BELIEVE IN ALGEBRA, NOT CALCULATOR

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blackpenredpen | 曹老師

[@aasyjepale5210]

i did some mental math, but hit a wall at trying to find the square root of 63,252,753,001

[@iamgroot3615]

that’s some impressive mental math assuming you’re telling the truth . Is there a trick or something

[@AngryAxew]

@@iamgroot3615 theres no trick hes probably lying

[@AngryAxew]

r/iamverysmart

[@marvinfung2050]

AngryAxew there's no reason not to be able to mental math those numbers Like 500(500+1) which is easier which is 250000+500 and it similar to the end

[@narayanankannan6787]

I mean it's OBVIOUSLY 251501.

[@Armbrust666]

Wow... this essentially proved that if you take the product of four consecutive -numbers- integers and add one to it, than it's gone be a square number.

[@ClimateAdam]

Awesome! Good spot!

[@fanyfan7466]

Gábor Tóth holy shit you’re right! That’s crazy man

[@blackpenredpen]

Yup!!

[@kingbeauregard]

The most pathological case I can think of is -1 thru 2, and yes indeed I get 1, which is a perfect square.

[@pcklop]

My professor had us prove a more general result: take the product of four numbers in arithmetic sequence, then add the fourth power of their common difference. Show that the result is a perfect square.

[@threadeater345]

1990: we'll have flying cars by 2019 2019: 2=1+1, wow I'm a genius

[@blackpenredpen]

LOL

[@ghotifish1838]

2+2 is 4, minus one that's three quick maths

[@Kyanzes]

Flying cars... you can't even have a sharpie that could change color. Say, red and black.

[@santinodemaria2818]

@@ghotifish1838 topical meme reference

[@unutentediyoutube3282]

Well it can also be 2=500-498

[@yuliaokhremenko6613]

I just started learning English, but the explanations are clear and interesting even at my levels of English. Thanks a lot 😁👍

[@blackpenredpen]

Юлия Охременко I am Glad to hear!

[@enhace15anos.83]

x2

[@corona8073]

U r indian Chinese korean or ....???

[@donovanholm]

@@harelavv8806 the name may seem obviously Russian to some but not all

[@siddharthsoni2101]

@@blackpenredpen hii

[@Hypoli]

My last words whispered in a final breath : "Don't forget the +1"

[@not_So_Random-Dude]

😂

[@68plus1.]

LMFAOOO

[@naiknaik8812]

He never resumed the video

[@user-tk4dj4il5s]

He added the one wdym 🧐

[@sethdon1100]

Olympic math taught me that insanely hard problems often had elegant solutions, this is no exception.

[@blackpenredpen]

: ))))

[@hafizh8461]

@@leif1075???

[@hemandy94]

@@leif1075 people like these are called problem solvers...

[@drudi1]

@@leif1075 well it took me about 5min to solve it so I think is not impossible to solve. All of this types of equations where you have 4 consecutive numbers multipled are done like this

[@jayasri6764]

Lol,This problem is actually super easy,(Every single Olympiad contestant would have solved this question,at some point of their life) .Insanely hard problems need not have simple solutions . That's a downside of the math Olympiad .They make you expect difficult problems have simple solutions.(Although,most imo contestants don t fall for this fallacy).Real insanely hard problems have not been solved by anyone,yet.

[@BryanLu0]

Instead distributing at 4:18 u = x^2 + 3x + 1 (u - 1)(u + 1) + 1 = u^2 So the root is x^2 + 3x + 1 = 251501

[@blackpenredpen]

Bryan Lu omg that cat!!!!

[@AmitBentabou]

Or even u=x^2+3x, then u^2+2u+1

[@matias12381]

digno de nyan cat, jajajajaj

[@mattat3847]

My life is a lie. I thought u subbing was only for integrals

[@RunstarHomer]

@@mattat3847 nah man, sub whenever it makes the problem simpler

[@leagueplays2100]

i put it in my calculator and got 251501, that was easy

[@bowtangey6830]

Boo!

[@Netherexio]

@The Balton American calculators are beasts

[@paradox9265]

@@Netherexio Agreed but Americans aren’t

[@Netherexio]

@@paradox9265 What do you mean?

[@user-yp6eb5wj4w]

It was easy, but not so beautiful like this)

[@ganaraminukshuk0]

"If you're using a calculator, why are you watching this video?" Sanity check.

