This integral will have you on the floor 🤣🤣

Рет қаралды 58,054

Жыл бұрын

Integral of 1 over integer part of 1/x from 0 to 1. I calculate a nice integral with a floor function by using Riemann sums and using telescoping series, it’s a must see for calculus lovers. The answer involves the Euler Mascheroni constant
Integral fractional part of 1/x
• Integral of the fracti...
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Пікірлер: 166

@thegrandmuftiofwakanda Жыл бұрын

I didn't recognise this "floor" notation and thought you were asking me to find the continuous integral of x dx between 0 and 1. Interesting problem with an interesting result, though.

@gabrielhuan7482 Жыл бұрын

Any math problem will have me on the floor in a fetal position

@averageboulderer 4 ай бұрын

Glad I got it right. Euler's solution to the Basel problem is a fun topic by itself. I love it when continuous and discrete domains are mixed. Reminds me of signal processing math.

@Yolwoocle Жыл бұрын

I love your enthusiasm!

@Classy Жыл бұрын

I accidently discovered what a floor function was trying to find the wave equation of a triangular traveling wave. It scared me and so does this integral.

@roberttelarket4934 Жыл бұрын

No it had me on the "ceiling"!

@aneeshsrinivas9088 Жыл бұрын

WTF happened to the floor.

@tyruskarmesin5418 Жыл бұрын

Floor is the piecewise function that let him break it into a bunch of simple integrals.

@aneeshsrinivas9088 Жыл бұрын

I was making a pokemon talk reference.

@yogi30051972 Жыл бұрын

@@tyruskarmesin5418 can you give more info on this please? I'm new to these functions. I looked it up on wiki but it described it in a very strange way and I don't understand

@adrienanderson7439 Жыл бұрын

@@yogi30051972 So for each interval of x that he integrates 1/x over, (1/2 to 1 and 1/3 to 1/2 ect), the floor of 1/x for x being in that interval is just a constant. For example 1/x where x is a value somewhere between 1/3 and 1/2 means 1/x is between 2 and 3 so the floor of 1/x is just 2 (decimal part removed). Which allows him to integrate the floor of 1/x as just 2 over that interval. If you add all those pieces together you get the whole integral from 0 to 1.

@Djdjennerd Жыл бұрын

memory unlocked

@user-yd1yi3gm5e Жыл бұрын

Great video as always! I am pretty sure that the last integral, of the frational part of {1/x} from 0 to 1, is actually positive and equal to 1-gamma. I transfered the integral to a pretty weird sum, and as wolfram alpha said, the sum equals 1-gamma.

@BadlyOrganisedGenius Жыл бұрын

Great video! Vear clear explanation, and a cool final result. If you want a challenge, you could try: Integral from 0 to infinity of {e^x} - 1/2 i.e fractional part of e^x, subtract 1/2

@jacobtopp6356 Жыл бұрын

Great video!

@michaelbaum6796 Жыл бұрын

Very cool solution - thanks 👍

@nedmerrill5705 Жыл бұрын

That right-hand expression collapses like a cheap suit.

@user-iz6gi1rf4t Жыл бұрын

The last integral is positive: 1-γ.

@seanfraser3125 Жыл бұрын

I derived the series at 1:47 using integral substitution. If we let u=1/x, then dx = 1/u du (note I didn’t mix up x and u here, the two formulations are equivalent). From there it’s a relatively straightforward forward sum of integrals.

@davidbelk46 Жыл бұрын

I literally just saw this problem on Brilliant yesterday and thought... "Huh!? Where do I even begin??" Thanks.

@ema8889 Жыл бұрын

very good video, it could help me to solve an integral with matrices or with linear algebra, greetings from Mexico

@MultiNeurons Жыл бұрын

Always looks like there is something mysterious when we manage integer numbers and suddenly appears Pi

@denismilic1878 Жыл бұрын

Fun math, fun video and fun guy.

@johnloony68 Жыл бұрын

The calculation you did in the video uses the floor function, which is different from the image shown in the thumbnail

@drpeyam Жыл бұрын

The thumbnail shows the floor function

@a.b.c.d.e... Жыл бұрын

@@drpeyam the Thumbnail shows square brackets

@maitri1656 Жыл бұрын

@@a.b.c.d.e... Greatest integer function

@johnloony68 Жыл бұрын

No it doesn’t. The thumbnail shows square brackets [ ]

@drpeyam Жыл бұрын

It’s another notation for the floor function…

@Khalid_7477 3 ай бұрын

Thank you so much ❤

@davidhoekstra4620 Жыл бұрын

I must be missing something. Isn't [1/x]expn(-1) just x? So that would be Integral(x) from 0 to 1. And wouldn't that just be 1/2?

@drpeyam Жыл бұрын

No it’s the floor function

@abhinavk0929 Жыл бұрын

I felt smart when I figured this out without watching the video, then I felt even smarter after reading the comments lol!

@jameeztherandomguy5418 Жыл бұрын

bruh look at the thumbnail its not the floor f unction there, so people solved it on the thumbnail

@mr.nobody.01 Жыл бұрын

Very cool!

