# Carlo Mariconda

I am professor of mathematical analysis at the University of Padova. Among my interests: magic, tango, photography, innovative teaching, technology and e-learning.

## Activity

• Thank you Matthew, we believe that such a rigorous approach helps a lot in avoiding mistakes, quite frequent in combinatorics. let us know if you "internalized" the principles after a second reading!

• Thank you Maria! it is a pleasure to learn that you are now able to solve problems that were a challenge before studying the Mooc

• Thank you very much Renate, we are happy that you enjoyed the Mooc!

• You are right Slawomir!. The mathematical structure of "sequence" has the advantage to regroup words, telephone numbers, ordered lists of names or numbers, etc... into a single notion.

• Hi maria, these topics require for sure hard work, especially at the beginning. We are trying to force you to work on these objects, afterwards, you'll take advantage of this effort! Ciao

• Hi Maria, everything is right, BRAVA (=good)!

• Hi Ian, this is combinatorics from a mathematical point of view. Of course there are lots of applications (statistics, crypthography,..). Here we just focus on the basic principles and structures. Once you learn how to count, you will be able to do whatever you wish involving combinatorics.

• TY! We hope you'll enjoy the Mooc.

• Carlo Mariconda made a comment

Hello. I am a mathematician, and advisor for elearning, of the University of Padova, Italy. I already have some Moocs on FL (Precalculus, advanced precalculus) and I am preparing a new one (combinatorics). I will suggest this course to my colleagues, that are now all going online.

• There are several instruments for doing that. For instance if you are using a LMS (say Moodle) there are activities like quiz/homework that allow to test your students, or ask them to submit a homework. I usually use quiz even in my "normal" lessons: every week end I ask students to answer to 3-4 questions about the subjects of the week and count it for the...

• Thank you!

• Hi Habeeb, we really hope that you will enjoy the course!

• Hope you will be able to hear me! I am going to ask in order to solve this technical problem. TY

• Thank you See Min Lim. We are going to schedule the advanced part as soon as possible. We are preparing a new Mooc on a different topic (hopefully beginning in June), it will be a surprise: hope you are going to follow us.

• Looks correct. Bravo!

• Hi Michael, you can find something in wikipedia: https://en.wikipedia.org/wiki/Greek_alphabet

• Don't worry Aniko, Precalculus will come back again ("Precalculus, the revenge")

• Welcome Ian, hope it will keep you busy enough.

• Hello Ron, we hope that you will enjoy the course. We are curious to know what are the main differences in the methodology with respect to what you learned.

• Carlo Mariconda made a comment

Hi Murray, we hope that you will enjoy our course anyway.

• I agree with you Maria!

• Any answer to the final question?

• Sure! TY Ged. Maria: hope it is clear now.

• Thank you Comsa!

• Thank you Maria!

• Hi Caitlin, this is a very good approach to a Math course!

• Thank you Gill!

• Of course, the language is one of the main difficulties, we try to do our best. Keep going and do not hesitate asking us.

• Hi Maria, thank you! Hope you'll follow everything easily.

• Hi Stephen, welcome to the course! The union of two sets is the set whose elements belong to at least one of the two sets. The intersection of two sets is the set of elements that belong to both of them. Does this clarify your doubt?

• Carlo Mariconda made a comment

Hi Caitlin and welcome to the course. As Ken says, P(100) is the proposition:
1+...+100 equals 100 X 101/2

• Hi Gordon and welcome to the precalculus course. We added the meaning of these symbols to the glossary. Thank you and do not hesitate asking us, we understand your difficulties.

• Thank you Deborah!

• Thank you Mike, hope you will be able to finish week 2 in time!

• Sorry Mike we have been less present due to hundred of students under exams in our calculus courses, it's really a bad period. Also Francesco and Valentina are not present for different reasons (research, ...) You are not alone in any case! Ciao, C

• Hi Anne-Marie. Depending on your background you may face some difficulties if you didn't follow our first precalculus course. Good luck and welcome.

• Dear Mike, welcome back, we are happy to see you again!