[@blackpenredpen]

hahahahaa

[@andrel8243]

I am a calculator, not a person

[@mysticdragonex815]

*laughs in Shakuntala Devi

[@TheS1lentX]

Jeez thats smart *proceeds to use the calculator to prove that 251501 is the right answer*

[@blackflash9935]

5:48 “Back in my day kids would use *ALGEBRA* but now their brains are rotting from these darn *CALCULATORS* ”

[@mr.n1933]

Dafuq did i jusf watch.i lost it when the 2=1+1

[@executorarktanis2323]

You are not nerdy enough

[@blackpenredpen]

Did you know that 2 = 1 + 1?? I bet not! jk : )

[@williamadams137]

blackpenredpen No i don’t, i need a calculator to this

[@snejpu2508]

That's pretty funnt, but sometimes such things are the most difficult to see, for example: we have f(x)=x^4+8x^3+18x^2+8x+17, and a question, for which x, the function f(x) is a prime. You can check infinitely many cases and never know the answer, but what makes this question easy (but on the other hand is not so obvious), is that 18=17+1. Because then we have (x^2+1)(x^2+8x+17), which has to be a prime. One of them has to be = 1, the other one has to be some prime then... We are left with only 2 cases, because we know, that 18=17+1. : )

[@theolbiterator5408]

No but I knew 2= 0.9+1.1.

[@chaitanyagadekar5025]

I Known 2+2 = 5

[@clubstepdj]

What i know is 5/2 = 2 with int data type

[@maheshagrawal7779]

no 2=1+1/2+1/4+1/8+1/16... you have got many misconceptions blackpenredpen!!!

[@iabervon]

When he writes 1, he's obviously just abbreviating 1/2+1/4+1/8+1/16+...

[@InDstructR]

@@iabervon and when he writes 1/2 he's abbreviating for 1/4+1/8+1/16+...

[@agces2001]

@@InDstructR And when he writes 1/4 he's abbreviating 1/8 + 1/16 + 1/32+...

[@InDstructR]

@ki kus won't stop me, And when he writes 1/8 he's abbreviating 1/16+1/32+1/64+1/128+...

[@shounakghosh8595]

Whoa that converged quickly

[@tanmay8017]

I remember solving this exact question in my JEE ( Mains ) exam.

[@classicmelodyvetrivel710]

@Sanat R mains usually has easy questions

[@Avighna]

@Sanat R - Study Vlogs Sure, yeah, "easy question" 😬

[@Avighna]

@Sanat R - Study Vlogs Woah, really? What kinda questions do they ask? Could you send me a link?

[@dr3w199]

Nice. I did it this way: Assume that the expression is a square number so: x(x+1)(x+2)(x+3)+1 = n^2 x(x+1)(x+2)(x+3) = n^2 - 1 x(x+1)(x+2)(x+3) = (n+1)(n-1) What I did then is realise that the factors of the product on the right differ by 2. Playing around you can find that: x(x+3) = x^2+3x = n-1 (x+1)(x+2)=x^2+3x+2 = n+1 So n = x^2 + 3x + 1 Not as neat as your method though! Thanks for the video

[@juanbomfim22]

OMG ive almost done it completely. i just stopped at (n+1)(n-1) lol WD! i mean 'not that almost' lmao

[@joshuamason2227]

How do I play around with it

[@dr3w199]

@@joshuamason2227 Well you have the product of 3 binomials and a monomial for which we can multiply in any order. If you try a few cases, or think about it you spot that x(x+3) and (x+1)(x+2) have a difference of 2.

[@joshuamason2227]

@@dr3w199 okie

[@sunnykarwani3556]

Damn... It's a great method. Neat work. 💯

[@steve2817]

1 + 1 = 3 And sinx/n=six=6.

[@rio_agustian_]

You stupid, 1 + 1 ≠ 3 3 = 2 + 1 π = 2 + 1 π - 1 = 2

[@CookieJar2025]

@@rio_agustian_ so π = 3 lol nice discovery

[@Kevin-14]

@@CookieJar2025 e = 3 = π

[@SameerKhan-nd5qb]

@@rio_agustian_ noob

[@SameerKhan-nd5qb]

@@Kevin-14 lool nooob

[@rd8396]

Take x ^2 + 3x = a Then in step 2 a(a+2) + 1 a^2 + 2a + 1 = (a+1)^2

[@mat1305h]

Yes much easier, and you see it imediatly too.

