Well-defined & Wonderful

This episode is concerned with the exponential growth and the exponential function. In our course on mathematical analysis, we introduce the exponential function via an absolutely convergent series. This helps us, using the material from earlier episodes (take also a look into the notes on that), to see that the exponential series/function transforms addition in multiplication; a fact we have seen...

In this episode, again with the help of Fabian's, we discuss changing the order of summation in an infinite, convergent series. As a possibly surprising effect we have that for some series changing this order of summation leads to a different limit or even to a divergent series. We discuss this effect in detail looking at the harmonic series with alternating signs -- a series we know converges by...

This episode is the accompanying episode to the previous one. Here we dig deep into the different type of infinity of the real numbers. In fact as we set out to prove in this episode there are significantly more real numbers than naturals. More precisely, there is no way to label all the real numbers with distinct natural numbers and to reach to every real number. An absolutely amazing fact showin...

In this episode we learn about the concept of quantifying infinite sets. The first infinite set that comes to mind is the set of natural numbers introduced before. In consequence, we single out this size of infinity as the one we are most comfortable with. This yields the definition of countability. We also explore a little the intricacies of infinity by looking at a fully booked hotel still capab...

In this episode we start with a little correction needed for the episode on the supremum of sets. Then we embark on the cruise to a special type of sequences: series. We define this notion provide the most prominent examples (geometric and harmonic) and some convergence tests. Among these the most important in turn are the comparison test, the ratio test and the root test. As an application of con...

In this episode we apply our knowledge of real numbers to obtain that for any non-empty bounded above set of real numbers there exists a least upper bound in the real numbers. This helps to construct the square-root or, in fact, any integer root, of non-negative real numbers. Furthermore, we introduce the Bolzano--Weierstraß Theorem, that is, we prove that any bounded sequence contains a convergen...

In this special episode we have a guest mathematician. Gabriel Barrenechea joins us talking about sequences, convergence and completeness. We identify the last remaining axiom we need to uniquely identify the real numbers. Asking for completeness, that is, any Cauchy sequence is supposed to converge, we provide a means to fill the holes of the rationals. With this property we can show existence of...

This is the second part dedicated to § 4. In this second part, we draw the attention to some consequences of the definition of order. Namely, the existence of a modulus. We provide elementary properties of the modulus and, most importantly, mention both the triangle and inverse triangle inequality, the name of which will be more properly justified later on. The other half of this episode in concer...

The field axioms from the last episode are not enough in order to have a good enough basis to do analysis. A striking fact underlining this lack of definedness can be seen in that the field consisting of 0 and 1 only is fine with the field axioms and surely does not contain the natural numbers. So what's the fix then? We introduce the existence of positive elements. These positive elements are clo...

In this episode we introduce and describe the basic rules of computing numbers. We formulate the axioms of addition and mutiplication and draw some interesting consequences from these rules. Interestingly, all the axioms are satisfied by a set with 2 elements only. In our quest to get a hand on the nature of the real numbers we're thus quite far away.

In this episode we will focus on the core properties of natural numbers and the principle of mathematical induction. We detail the four axioms of Peano's thus characterising natural numbers and the foundation of addition and mutiplication.

We provide a very brief overview of some basic notions in set theory and how to properly define mappings. At the concluding part of the episode we state and prove the well-ordering theorem for the set of natural numbers.