I always find Knuth's motivation for his concrete mathematics book funny. It was "wanted it without the dryness that most math books have" yet he produced the same dryness he spoke of that is appearent in that book

@EJP286CRSKW

7 күн бұрын

But Knuth has jokes ...

Donald Knuth is the primary co-creator of "surreal numbers" -- with Game Of Life's John Conway collaborating with him as secondary co-creator

@AntoshaPushkin

7 күн бұрын

I thought surreal numbers were John Conway's idea, and then Donald Knuth wrote a book about them

@shruggzdastr8-facedclown

7 күн бұрын

@AntoshaPushkin : I suggest that you watch the Numberphile video on surreal numbers -- in which Knuth is interviewed on the subject of the creation and development of this area of math(s)

Knuth's books are really the best. Super dense, but he tried to include everything known, at the time, of a any given topic.

I actually did it the other way around, I find it more natural without having to think about inserting that integral

Ah to go back to the days when programming was recognized as a mathematical discipline. Most modern CS grads find concepts from Knuth puzzling, and yet, they are *still* foundational

This book is where I learned a lot of math topics like induction, series manipulation, and basic number theory. It has an amazing selection of problems with solutions to every one in the back. “Concrete Mathematics” is another great book by Knuth that covers these topics in more detail.

That was a pretty cool application indeed

As soon as I saw the integral, I knew this was from Knuth's book

@ACium.

9 күн бұрын

why is that?

@basqye9

7 күн бұрын

You “knuthit”?

Donald Knuth ... An amazing scholar. I majored in mathematics with a minor in computer science and was always fascinated with the mathematics of computer programming. To me some of his most interesting recent work deals with the nested summation of powers. He has made many contributions to the online encyclopedia integers sequences as well, a source that seems to be rarely mentioned in your videos. You are an amazing mathematician and teacher thank you

Knuth also wrote a short novel about Conway's Surreal Numbers

I got TAOCP 35 years ago. Still have the book. Still haven't finished. Still love every bit of it. Still want all other volumes.

ACP may be old, but it is still the bible for Computer Science. Nobody has even tried to supersede it. There are lots of other good math problems in it.

The last thing is the dirlecht eta function of 2 which is equivalent to the riemann zeta function 2 times 1/2 (you can verify using the relationship formula) which why you get pi^2/12.

Left hand side of the board is going out of frame fyi

@tomholroyd7519

9 күн бұрын

Sometimes I miss "Future Michael" in the edit saying oops about the left but it doesn't matter because you can infer what is there

nice problem from the inventor of LaTeX!

@Mathemagical55

9 күн бұрын

Just to be precise, Knuth wrote TeX and Leslie Lamport wrote LaTeX as a collection of macros which sits on top of the Tex engine.

9:30

Thanks!

Why is this sort of problem in an art of computer programming book?

@datamatters8

9 күн бұрын

Knuth's book contains lots of material on algorithms, computational complexity & other computer science material. I'm guessing he started the series before the term Computer Science was widely used. He is also a Turing award winner. See the bio on Wikipedia.

@roberttelarket4934

9 күн бұрын

@@datamatters8: Yes I know but it still doesn't answer my question. This problem with Mike has nothing to do with computer algorithms etc.

@datamatters8

9 күн бұрын

The problem is homework problem number 20 page 78 in chapter 1 - "Basic Concepts" of Vol. 1, copyright 1968. This book is 634 pages divided into 2 chapters, "Basic Concepts" and "Information Structures". He illustrates code using assembly language for a mythical computer called MIX. The popular languages of that time period were BASIC and FORTRAN IV which he avoided as new languages were coming on the scene almost yearly, e.g. COBOL, PL\I, ALGOL, SNOBOL, etc. Chapter one has sections for Math Preliminaries --- Number systems, Permutations and factorials, Binomial coefficients, Harmonic numbers, Fibonacci Numbers, Generating functions, O-notiation, etc. This problem is in the section on Harmonic numbers. "There is an analytic way to approach summation problems such as the one leading to Theorem A in this section: ...". Knuth included the section on Harmonic Numbers, stating "...in analysis of algorithms it pops up nearly every time we turn around..." where he defines the notation Hn. So the problem is related to math foundations to prepare the ground for subsequent analysis & comparisons of algorithms. The book is in its 3rd edition. I suggest you pick up a copy if you are interested in programming. There are currently 5 volumes in the series that can be bought in a boxed set. Knuth has also co-authored a book titled "Concrete Mathematics".