[@Polarspy]

was about to say this, i think it's a lot more intuitive

[@milanmitreski7657]

Isn't it beautiful how one problem can be solved in diffrent ways, even if the idea and the method are nearly the same. That's why we love maths.

[@sanjaisrao484]

@@milanmitreski7657 Yes

[@akshetpatial5466]

You extra smart boy the time required here will be same

[@pfever]

Sorry...Time over! give me your exam!

[@TheGaberGuy]

Blew my mind! Earned yourself a new subscriber! Keep up the good work!👍

[@Inujasa88]

0:10 is this a pewdiepie reference? 😂😂

[@albel2094]

He liked it!!!

[@randomdude9135]

@@albel2094 yupp

[@randomdude9135]

Yupp

[@jekoddragon6227]

now do it with CALCULUS

[@davidappell3105]

Why do you think this is funny?

[@gabrielpinhal8325]

@@davidappell3105 because suffering is funny

[@sanchit6107]

@@davidappell3105 Its FUNI

[@agabe_8989]

0:01 that's my life philosophy now

[@DatSwif]

This is beautiful. I've been looking at it for five hours now

[@chaitanyagadekar5025]

0:09 That was PowerFul

[@lucasxue2211]

i remember my math teacher asking me to prove that n(n+1)(n+2)(n+3) + 1 is always a perfect square given that n is an integer

[@ericzgrey]

I love this. You did a great job of laying out a good challenge.

[@tiborgrun6963]

Not only 2 = 1+1, but also 0 = 1-1. From the second row: (x^2+3x+1-1)(x^2+3x+1+1)+1 and per the third binomial equation = (x^2+3x+1)^2 -1^2 +1 = (x^2+3x+1)^2

[@etemkaandelibas3649]

I didn't understand. Where did you use binomial expansion

[@jinja3113]

0 = 1-1 1 = 1*1 2= 1+1

[@yogeshpathak73]

I didn't see any binomial here... But what i see is that you used the form (a+1)(a-1) + 1 = a^2 - 1 +1= a^2

[@Bayerwaldler]

@@yogeshpathak73 I think Tibor Grün is from Germany. In German school curriculum the formula (a+b)*(a-b) = a^2 - b^2 is known as 3. binomial formula. b=1 is a special case.

[@yogeshpathak73]

Oh ok... Didn't know that. Thanks.

[@mhm6421]

Continueing from: sqrt( (x^2 + 3x) (x^2 + 3x + 2) + 1 ) let y = x^2 + 3x sqrt( y * (y + 1) + 1 ) = sqrt( y^2 + y + 1 ) = sqrt( (y+1)^2 ) = y + 1 = x^2 + 3x + 1 = (x + 1) (x + 2) - 1 = 501 * 502 - 1 = 251501 much easier to multiply :p

[@razvy3827]

that is what i wanted ti type nice 👍

[@asa-ze1kn]

I'm only in 8th grade Algebra 1 but I was using variables to find how some of your factorizations works. You went from (x^2+3x)(x^2+3x+2)+1 to (x^2+3x)(x^2+3x+1)+(x^2+3x+1). What I did was set (x^2+3x) to a variable (a). (a)(a+2)+1 a^2+2a+1 (a+1)(a+1) Now substitute back in. (x^2+3x+1)(x^2+3x+1) When in doubt use variables..

[@trueriver1950]

Yes, that's using even more algebra than BPRP did.

[@zocker2586]

Well yes because using the variables is actually the logic behind the solution, it's just that it was invisible throughout the process :D

[@baranibarani4970]

Where r u from?

[@sanjanabiswas9774]

Agreed! Variables always help to proceed the solution.

[@enricomassignani]

I put x=500 but multiplied everything. In the end i got to sqrt((x+y)^2) with x=500 and y=501^2

[@ugursoydan8187]

a very good perspective and a very good solution. thank you!!!

[@SawkTheFighter13]

Why are your videos so entertaining? I'm so glad I came across this channel.

[@isaacdeutsch2538]

I chose to make x = 502, which ends up yielding a nice difference of squares and a two term quadratic, which is much easier to distribute. The quartic you get has a palindromic pattern reminiscent of pure binomial coefficients, making it tempting to say the golden ratio is a root. It is, in fact, a root, so synthetically divide the quartic by the golden ratio identifying polynomial, x² - x - 1. You end up with the golden ratio identifying polynomial again, meaning that the original quartic in that square root is (x² - x - 1)², so cancel the power and the root. Plug 502 back in for x, some quick multiplying and subtracting by hand and you've got 251501.

[@msmmath87]

Really good solution! GOOD Teacher👍

[@mathstraining6996]

Really good video. Thanks for inspiring students. Keep it up

This was fantastic. I wish to thank you for your videos.

[@MrShad]

What a incredible content. Im a student of math (i'll be a teacher in the future) from Brazil. Thank you so much for sharing knowledge!

[@detachedmars158]

You can also put a +1-1 inside the x^2+3x bracket and it'll be in the form of (a+b)(a-b).

[@kilindogma9711]

that's what i thought he was gonna do as well but what he did was cool as well.

[@ssdd9911]

why?

[@iabervon]

Yeah, (x-1)(x+1)+1=x^2-1^2+1 seems easier to find than multiplying out exactly the right portion of the big expression.

[@-Mohammed_S]

Your explanation is awesome . I like your teaching very much. Thanks

[@makotoca4429]

This just blowed my mind!!! Love this

[@workout9594]

3:20 I solved it differently. Let y= x^2+3x. Then substitute y into the expression making y(y+2)+1, distribute so y^2+2y+1 and that is a perfect square of (y+1)^2. Here, the square root and exponent cancel each other leaving y+1, sub back in x and then easily find the answer :)

[@matthewmanzanares6798]

this is also what I did and I think that this is a bit better because you don't have to split 2 into 1 + 1 and do the rest

[@cheesecircle3033]

That's what I did as well

[@eannacoleman957]

I love the explanation, though I did it a bit differently. When I got to the second line, I substituted (x²+3x) as y and found that that worked much simpler than distributing 2 as 1+1.

[@apanapane]

That was beautiful. Thank you.

[@zoroarklover363]

I love you, you make me remember stuff I had forgotten!

[@almightyhydra]

3:20 just put y = x^2 + 3x, then you have y(y+2) + 1 = y^2 + 2y + 1 = (y+1)^2. So the answer is y + 1, or x^2 + 3x + 1.

[@cypherx7247]

I also did it in this way...but that way was also fine...its all about which method comes in your head first

[@lasergamer2869]

Dang that’s genius

[@LudwigvanBeethoven2]

Every body knows 1+1=2 but i know 1+1 =/= 3

[@blackpenredpen]

♫♪Ludwig van Beethoven♪♫ Hahahhaha

[@jgsh8062]

I’ve got you all beat with 1+1 > 0

[@JDguy11222]

@@jgsh8062 nah mine's better 1+1≠1+1

[@bucinoulje7505]

i watched this video this video right before my math competition and the same type of question came up on the task sheet. Thank you very much!

[@bucinoulje7505]

for those wondering the question was 202120212019(202120212021)(202120212023) all over 100010001 x (202120212021 squared +4)

[@thatssokwekwe]

I expressed it as sqrt((501.5-1.5)(501.5-0.5)(501.5+0.5)(501.5+1.5)+1) You get two a^2-b^2 expressions that you can multiply out, add the 1, and then factor into a squared quadratic expression Very neat and, as someone mentioned elsewhere, it generalizes to “1 plus the product of any four consecutive integers is a perfect square”

[@snatchngrab8262]

The world needs more teachers like you. I'm more impressed by your teaching skills than any math. Much respect.

[@paawanjethva]

Everybody knows e^{iτ}=1 . . . . But I know 1=e^{iτ}

[@blackpenredpen]

Nice!!!

[@paawanjethva]

@@user-bd9mu3ee1i That's e^{iπ}. τ=2π

[@fgvcosmic6752]

My mans using tau! Up top!

[@peterg644]

@@user-bd9mu3ee1i he's using tau not pi

[@naysay02]

man this was amazing to watch! so clever!

[@arkabhattacharya1792]

The entry was epic and the whole video is interesting.

