Introduction

In q-calculus we are looking for q-analogues of mathematical objects, which have the original object as limits when q tends to 1. In the q-analysis, we make formal proofs as follows. The validity of the obtained formula depends on the different parts of the proof. A typical proof in q-analysis looks as follows: Theorem ∗ $$ A=B \quad \mbox{\textit{or}}\quad A\cong{B}. $$ Here, ≅ means formal equality. We give a criterion for the validity of formula (∗). We introduce the fundamental concept of infinity (see Section 3.7 ) and make a comparison with nonstandard analysis. Throughout the whole book, we make a comparison with the units of physics to entice this important group of scientists. We make a complete list of analogies between the q-difference and q-sum operators and the differentiation or integration operator. We present the first q-functions in order to facilitate the description of the various schools.