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PODCAST

Well-defined & Wonderful

In Well-defined & Wonderful we embarque on the quest to understand the core principles of mathematical analysis. Being based on a lecture course aimed at first year students in Germany the podcast highlights the most important aspects of the individual chapters of the course. The corresponding lecture notes will be available through marcus-waurick.de

science mathematics

All Episodes

00:13:29

Theorems about Continuous Functions Part 1 - The...

Well-defined & Wonderful ·

2022/11/01

In the first part of a mini-series about properties of continuous functions we discuss the intermediate value theorem. We shall conclude that intervals are preserved under continuous mappings and provide another proof of the discontinuity of functions jumping from 0 to 1. Proving the intermediate value theorem, we have the occasion to revisit an argument we used to prove that the reals are uncountable. Picture: Steven Baltakatei Sandoval, CC BY-SA 4.0, via Wikimedia Commons

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00:19:40

Continuity Part 2 (with Fabian Gabel)

Well-defined & Wonderful ·

2022/07/19

This episode is devoted to the study of actual mathematical examples of continuous mappings. The arguably easiest example will be a constant function. We shall discuss a function having a jump at 0 in order to have a non-example at hand. Finally, we prove that the exponential function introduced earlier defined a continuous function. We will exploit this property later on, when we provide an answer on how to raise 2 to the power of root 2.

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00:19:17

Continuity Part 1 (with Fabian Gabel)

Well-defined & Wonderful ·

2022/07/11

In this episode we discuss one of the most important concepts in mathematical analysis -- the concept of continuity for mappings. With great patience and attention to detail we describe the exact definition for a map f to be continuous at some point a in a metric space into a possibly different metric space. We highlight some examples from everyday life and conclude with the property that continuous maps are precisely those maps mapping convergent sequences with some limit a to the same with limit

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00:15:55

Metric Spaces Part 2 (with Fabian Gabel)

Well-defined & Wonderful ·

2022/06/13

The second episode on metric spaces is focussed on a concept derived from the convergence of sequences of real numbers. Knowing what distances between elements in metric spaces are, we immediately realise that we also know, when two elements of a metric spaces are close. Namely, when the metric evaluated at those elements is small. Thus, we introduce convergence of sequences in metric spaces to some limit element by asking for the metric evaluated at the sequence elements and the limit form a null

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00:18:00

Metric Spaces Part 1 (with Fabian Gabel)

Well-defined & Wonderful ·

2022/06/06

In this episode of well-defined & wonderful, we introduce the concept of a metric space. In order to rationalise the definition of this abstract concept, we go through elementary examples from ``practice’’ to build up our intuition. The core concept we want to mathematically describe and understand is the notion of distance. A metric space is then a mathematical object where distances of the elements in this space can be measured using a metric. In order to be justifiable as a mathematical model

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00:14:30

Exponential Series (with Fabian Gabel)

Well-defined & Wonderful ·

2022/05/30

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 for power laws already. In consequence, we can also show that for integer n, exp(n) is the same as e

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00:15:53

Rearrangement of Series (with Fabian Gabel)

Well-defined & Wonderful ·

2022/05/23

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 the Leibniz criterion. As an upshot of this episode we also learn that absolutely convergent series can

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00:17:42

Uncountability of IR Part 2 -- Uncountability (with...

Well-defined & Wonderful ·

2022/05/13

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 showing that our process of filling all those gaps the rationals had was a nontrivial step.

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00:12:17

Uncountability of IR Part 1 -- Countability (with...

Well-defined & Wonderful ·

2022/05/13

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 capable of hosting additional guests. This famous building is called Hilbert's Hotel and helps us to also

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00:16:23

Series

Well-defined & Wonderful ·

2022/01/05

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 convergent series, we can show that any real number admits a decimal expression. (license of picture under

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00:11:47

Supremum of Sets and The Bolzano--Weierstraß Theorem

Well-defined & Wonderful ·

2022/01/03

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 convergent subsequence.

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00:11:38

Sequences and the Completeness Axiom (with Gabriel...

Well-defined & Wonderful ·

2021/12/06

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 square roots of any positive number, which will be done in the next chapter.

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00:10:43

Order Axioms Part 2

Well-defined & Wonderful ·

2021/11/18

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 concerned with the size or proportion of the natural numbers that have been identified last time in the set of

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00:18:05

Order Axioms Part 1

Well-defined & Wonderful ·

2021/11/11

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 closed under multiplication and addition and obey the rule that any real number is exactly one of the three

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00:08:51

Field Axioms

Well-defined & Wonderful ·

2021/11/02

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.

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00:07:02

The Natural Numbers & Mathematical Induction

Well-defined & Wonderful ·

2021/10/26

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.

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00:18:43

Sets, Relations, and Mappings

Well-defined & Wonderful ·

2021/10/26

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.

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37 results

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