Here, you find my whole video series about Start Learning Mathematics in the correct order and I also help you with some text around the videos. If you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:
Start Learning Mathematics is a video series I started for everyone who is interested in learning mathematical topics. Here, I want to present the necessary foundations you need to enjoy the whole world of mathematics:
So let’s immediately start with the first lection in logic:
We discuss what a logical statement is and how we can transform it into new logical statements:
Now you know that the mathematical logic we learn here is not so different from the everyday logic you use all the time. Here, it is more about to learn the correct symbols and always to be precise.
Okay, you see that the topics are not so complicated despite having some new words like disjunction and logical equivalence. However, you can easily visualise them with simple everyday examples to get a feeling for them. In the last part about logic, we learn some more notions:
At this point, you know a lot about the language of the mathematical logic such that we are ready to start with set theory:
So, sets don’t seem very complicated but you might not see yet how powerful they really are. All mathematics you ever know or will learn here with me can be formulated with sets. And that is the beauty of it: We only need set theory as the foundations of mathematics to construct the whole building on top! Okay, then let’s talk more about sets:
Now by knowing the basics in set theory, we are able to construct of known mathematics just out of sets. This makes everything stands on solid grounds. We start with the natural numbers.
Now we are ready to start talking about the real numbers, which are so important in all calculus:
The term Cauchy sequence, you now have learnt, is very important in your mathematical carrer. Such a sequence describes the notion that we get closer and closer to a point. Looking at such Cauchy sequences of rational numbers, we find the first problem:
We have seen that the to notions Cauchy sequence and convergent sequence are not equivalent in the rational numbers. This is indeed a heavy disadvantage, because it means that we can get closer and closer to a point of the number line by using rational numbers but this point is not a rational number anymore. The real numbers will fix this problem. However, before we construct them let us use the axiomatic method to calculate with them first.
Now we can talk about the complex numbers. They are needed to solve more equations and one way to motivate this is to discretise the multiplication: