A Readable Introduction to Real Mathematics
Authors: Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Publication: 2nd edition, 2019 (copyright 2018)
Publisher: Springer Cham
Series:Undergraduate Texts in Mathematics
Length: XVIII + 218 pages
Main areas: Number Theory, Geometry, Foundations / Mathematical Reasoning
Summary
A Readable Introduction to Real Mathematics is designed as a bridge from school mathematics to the style of mathematics encountered at university. Rather than concentrating primarily on computational techniques, the Rosenthals aim to teach readers how mathematicians think, formulate arguments, construct proofs, and recognize mathematical structure. Remarkably, the formal prerequisite is essentially only high-school algebra, making the book accessible to beginning university students and mathematically strong secondary-school students.
The first part develops elementary number theory and proof techniques through the natural numbers, mathematical induction, modular arithmetic, the Fundamental Theorem of Arithmetic, Fermat's Little Theorem, Wilson's theorem, and the Euclidean algorithm. These ideas are given a modern application through a short treatment of RSA public-key encryption, illustrating how apparently elementary arithmetic can underpin sophisticated technology. The discussion then expands to rational and irrational numbers, complex numbers, and the mathematics of infinity, including the comparison of the cardinalities of infinite sets.
The later chapters move toward geometry and more advanced mathematical ideas. The authors develop Euclidean geometry and the theory of straightedge-and-compass constructions, culminating in results concerning impossible constructions such as the trisection of a $60^\circ$ angle. The book concludes with introductions to infinite series and higher-dimensional spaces, giving readers a glimpse of analysis and higher-dimensional geometry. Throughout, numerous exercises range from routine reinforcement to challenging problems intended to develop genuine proof-solving ability.
Key takeaways
Springer book page
Authors: Daniel Rosenthal, David Rosenthal, Peter Rosenthal
Publication: 2nd edition, 2019 (copyright 2018)
Publisher: Springer Cham
Series:Undergraduate Texts in Mathematics
Length: XVIII + 218 pages
Main areas: Number Theory, Geometry, Foundations / Mathematical Reasoning
Summary
A Readable Introduction to Real Mathematics is designed as a bridge from school mathematics to the style of mathematics encountered at university. Rather than concentrating primarily on computational techniques, the Rosenthals aim to teach readers how mathematicians think, formulate arguments, construct proofs, and recognize mathematical structure. Remarkably, the formal prerequisite is essentially only high-school algebra, making the book accessible to beginning university students and mathematically strong secondary-school students.
The first part develops elementary number theory and proof techniques through the natural numbers, mathematical induction, modular arithmetic, the Fundamental Theorem of Arithmetic, Fermat's Little Theorem, Wilson's theorem, and the Euclidean algorithm. These ideas are given a modern application through a short treatment of RSA public-key encryption, illustrating how apparently elementary arithmetic can underpin sophisticated technology. The discussion then expands to rational and irrational numbers, complex numbers, and the mathematics of infinity, including the comparison of the cardinalities of infinite sets.
The later chapters move toward geometry and more advanced mathematical ideas. The authors develop Euclidean geometry and the theory of straightedge-and-compass constructions, culminating in results concerning impossible constructions such as the trisection of a $60^\circ$ angle. The book concludes with introductions to infinite series and higher-dimensional spaces, giving readers a glimpse of analysis and higher-dimensional geometry. Throughout, numerous exercises range from routine reinforcement to challenging problems intended to develop genuine proof-solving ability.
Key takeaways
- Main purpose: teaching mathematical reasoning and proof rather than merely calculation.
- Level: excellent transition from secondary-school mathematics to undergraduate pure mathematics.
- Major topics: induction, elementary number theory, RSA cryptography, complex numbers, infinity/cardinality, Euclidean geometry, constructibility, infinite series, and higher-dimensional spaces.
- Prerequisite: essentially high-school algebra, although mathematically mature readers will benefit most.
- Strength: it introduces genuinely important mathematics using relatively elementary tools and a large collection of exercises.
- Best suited for: first-year mathematics students, prospective mathematics majors, teachers, strong high-school students, or anyone wanting to understand what proof-based “real mathematics” is like.
Springer book page
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