Practical Mathematics 1: The Numbers
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This math course gives you (or reminds you) what is on the basis of mathematics: the numbers. And it is as useful as numbers are: how could you do without counting ability?
The way it leads you into the great wold of numbers is very practical and progressive.
You are guided into the different kinds of numbers, from natural integers to fractions, by the means of actions: count, subtract, divide.
By the end of that course, you will be an efficient calculator, with the theoretical background that helps you to integrate the knowledges.
This will allow you to perform your accountancy, calculate the gain of a financial placement, and generally to do the many things calculation skills allow you to do in everyday life.
You will also be introduced to the rest of our series about practical mathematics.
The course is made of 16 lectures with 3 hours video and many pdf additional files, that get deeply inside the matters and give solved exercises.
The lectures are accompanied by 6 quizzes that, together with the exercises in the pdf files, help you to assess the knowledges you got.
The course is structured in 6 sections that follow the progression into the land of numbers.
The section 1 is an introductory lecture that gives you the taste of getting into the course.
The section 2 shows the natural integers, the numbers we use to count. The counting action sustains all the section.
The section 3 shows how the subtraction leads to the negative numbers that, together with the natural integers, form the relative integer set Z.
The section 4 explores the multiplication of relative integers, in particular the signs rule.
The section 5 shows how the division leads to fractions that, together with the relative integers, form the rational numbers set Q. That section explores in detail the addition, subtraction, multiplication and division in Q.
And the section 6 is a conclusive lecture, that recapitulates the course and introduces to the rest of the series of courses about Practical Mathematics.
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|Section 1: Introduction|
This introductory lecture welcomes you to the course "Practical Mathematics I: The Numbers". We are together for rather much time, to learn the essential of numbers, the math foundation, from a practical point of view. Thus, you will overcome the difficulties you may have with mathematics.This course is the first one of a series about Practical Mathematics: more and more will come!
|Section 2: The Natural Integers|
We begin this mathematics course with the first mathematical activity: counting. In mathematicians' wording, it mean exploring the positive integers.
It is done progressively. We start from counting on the fingers to count the population of the World.Scilab is used as a tool to count further than by heart.
We count further and further in order to get all the natural integers, including 0 and all the great numbers, even greater than we can imagine. Then we define the addition by the means of counting process.
The video shows applications with Scilab session, and the attached pdf file goes deeper inside the theory.
|Quiz 1||4 questions|
Consolidation of the knowledge of natural integers and the limits of Scilab.
|Section 3: Addition and Subtraction between Relative Integers|
The subtraction of natural integers is illustrated and explored, which leads to negative numbers, in which we count in the reverse order. Both natural integers and negative numbers form the two-sides of the infinite relative integers line.
Relative Integers Exploration
The relative integers set 'Z' forms with the addition what is called a commutative group.
This fact is explored in Scilab before it is formally stated. An example of accounntacy is given as an application.
A pdf file is given as an additional material, where the rules about parentheses are detailed, and in which you get prepared for your first equations in the next lecture.
The group of relative integers
We conclude this section which is about the addition and subtraction of integers with two additional tips.
We first explore our first equations, with examples managing numbers of different signs. These equations are further explained and proved in the "FirstEquations" pdf file.
Then we recall the manual addition and subtraction, showing it in an example. The remainders mechanisms are generally explained and proved in the "Remainder" pdf file.
My first equations
|Section 4: Multiplication between Relative Integers|
Now, we deal with multiplication
We first multiply positive integers, considering the multiplication as a (long) addition.Then we learn progressively to multiply relative integers with different signs, beginning with showing it in Scilab.
Multiply Relative Integers
The relative integers set Z has, together with the addition and the multiplication, a so called structure of "ring". This lecture explains the characteristic properties of a ring, always beginning with examples.
There are 2 additional pdf files. "ZplusMultRing" explains more completely the structure of ring and its consequences, in particular the role of parentheses in a formula.
The second pdf file is a serie of solved exercises using the properties of the multiplication together with the addition.
This lecture shows additional tips about the multiplication of integers.
We first see the manual multiplication as an application of the distributivity law, with examples in the video and explanations and proof in the pdf file.
We finish with our first functions, the linear functions, which draw straight lines.
The powers of 10 are on the core of decimal notation of numbers, as well as on the definition of the orders of magnitude.
This lecture details these matters, with a video and a pdf additional file.
|Section 5: Division and Rational Numbers: the fractions and the decimal numbers|
This Lecture introduces the fractions, that are with the integers the rational numbers. The rational numbers are the exact results of the division of relative integers. The fractions are the way to manage sharing a quantity into portions.
The added pdf file examines the notions of multiple representations of a rational number as fractions and optionally as a relative integer. It introduces the notion of "canonical" representation.
|Quiz 6||3 questions|
Assessment of the new knoledge about the division of relative integers
As that lecture contains many formulae, not only numerical examples are given in the video, but an additional pdf file called ‘FractionsAddSubtractExercises’ is given with all the formulae set and exercises, with the solutions at the end.
Another pdf file, called ‘RationalNumbersAddSubtract’ shows that the different formulae given are equivalent for the different representations of the same rational numbers.
|As that lecture contains also many formulae, numerical examples are given in the video, and an additional pdf file called ‘FractionsMultDivExercises’ is given with all the formulae set and exercises, with the solutions at the end.|
The rational numbers set Q has, together with the addition and the multiplication, a so called structure of "field". This lecture explains the characteristic properties of a field, always beginning with examples on fractions.
There is one additional pdf files, "QplusMultField", that explains more completely the structure of field and its consequences, in particular the definition and properties of the division between 2 rational numbers.
We end this course with three additional matters related to rational numbers: proportionality calculations (the so-called ‘rule of 3’ for French scholars), the decimal approximations, and the percentages calculation.
All these matters are driven by examples, with a Scilab session for the decimal approximations.
|Quiz 7||3 questions|
Some questions to assess the last lecture about proportions, decimal approximations and percentages.
|Section 6: Conclusion|
This is the end of the first course of the series about practical mathematics.
We constructed different types of numbers, in a progression from the simplest (‘count’) to the most complex (‘divide’).
We now have the rational numbers, that are nearly “all” the numbers.
What is about to come for the next course of the series? The last numbers we can set on a straight line, the ‘real’ numbers. We shall introduce them very progressively, always starting from a practical point of view!
Fabienne Chaplais is 57, and lives in Paris, France.
She is married and has three children.
She obtained the French highest degree to teach mathematics at undergraduate level. This means that she is very accurate in mathematics and in teaching them to anyone from the beginners to the undergraduate level.
She then turned to an Engineer’s career for about 30 years.
She became an expert in R&D, especially when using applied mathematics and scientific programming in high level languages such as Matlab and Scilab.
She notably worked on satellites guidance, shuttle accosting and reentry .She applied her exoertise to implementing various complex algorithms such as Kalman filters and fuzzy logic.
After that, she worked in the railway industry, on automatic urban transportation systems for Paris and New York. After various R&D projects including error correcting Viterbi encoding and decoding as well as formal method based B language, she became an expert in safety analysis involving many specialized sharp inductive and deductive approach, including probability calculations.
She then founded Mathedu with her husband, a Researcher in Control Science.
Mathedu aims to teach mathematics from a practical point of view.
The idea is to let the students be in action with a very pragmatic approach, using its computer with Scilab installed as a laboratory.
Then and only then, the link with theory is done, in a very progressive way.
Learning maths with us will let you find the subject easy, so that you will no more understand why mathematics were so hard to understand before…