundations while minimizing
the use of technical language. He expertly guides the reader to deep
fundamental analysis results, including completeness, key differential
equations, definite integrals, Taylor series for standard functions, and
the Euler identity. This pioneering book takes the sophisticated reader
from simple familiar algebra to the heart of analysis. Furthermore, it
should be of interest as a source of new ideas and as supplementary
reading for high school teachers, and for students and instructors of
calculus and analysis.