@kenwaln4508

9 күн бұрын

I took his class in 1985 or so with this book as the text. Where he went deep in the math it was usually to prove the complexity or correctness of an algorithm. I'd have to go to the text but it really put the science into computing.

@roberttelarket4934

8 күн бұрын

@@datamatters8: No I'm not at all interested In programming!!! A detestable sideline!!! Math is where it's at!!! The COBAL you mentioned was well established along with FORTRAN in the mid-1960's(I had it then as a senior in U.S. high school). They were probably the only languages. BASIC didn't come in I think until the early 1980's.

thanks sir

can you put the sum inside without beppo-levi's theorem?

FYI - Donald Knuth is a famous computer scientist (now 86) at Stanford U. This book is part of a series and still in print with a 3rd ed. Wikipedia has a nice bio.

Thanks for pronouncing Donald Knuth’s name correctly - your usual level of care!

@stefanalecu9532

9 күн бұрын

his level of care also includes constantly making typos and ignoring his community, so I am not sure I'd be that kind :)

@divisix024

8 күн бұрын

@@stefanalecu9532 His discord server has been smothered by spam

Nice, followable.

That's a pretty complicated way of computing pi^2/12 :-)

In what context it was used?

@jjtt

9 күн бұрын

My guess is that it's for solving/analyzing recurrence relations using generating functions. For example if you wanted to compute the asymptotic worst-case runtime of an algorithm, in general it may be described by a non-linear recurrence relation, in which case you may need to use rather heavy mathematical artillery just to get some upper bound, say

@SuperSilver316

9 күн бұрын

No idea what its computer programming context is, but this sort of stuff is seen in all types of Euler Sum problems, more specifically used for generating functions.

Can you use the geometric series' formula here, given that it blows up when t = 1?

@LinaWainwright

9 күн бұрын

Technically we just take the limit as the upper bound tends to 1 from below, so it's fine to use the geometric series formula.

@renedelatorre2138

9 күн бұрын

@@LinaWainwright I wish Prof. Michael discuss the Dominated Convergence Theorem in detail as this has appeared in a lot of his videos.

@LinaWainwright

9 күн бұрын

@@renedelatorre2138 I concur, but this doesn't have anything to do with the DCT.

8:47 Why can you swap the sumation and the integral?

@SuperSilver316

9 күн бұрын

There’s Probably more than one answer to this, but since (0,1) is the interval of integration, we are within the region of convergence for the power of series of ln(1+x). This means we are allowed to change the order of summation and integration. There are probably stronger forms of convergence here that can be declared, but (0,1) tends to be good enough for a lot power series that are like this.

Nice!

Wow

4:15 I know you like graphic design and colored chalk and stuff, but I am colorblind (dichromat), and I'm sure some of your other viewers are as well, so I want to say, there is NO visible difference between peach and green or whatever you were using there. Pastels are eh. Contrasting primary colors are good. Except don't use red. Colorblindness sucks sometimes. Why is RED the emergency color! It's so DIM

@plumbr13

9 күн бұрын

If you omit red, there are only 2 other primary colours.

@datamatters8

9 күн бұрын

Red is the color of blood. If you see lots of blood, not a good sign. :)

@Hiltok

9 күн бұрын

For those who are not colour blind, when in a natural environment like a forest/bushland, then red catches the eye better than any other colour. For that reason, it is often seen in nature as a threat or alarm colour. That is why red is the emergency colour. The fact that a not insignificant proportion of people have red/green colour blindness shows that it is not highly disabling at a biological level, as inconvenient as it may be. For what it is worth, I am not colour blind but still find the contrast between the various chalks to be insipid because the light/dark contrast with the board is overwhelming. The 'green' chalk overline above the a_n(xt)^n sum looks almost indistinguishable to the white chalk to me, while the 'peach' chalk overline above the a_n(x)^n sum is just a subtle tone difference. In other words, you aren't missing much.

Now Euclid's Elements is what I would call a jolly old Maths book! 🙄😁😂

I can escape Euler Sums no matter what

Cool

interverting sum and integral without any explanations is not math : it's physics, its programming, but it's not math anymore

Why is this video nine minutes long? this problem is trivial if you start with the RHS (the "contrived" side), then use the power series definition, then use convolution to turn the product of sums into a double sum, then pull the sums out and do the integral term-wise, then use that the fraction difference reduces to Hn over the m-sum. If this sounds long-winded, just do it yourself and it's not hard at all. A very "bugman" (i.e., mechanistic, no intuition) problem.

first!