[@DavidS-qn3jm]

I did assume there was a nice solution, but expanding under the root to get x^4 + 6x^3 + 11x^2 + 6x + 1 was pretty easy, and then matching coefficients in (x^2 + ax + 1)^2 was straightforward too. But yeah, the main thing is to replace 500 by x. I don't think I could intuitively see which two of the brackets would make it easier, and I'm not sure that's a better method than expanding the whole thing to only 4 terms (plus the one on the outside).

[@sadeekmuhammadryan4894]

There are things to learn from each of your videos 😁❤️

[@deadvirgin428]

Well yes, that's the point.

[@neetabhojwani9924]

Nice video . I learnt a lot

[@star_ms]

Seemingly elementary problems can have wonderfully elegant solutions! All we need is to substitute a number with x, and the magic begins.

[@martinepstein9826]

Nice factoring method but it might have taken me a while to spot. Multiplying out and factoring isn't so bad (x - 1)x(x + 1)(x + 2) + 1 = (x^2 - 1)(x^2 + 2x) + 1 = x^4 + 2x^3 - x^2 - 2x + 1 = (x^2 + bx +- 1)^2 = x^4 + 2bx^3 + (b^2 +- 2)x^2 +- 2bx + 1 We see this works if b = 1 and c = -1 so the answer is 501^2 + 501 - 1 = 500^2 + 2*500 + 1 + 500 = 251501

[@jdsingh3607]

Now this video makes me like algebra

[@FermionClasses]

kzbin.info/www/bejne/jomVdXxvepatjJo

[@kodichandrashekher7213]

It is so interesting and good to remember. Thanks.

[@balumathtech2022]

Nice... Great method of solving

[@aikenkazim5318]

please give an example differentiation of complex functions

[@ozonejgs2887]

I am so impressed with myself, I actually used the same method you did before watching the video =D

[@blackpenredpen]

Jan Wrobel nice!!!!

[@kartikeyasingh8592]

You are an absolute genius bro🤟🤟 I was surprised by your last step result 😱😱😱

[@bernardelmargi9733]

This is very useful, I’m trying to think more outside the the box for hard mathematical equations either for proofs or something like this, I didn’t thing to make 500 = x, but it simplified it so much. This is a useful skill and I will be sure to use it

[@imadkhan1825]

You can also this as x^2+3x=t and expression would become t(t+2)+1 =(t+1)^2 this que came in practice test for jee last week And guess what i solved that 😎😎😎👍👍

[@laudine878]

Doing a PhD in Literary Studies, but stuff like this is why I absolutely love maths ♥

[@UberPretzel101996]

Man, you deserve more subs!

[@smragibrezwan2778]

WOW! Now that's an amazing way to solve it!

[@whyit487]

I love your videos about not using calculators (Like the Wolfram-Alpha video)! They're the best! Keep up the good work! It's nice going back to algebra sometimes...

[@blackpenredpen]

Why It? Yea me too. I try to mix things up a bit.

[@mango417]

"And now, here's the deal"… You know that when he pronounces that phrase things are 'bout to get complicated.

[@varunkashyap6571]

Awesome solution to the question

[@lisamariefan]

Actually trying it out was fun, and there was some surprisingly elegant generalization going on behind the scenes.

[@Mothuzad]

Shout-out to my colorblind fam who can never tell when he switches pens

[@ammardaffa]

I know this kind of the prob, i use (n+1)(n+2)-1

[@yannisdekonoha]

When he drops the "Check this out", you know crazy stuff will happen on the board

[@AdityaTiwari-wf2gi]

Brilliant solution .keep it up

[@moosemoosington1441]

Dude, I always had a good grasp on algebra as a kid and in highschool I always aced most algebra, but somehow my teachers (and I) missed this property of algebraic equations. So freaking cool. It has been nigh on 15 years since high school, but I am still learning new and cool algebra. Thanks so much blackpenredpen!

[@ishmeetsingh1146]

Wow your method and my method are similar .....I had take the whole expression as y and then square it and then assume x to be 500 and multiplied and had taken x^2+3x to be z and at end I got y=z+1 that is y = x^2 + 3x + 1 and it's done

[@ciahartsrodrigues8823]

3:19 The "WOW". LOL

[@ShaunakDesaiPiano]

3:32 perhaps an easier method to spot is if you partially expand the brackets so that you end up with (x²+3x)² + 2(x²+3x) + 1, which is precisely (x²+3x+1)²

[@kinyutaka]

I'm doing this on the toilet, so I only hope I'm starting correctly, with (500)(502)=(501²-1) and (501)(503)=(502²-1) But then again, we could cheat and go with (501)(502)=(501½²-¼) and (500)(503)=(501½²-9/4)?