• Dear Valeriya, we hope that this course will be useful for your purposes!

• True, both are valid. We tend to write ln x instead of the formally more correct ln (x) , it is just a matter of notation and common use.

• Good luck Mehari!

• Welcome Ivana!

• Hi Mike. The intensity of the jet noise turns out to be 10^{4.8} =63 096 times than that of a motorbike. It is consistent with your calculation, since 2^{16}= 65 536...

• TY again Ged!

• True Ged! But I still much prefer the venitian notation [0, +∞[...

• Now you should see it correctly. The problem is that the automatic conversion from presentation to pdf does not show the hidden parts that appear in the middle of the presentation. TY Ged

• @BrianDowning Dear Brian, n bar stands for the first integer for which you are checking the validity of the proposition. It is not an arbitrary integer, in any case. For instance if you want to prove a proposition P(n) from n=3 you then n bar =3.

• @GedLangosz Dear Ged thank you now you should be able to see the missing parts!

• Thank you Ged, now you should see the missing part!

• Ged's definition is right. It means that, given a real number, you can approximate it as well as you want with a rational number.

• Absolutely true Ged. The codomain is the ambient set where the function takes its values (say real numbers, complex numbers, rationals, positive real numbers...). The range is exactly the set of the values of the function. For instance if f: N ->N defined by f(n)=n^2, the codomain is N but the range is the set of squares of natural numbers: {1,4,9,...}

• Nice! TY

• TY Ged!

• The fact here is that Francis is using the terms of the decimal expansion, showing their meaning. Best in which sense? I ensure you that 1570796327/500000000, obtained by means of the 9 first decimals of Pi=3.141592654... is even a better one!

• Dear Ged, you may be true. Nevrtheless we warmly suggest to follow the course mostly with your left side of the brain...

• Dear rasmita, it's normal! Enjoy the course!

• Let us denote here sqrt(a) the square root of a, a^b the b-th power of a. We know that sqrt 2 is irrational . Consider (\sqrt 2)^{sqr 2}. If it is rational we are done. Otherwise, set r=(\sqrt 2)^{sqrt 2}, that we assume to be irrational, and s=sqrt 2. Then r^s=sqrt 2^2=2 is rational. [Notice: we do not establish whether (sqrt 2)^{sqrt 2} is irrational....

• Dear Ged, both ( or ] are admissible following the [International standard ISO-31](https://en.wikipedia.org/wiki/ISO_31-11). The notation with ***,b[ or ]a,*** was introduced by Bourbaki and is widely used in Europe: just travel around and give a look!

• Does it look better to you now? TY, carlo

• Dear Benard, thank you!

• TY Graham!

• Dear Benard yiu are right. TY!

• True! TY

• Dear Ajay,
the first component of the polar coordinates of the point (x,y) represents the distance from the point to the origin, and is thus positive.

• Hope you ended to pay your mortgage!

• Dear Ged, there are some exceptions! The exponential e^x may be extended to the complex numbers, with the property that e^(z+w)=e^z e^w for every pair z,w of complex numbers. Calling i the imaginary unit of the complex plane (i has the property that i^2=-1), when z=x+iy with x,y reals one has e^(x+iy)=e^x e^(iy)=e^x(cos y+ i sin y). Thus if you take x=0 and...

• Dear Glenda, thank you for your support. We still don't know if this course will be back again, it depends on Futurelearn.. Ciao.

• True! Ciao

• @GrahamGardiner Dear Graham, it is enough to remember that roots are defined just for positive numbers. If yoy deal with an expression of the form x^b just think at its definition. What is x^5? x multiplied by itself for 5 times, it makes sense for every real x. And x^{-5}? 1/x^5: it makes sense for all x different from 0. And x^{pi}? Well it's e^{\pi ln(x)},...

• @FabienSimonis Now the pdf is available. Thank you!

• Sometime we remember useless things forever, while important ones vanish...

