# MatheduReengineering Mathematics

• Section 1: Introduction

### Lecture 1: Introduction

In that introduction, we expose how counting further and further, added to a bit axiomatic construction, leads to define the natural integers line, as an increasing sequence going from 0 to infinity (+$$\infty$$)
• Section 2: Count and construct the Natural Integers set

We will start with the beginning of mathematics: counting, first on the fingers, then with marbles, in order to construct the root of 10 basis numeration we use.

We end the course with Anaconda and Spyder lauched for a Python session, that allows us to go further in counting.

### Lecture 3: The Natural Integers Line

Here, we propose two approaches of the construction of the natural integers set.

The first one is visual, with a line carrying the natural integers. That line is travelled from one natural integer to its follower, starting from 0 and with no end.

The second approach is formal, with the enonciation of the so called Peano Axioms, that rigourously constructs the natural integers set.

• Section 3: Order the Natural Integers Set

### Lecture 4: Compare Natural Integers

Now that we have constructed the natural integers line from left to right, we use the relative position of the numbers on that line to compare them.

We study all the comparison operators <, >, ≤, and ≥, atogether with their mutual properties.

An we explain how to compare two natural integers displayed in the decimal notation.

### Lecture 5: The totally ordered set $$(\mathbb{N},≤)$$

We study here more advanced topics, the property of ≤ as an order relationship on the natural integers set.

In fact, it is a 'total' order, as any two natural integers may be compared at least one way.

This allows to be able to consider N as an increasing sequence, with minimum 0 and no maximum, reinforcing the fact that the natural integers line "go to infinity".
• Section 3: Conclusion

### Lecture 6: Recapitulative Document

You find that the course is rich in information, and complicated to fix in your memory?