@armanavagyan1876 Жыл бұрын

Hi PROF nice to see U back☺

@sadiqtheog2265 4 ай бұрын

Perfect❤❤ i love your show

@theproofessayist8441 Жыл бұрын

The integral of the integer part does not necessarily need to be an integer because the integer part is just in the integrand correct and it becomes an infinite series as we saw what you did. - Good to see Euler Mascheroni constant again!

@decare696 Жыл бұрын

how is that second sum telescoping? Did you mistake it for the sum of (1/n - 1/(n+1)) or did I miss something?

@drpeyam Жыл бұрын

It is telescoping, you can use partial fractions

@decare696 Жыл бұрын

Aaah I see, the numerator is 1 + n - n.

@ekadria-bo4962 Жыл бұрын

My face in the floor when doctor peyam make a video ❤😊

@drpeyam Жыл бұрын

Awwwwww thank youuuu!!!

@etiennebasset7493 Жыл бұрын

Beautiful result out of ugly integral 😂

@General12th Жыл бұрын

Doubly cool!

@newwaveinfantry8362 Жыл бұрын

After reading the comment section I am convinced that a lot of people, even in the math community, don't know what the floor function is.

@drpeyam Жыл бұрын

I know right?? I’m so shook 😂

@jameeztherandomguy5418 Жыл бұрын

@@drpeyam thumbnail doesnt CONTAIN it

@thegrandmuftiofwakanda Жыл бұрын

I've used it a lot over 20+ years in MATLAB / C / C++ where we just see it as floor() but have never seen this "partial square bracket" notation.

@thegrandmuftiofwakanda Жыл бұрын

@@jameeztherandomguy5418 Yes that's why I thought the problem was to integrate x dx between 0 an 1, which of course is trivial. Then on the white board it looks different, if you spot it.

@softwarelivre2389 Жыл бұрын

People do know what the floor function is. The problem is the thumbnail did not show the floor function.

@kendeorosz4811 Жыл бұрын

Hello! What is the convolution of two cauchy distribution?

@drpeyam Жыл бұрын

No idea

@kendeorosz4811 Жыл бұрын

@@drpeyam I tought you could help me.

@mwesigwadaudi5718 8 ай бұрын

@@drpeyam😂😂😂your lying

@ElifnurKale-fj2ex Жыл бұрын

Amazing 🥰🥰🥰🥰

@MultiNeurons Жыл бұрын

What about studying the triples as [2,3,5] with one even and two odds numbers such that 2*3+5=11 (prime); 2*5+3=13 (prime); 3*5+2=17 (prime) ? Never heard about this topic. Seems quite interesting excluding symmetries. Another example could be [3,7,10] ... and so on and so forth...

@drpeyam Жыл бұрын

I’m not a number theorist

@conanedojawa4538 Жыл бұрын

on 0:29 you start with the lower bound of the first integration with 1/2 not 0 why?!

@shivanshnigam4015 4 ай бұрын

Because for x being 0 to 1/2 (floor(1/x))^-1 will be zero

@averageboulderer 4 ай бұрын

Because if x belongs to (1/2, 1], floor(1/x) = 1. The next interval that results in the same operator being 2 is (1/3, 1/2] and so on until (countable or discrete) infinity. Hope that gives you clarity.

@naveen.v4734 3 ай бұрын

actually he goes till that , its just that he has started with 1/2 to 1 then 1/3 to 1/2 soo eventually it will reach 0 and be included (which we will take care of in the summation)

@mokouf3 Жыл бұрын

When you talk about π²/6, you showed a picture of π/6! Maybe...that was actually π²/6? Please tell me!

@drpeyam Жыл бұрын

😂😂

@markrosenthal9108 Жыл бұрын

Pi squared? No way! Pi are round. Cornbread are square.

@buddydog1956 Жыл бұрын

Pie a la mode ~

@lorax121323 Жыл бұрын

Potato casserole squared

@darshbir Жыл бұрын

Never seen someone so excited to see pi^2/6 as an answer. P.S. where could i find the solution to getting pi^2/6

@raonimesquitadossantos7175 Жыл бұрын

The videoxof 3blue1brown about it is nice

@BridgeBum Жыл бұрын

Google "Basel Problem"

@devd_rx Жыл бұрын

fourier series

@erinsgeography3619 Жыл бұрын

@@BridgeBum holy hölle!

@amineb9933 Жыл бұрын

Fourier series

@AnhNguyen-ul4hl Жыл бұрын

(1/x)-¹ = x

@curtiswfranks Жыл бұрын

But not if there is a floor in the middle.

@mihaleben6051 Жыл бұрын

What.

@felixbrandt6419 Жыл бұрын

I don't understand this at all. I thought the answer would just be 1/2??

@sanjayg1728 Жыл бұрын

The answer 1/2 is true if the expression in the brackets was just 1/x. But it isn't. Look carefully, 1/x isn't surrounded by square brackets (although it looks like one). It is actually a floor function applied on 1/x. Just google the symbol for floor function and you'll understand.

@felixbrandt6419 Жыл бұрын

@@sanjayg1728 Thanks, man.