• Hi Nadarajah, did you upgrade your Os? The problem with audio/videos is that they usually need recent codecs/software versions. Try to download the video and use the last version of VLC (free) to watch it

• Dear Francis, x^3 is injective and its image is R: it is thus bijective as a map from R to R. Let us look at its inverse.
Let y be real: we look for the (unique) x such that f(x)=x^3=y: such a x will be f^{-1}(y). In some books x is called the cubic root of y. However, for many good reasons that we do not recall here, we define cubic roots just for positive...

• TY Ajay!

• The video sound seems correct for us: did you update your operating system?this is usually the problem. Also, did you check your internet connection? Concerning the "magic board" we really built it with the help of a...magician. We called it Board On Air, it is based on the Lighboard by M. Peshkin.

• Dear Svargo, do you see at what stage you missed a minus sign?

• Dear Tanya, here it is! TY.

• Relax, it's just the opposite!

• Dear Vitali, your bank does meet these functions! Take care.

• Dear Fabien, it seems to me that you are saying that y3 = t*y1 + (1-t)*y2 for some t, this is equivalent to what Francis writes (just replace t with 1-t). Then you realize that t=1/3, whence 1-t=2/3. The way you realize that t=1/3 is clear in this case. Imagine, however, that we asked to find the point of the form (-1/pi,y) along the segment: does your...

• Thank you Fabien and Rebecca, now everything should be fine (pdf + video).

• Thank you Ajay!

• hi Ajay, true in general. Here however we take E as the codomain of f. Any function, when you restrict its codomain to its image becomes surjective. In that case you just need to prove injectivity.

• Hi Clare, T
there are several "intuitive" trigonometric identities, we decided not to put them in a long list. By intuitive I mean that you can guess them by means of the interpretation of the sine and cosine in the trigonometric circle. The proof can then be obtained eaaily by means of the formulas sin(a+b)=... or xis(a+b)=... for instance cos (180-t) = cos...

• Hello James, how do you get c^2 = a^2 + b^2 + 2abCos(theta)? It may interest many students. TY.

• Dear Clare, see my answer to Rebecca. Ciao, C

• Dear Rebecca, the most difficult cases that you may encounter in the previous pre-calculus involve the absolute value or roots, they should not be present here. You will be able to follow easily here the inequalities that involve exponentials or logarithms. The difficulty in those that involve trigonometric functions depends only on the knowledge level on the...

• @GlendaLeeming Dear Glenda, please give a look to the word "Interval": is it more clear now? TY, ciao.

• yes, this is the way. Just a remark: do not forget on which intervals the various affine functions involved are considered, and which is their image, i.e., the sets of their values. For instance f: [-1,3]->[-2,3]. If x is in [-1,1] then y=f(x) is in [-2,-1]....f^{-1}:[-2,3]->[-1,3]. If y is in [-2,-1] then f^{-1}(y)=... etc.

• Hello Cruz, thank you for following us! Hope you will enjoy this mooc too.

• Dear Paul, You may be too young for this course ;-)

• Dear Vickie, welcome back. Of course, we suggest to follow the first precalculus mooc before going into this one. However you may in any case appreciate some steps concerning exponentials, logarithms and trigonometry.

• hello Svargo, welcome back again!

• TY, we are happy to see you here.

• Hope that these .... will become ***** TY.

• Dear Vickie, we hope that you will enjoy the Mooc. I understand it will not be an easy task to follow all the steps, but it's really good for brain, as Francis says in the trailer.

• Dear Khadijah, I am sure it will work. Have a nice time with us.

• Dear James, thank you. The next course begun just yesterday, hope it will not interfere too much with your holidays! have a nice time in Scotland.

• Dear Rachel, take your time. We hope to see you again!

• Dear Katerina, it is useful whenever you can see the coefficient b of x as 2 times something, e.g., 6= 2 X 3, or 2 Sqrt[3], or 2 Pi,..: the method is useful because in such a way you work with smaller numbers. For instance if you solve x^2+20 Sqrt[5]x+100=0 in the standard way you deal with a discriminant equal to 1600. With the trick n.2 the "reduced...

• Dear Ajay, thank you from all of us!