Banica T. Introduction to Arithmetic 2025

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Banica T. Introduction to Arithmetic 2025

Textbook in PDF format This is an introduction to numbers, fractions, percentages and arithmetic. We first discuss what can be done with integers and their quotients, namely basic arithmetic, a look into prime numbers, all sorts of counting results, and with a look into percentages and basic probability too. We then upgrade our knowledge by introducing the real numbers, and exploring what can be done with them, in relation with number theory questions. Then we further upgrade our methods, by introducing and using the complex numbers. Finally, we provide an introduction to modern number theory. The book is organized in four parts, as follows: Part I deals with numbers, counting, fractions and percentages, all you need to know, with the basics explained, and all sorts of useful tricks and formulae, in the spirit of the above Stone Age Theorem. We will often rely here on intuition, and previous real-life experience with the numbers, and be sometimes a bit philosophical. Part II deals with the real numbers, and what can be done with them, in relation with arithmetic. In particular, we will get to know more about the prime numbers, via the Euler formula, and other analysis tricks. We will also discuss some more advanced algebraic aspects, such as the fi nite groups, fi eld theory, and quadratic residues. Part III deals with the complex numbers, which are something more complicated, and far-reaching, and what can be done with them, again in relation with arithmetic. As main topics here, we will talk about the Cardano formula in degree 3 and 4, higher degree curves, Gauss sums with their sign computed, and the transcendence of e and π. Part IV goes back to questions from basic arithmetic, notably regarding the prime numbers, with more on the subject, by benefi ting from the knowledge of real and complex numbers. We will talk here about Mertens theorems, Chebycheff estimates, the Riemann zeta function, the Prime Number theorem, and the Riemann hypothesis. In the hope that you will fi nd this book useful, and get to love numbers and their theory. And for more, we will provide some references at the end