Mastering the construction and manipulation of the Natural integers
The first step towards Numerical Analysis
What the course will do for you
By the end of that course, you will be a master in:
- Construct the natural integers totally ordered set (N,≤)
- Prove theorems by recusion, a powerful method of proof
- Become a master in calculating expressions of any size Solve additive and subtractive equations in N
- Understand the exponential notation as related to the powers of 10
- Discover the polynomials as related to the exponentiation of the natural integers
- Construct the oredered mo,oid (N,+,≤), and win mastery in the concurrent use of addition and subtraction, and the comparison operators <, >, ≤, and ≥. *Build the ordered semi-ring (N,+,x,≤), and win mastery in the concurrent use of multiplication and exponentiation, and the comparison operators
That course includes:
- 54 high quality video lectures,
- 10 downloadable recapitulative documents, one per section, with all the proofs.
- 10 intermidiate tests, one per section, with a process to have them corrected
- One final assessment, to submit to correction and grading
And when the final assessment is successfully passed, the students will get a module cerificate.
Construct and Compare the Natural Integers set
That section is made of the content of the free math course Constructing and comparing natural integers., plus a bonus
We start the construction of the natural integers set by the very beginning: counting.
We first count on our fingers up to 10, then further, counting marbles, and then even further, with the help of Python coding session.
Then we construct formally the natural integers set, with a little bit theory: the Peano Axioms.Linked Text
The first activity that is to be done with numbers is then to compare them, answering for instance the question: "Who has the most marbles?".
So, let's compare natural integers to one another, introducing for the the comparison operators 'Less Than' <, 'Greater Than' >, 'Less or Equal To' ≤, and 'Greater or Equal To' ≥.
The intrinsic and mutual properties or these operators are studied, in order to efficiently compare two natural integers, and order a list of them.
We continue with some theory: the structured of totally ordered set of the natural integers set, together with the comparison operator 'les or equal to' ≤.
And we finish with a stepwis recapitulative document, which fixes and amplified the knowledge of the previous videos, with rigorously proven theorems.
BONUS (not included in the free course)
As an application of the Peano axioms, we introduce here a powerful way to prove theorems, the proof by recursion.
That method of proof will be useful many times in the rest of the course, as well as in the other courses of the series.
And you will see it many times in mathematics practice.
Add natural integers
The next thing we do with natural integers, is to add them to one another.
We start from a practical point of view, with an animated graphic along the natural integers line, as well as a real life example, with the French presidential elections.
After that, we have a look to the addition commutativity and associativity properties, in a Python coding session.
These properties are then formalized in our first algebraic structure, the commutative monoid (N,+).
And we finish with a demonstration of how to master addition, with the ways of parenthesing long additions, and the Sigma operator for very long additions.
Subtract Natural Integers
Subtract Natural Integers
After the addition comes the subtraction!
Explored with an animated graphics on the natural integers line, it is rapidly seen as the reciprocal operation of the addition.
That characteristic is on the basis of… our first equations!
But the reciprocity is not the only mutual property of addition and subtraction. The subtraction is also distributive on addition, a very important thing to know when we want to handle formulae with mixted additions and subtraction.
And we finish that section with a real life application of both addition and subtraction: the balance computation of an accountancy table.
Multiply natural integers
And now, the multiplication!
We start with a practical illustrative example from domestic life, then formalized with the definition of the multiplication as a long addition.
After that, we have a look to the multiplication commutativity and associativity properties, in a Python coding session.
These properties are then formalized in a newalgebraic structure, the commutative monoid (N,x).
Then we demonstratie how to master multiplication, with the ways of parenthesing long multiplications, and the Big PI operator for very long multiplications.
Mastering computation in the natural integers
That section is dedicated to the mastering of combined addition, subtraction, and multiplication.
We first construct the semi-ring (N,+,x), with a specific attention to the distributivity laws of the multiplication over the addition and subtraction.
Then, the parenthesis handling is developped for these operations, with the priority law of the multiplication versus the addition and subtraction.
And we finish that section with a special tip, the linear functions definition and drawing.
Powers of 10 and decimal numeration
The decimal numeration may be formalized with the introction of the so-called "Powers of 10".
With that formalization, we may justify the manual additions, subtractions, and multiplication, with the use of "carries".
But the powers of 10 are also related to orders of magnitude, and to the scientific, or exponential, notation.
Note that this lecture is an intrusion into the decimal numbers, with the decimal point. That will be further explained in the forthcoming course about the rational numbers.
Exponentiation in N
The powers of 10 are a particular case of exponentiation, the exponentiations of the number 10.
But the exponentiation of any natural integer is an economical way to denote a repeated multiplication.
And it leads to the polynoms definition and drawing, as well as to the proof of the formula giving the sum of the n first positive integers.
And, together with the ditributivity law, it gives birth to the so-called "remarkable identities", that enpowers someone's computing productivity.
The ordered monoid (N,+,≤)
Now we mix together the natural integers comparison, with their addition and subtraction.
Namely, we set that the ordering of natural integers is stable by addition and subtraction.
And that will lead us to our first inequations, of the type x+a≤b.
The ordered semi-ring (N,+,x,≤)
The last topic of that Natural Integers Masterclass is the way ordering of natural integers react to multiplication and exponentiation.
That will allow us to learn how to compare two natural integers, both in decimal notation, and in exponential notation.
And we go back to polynomials, to order their values at a given point in N.
Final Project: The Combinatory and the Binomial Coefficients
And we finish that course with a challenging final project, learning how combinatory issues gives us the coefficient of the development of the binomial (a+b) to the power n.
These coefficients are called the binomial coefficients, and are related to the number of subsets of cardinal p in a set with n elements
And they are recursively computable with the help of the so-called "Triangle of Pascal".