[@-a5624]

I know this is not related to this video but I wanted to post this on a new video so you might see it :) your trick for integrals of thinking "wouldn't it be nice if..." has helped me so so much, so thank you :) love your videos!

[@blackpenredpen]

Awww thank you!!!!!

[@Rishith198]

wow, just wow!! Fascinating !!

[@PedroMartins-ux7mv]

Very elegant solution

[@noverdy]

Everybody know e^2.pi.i = 1 . . But I know 1 = e^2.pi.i

[@mundane3809]

Wrong it's - (e ^ pi × i)

[@nikolas9105]

@@mundane3809 Nice try but thats -1 ignoring your name

[@mundane3809]

@@nikolas9105 no e ^ ( pi × i ) = -1 So if you make -1 negative, it become positive.

[@RunstarHomer]

@@mundane3809 you are correct but the original comment was also correct. e^2πi = 1.

[@mundane3809]

@@RunstarHomer oof yea it's actually correct. sorry for the mistake!

[@alexsandroagustini714]

Him: Starts the video with 1+1=2 Me: *ok we are getting somewhere now

[@science-y9209]

This was a great explanation

[@G_sen_sei]

Very interesting calculation!

[@hellopeter121]

What Everybody knows : 1+1=2 What BPRP knows : 2=1+1 . . . . . What I know : 1+1=2 and 2=1+1 😇😇😇😇😇😇😇😇😇😇😇

[@dev.imperatus6105]

Hmmm ive got an easier way when you use 1 + 1 instead of 2 Here is how I do according to you: (x^2+3x+2)(x^2+3x+1) =(x^2+3x)^2 +2 then √((x^2+3x)^2 +2) = x^2+3x+1

[@XWurstbrotX]

You can't solve squareroots of sums like that, eventhought your result is correct.

[@carterwoodson8818]

@4:15 wow I did not see it coming! Really good thank you!

[@SpongeDude8]

More impressed with how someone came up with the question

[@user-fr5cm7gs2m]

Молодец! Хорошо объясняет, все понятно. Спасибо

[@akhmadsw54]

Ага, он знаток

[@LiegeNorth]

this guy is a genius!

[@rahuldambo]

Thats reall nicee man!!

[@Geo25rey]

I need to start a tradition of watching this video every year. It's just that good

[@DriffPL]

I think you forgot about the absolute value; Square root of a 2nd power produces absolute value result because both positive and negative values are true.

[@bttfish]

Dunkoro it is obvious that the number inside the square root is positive，so ignore the absolute value symbol

[@enzoTHEferrari]

I understood everything until the bit at 4:35 - 5:04 What do you mean by "factoring out"?

[@AE-rg5rc]

When a number repeats itself in an addition you can factor it out, basically do the inverse of distributive property. So we have x²+3x+1 repeating in both therms. You can factor it out and you will be left with x²+3x+1 ( x²+3x +1), equivalent to x²+3x+1( x²+3x) + x²+3x+1 (1)

[@leif1075]

@@AE-rg5rc but thats just squaring it and you don't have two of the sake expression..you don't jave twobx squared plus 3x plus 2 you only,have one

[@ViratKohli-jj3wj]

@@leif1075 please Learn some math, this is for grade 6 atleast in asian countries.

[@leif1075]

@@ViratKohli-jj3wj I know some math, thanks very much..I had a valid question

[@vudomath]

At 3:45, you could also treat the first factor as (x^2+3x+1-1) so together with the second factor you have (x^2+3x+1-1)(x^2+3x+1+1) = difference of squares ((x^2+3x+1)^2 - 1. Plus the extra 1 on the outside you get the perfect square.

[@mesballo2224]

Your way to solve this is pretty AWESOME!! First I multiplied all together, i got x^4 + 6x^3 + 11x^2 + 6x + 1 then i calculate this polynomial for x=1 x=2 x=3 ....all the time i got a square!! I was really surprise!! I didn't expect x(x+1)(x+2)(x+3)+1 to be a perfect square for all x at all!!! This is incredible!! Thanks for sharing your knowledge you are very inspiring to me