@NKY5223 Жыл бұрын

@@sanjayg1728it's square brackets in the thumbnail lmao

@sanjayg1728 Жыл бұрын

@@NKY5223 Yeah, that's right.

@raonimesquitadossantos7175 Жыл бұрын

@@sanjayg1728No, it is brackets because it's the round not the floor function.

@armanavagyan1876 Жыл бұрын

PROF why do U write the N capital ? Thanks)

@drpeyam Жыл бұрын

My n are illegible

@mwesigwadaudi5718 8 ай бұрын

@@drpeyam😂😂😂😂eh

@soumyaghosh8823 Жыл бұрын

great

@theelk801 Жыл бұрын

your thumbnail shows nearest integer, not floor

@drpeyam Жыл бұрын

No that’s the notation for floor

@ErhardNeher Жыл бұрын

@@drpeyam no its not, you put in brackets

@rupeshmishra-vnc Жыл бұрын

I'm from India ❤ I like your video

@NurHadi-qf9kl Жыл бұрын

1/2

@drpeyam Жыл бұрын

Nope

@dienosorpo Жыл бұрын

Isn't it 0.5 bro?

@danieldanieldadada Жыл бұрын

I see

@saarthakjoe2154 Жыл бұрын

i initially got confused with the floor functions, but then i realised it is actually the least integer function (thats what we call it in india) and your ceiling function would be Greatest Integer Function

@vask5500 4 ай бұрын

It's the other way around. Floor is the greatest integer function

@mwesigwadaudi5718 8 ай бұрын

Can't that be a beta function😅😂

@mokshamahey Жыл бұрын

Isn’t the answer 1/2

@minecraftxd4996 Жыл бұрын

No because this is the floor(1/x)^(-1) not (1/x)^(-1)

@mokshamahey Жыл бұрын

@@minecraftxd4996 you just said the same function

@minecraftxd4996 Жыл бұрын

@@mokshamahey how is floor(1/x)^(-1) equivalent to (1/x)^(-1)?

@newwaveinfantry8362 Жыл бұрын

@@mokshamahey Keyword: floor.

@shahidpervaiz5216 Жыл бұрын

If we substitute p = 1/x ==> dx = - p^-2 dp Also If x = 0, then p range = infinity and x = 1 then p range = 1 Now we integrate (- p^-3) dp within the p range infinity and 1 Is it workable ? and once evaluated I get the result 1/2 + C !

@drpeyam Жыл бұрын

No that doesn’t work

@manny4148 Жыл бұрын

@@drpeyamtrue may that be, simply stating the falsehood of a solution without providing correction or guidance is both disheartening to someone who was excited about a problem you posed, as well as being unhelpful in any potential aspiration for improvement.

@kappascopezz5122 Жыл бұрын

Okay but why do a substitution like that? If you interpret the task like you just did, you can just use the power rule (x^-1)^-1 = x^(-1 * -1) = x, and then the integral becomes int_0^1 x dx which is 1/2 x² from the bounds 0 to 1, so 1/2. But all of this isn't reading the problem correctly: There is a floor function around the 1/x, which doesn't let you do any operations like this. (floor 1/x is not the same as 1/floor x)

@drpeyam Жыл бұрын

@manny4148 Ok

@martinrosol7719 Жыл бұрын

@@drpeyam why does your head look like a strawberry?

@mfelix3511 Жыл бұрын

cool

@AlphaNumeric123 Жыл бұрын

Fun video!

@engineer4906 Жыл бұрын

i! ???

@hsinghal1 Жыл бұрын

Its just x^2/2 and integral of that is 1/2

@drpeyam Жыл бұрын

Nope

@niklasnowak9010 Жыл бұрын

Why isnt (x^-1)^-1 just x?

@andreyfom-zv3gp Жыл бұрын

Because this square brackets means floor function. The Floor function returns the smallest integer not greater than the argument. This aspect makes the question harder and, of course, more interesting.

@pandurangaraonimmagadda9966 Жыл бұрын

Answer:1/2

@drpeyam Жыл бұрын

@thegrandmuftiofwakanda Жыл бұрын

@@drpeyam Well per the problem presented on the thumbnail, yes. On the whiteboard however, you're right, no.

@wakeupthewublins69 7 ай бұрын

@@thegrandmuftiofwakandathe thumbnail is dependent on the country

@thegrandmuftiofwakanda 7 ай бұрын

@@wakeupthewublins69 I can only comment on what I can see, and what I can see is ambiguity.

@jhg8464 Жыл бұрын

aint no way bro used an integral to find a triangle

@drpeyam Жыл бұрын

It’s not x

@jameeztherandomguy5418 Жыл бұрын

@@drpeyam bro got the supreme clickbait. Are you trying to get pepole to say its 1/2 because you didnt show the floor function on your thumbnail???

@drpeyam Жыл бұрын

I showed the floor function in the thumbnail

@jameeztherandomguy5418 Жыл бұрын

@@drpeyam brackets can mean anything. That is not commonly accepted notation -- especially since you showed it correctly in the video sems like clickbait

@jhg8464 Жыл бұрын

@@drpeyam brackets are not floor, floor notation misses the top part of bracktes