@@normanmai52 Well in their era there wasn't that much to do compared to us now so I'm guessing his just discovering new methods the entire day and the day after that.
Wow, pure magic. I didn't even notice the moment when x^2+y^2=1 turned into a wall of formula and I have no idea where it came from. Thanks for explaining it so well.
So clever. Also, remember the guy literally came up with the process of integration all by himself, so it's not only that he was the first to do this, he was one of the first to BE ABLE to do this, and because of his own work. Say what you will about Newton, but this is the work of a genius.
“Nature and nature's laws lay hid in night: God said, 'Let Newton be! ' and all was light.” The great English poet Alexander Pope (1688-1744) wrote this epitaph to be inscribed on the tomb of Sir Isaac Newton (1643-1727), and captured in two lines the immense importance of Newton to the history of modern science.
Whats also interesting is that he further expanded on the method to use a smaller region because it would converge faster (i think 1/3 of the region?). It totally makes sense to not include that in a short but I think this point adds to the ingenuity of the method.
in order to get expand (1-x^2)^1/2 Newton broke the previous rule of only using the binomial expansion formula for powers that were counting numbers (positive integers) Newton was the first to realise that the binomial expansion formula applies to all real numbers, if you're comfortable with infinite series. Newton took it a step further: He integrated the circle fromula between x=0 and x=1/2, which gave him the area of a sector of angle Pi/12 radians + a triangle with base 0.5 and height (root 3)/2 root 3 is equal to the binomial expansion 2(1-1/4)^1/2 Therefore Pi = 12 x(integral ((1-x^2)^1/2) - (root3)/8) This value converges quite quickly
he estado buscando durante bastante tiempo como calcular Pi, era una de mis mas grandes dudas, el saber por que se llega a ese resultado tan preciso en los decimales, vi muchos videos y casi ninguno solucionó mi duda, todos dando explicaciones largas y se desviaban del tema, sin embargo, he aqui en un video de tan solo 53 segundos por fin sé como calcular Pi, ya puedo descansar en paz, muchas gracias amigo.
I really love math and I hope to be a mathematician when I grow up. Your videos always explain things easily which we all really appreciate, thank you!
Just using Pythagoreans theorem, I was able to calculate pi to ten decimal places... I could have gone further but I did it by hand. It took several days and I got bored. It makes me wonder why it was so hard to calculate before Newton when the theorem has been known for centuries. Another thing that blows my mind is that if we know the radius of a circle, we can never measure the circumference and vice versa. We can see the beginning and end but it can't be exactly measured... 🤯
an important thing is that he was the first to apply the binomial expansion to the circle because for that he needed to use a fractionary power, and no one before had tested the binomial expansion for a non-integer n, but newton did and proved he could, that's the beauty of what newton did, expanding a theorem, proving the expansion is possible, start playing with it and using the other knowledge he "created" to revolutionize pi approximations
Thanks mate! Just found your channel and I am loving it.. I am not an academic nor a math ninja but I love seeing these "behind the scenes"! I was actually looking for something you might be able to help me (and many more, I hope) with... I am trying to "understand" the thought process behind reciprocal and reciprocal formula. I have been learning about circuits and was faced with the Parallel Operator/Sum which strikes me as something very "deep" in derivation of formulas and algebra/math itself but 99% of people will only ever copy the formula and solve it without ever understanding why.. How I see it is that by using the reciprocals of parallel domains (in this topic, circuits with resistances) the "formula" allows us to "sample" a new kind of "unit" of that abstract domain where we can then calculate the participation (the % contribution of the total) of each "parallel resistance" in the overall system and the final step (multiply by the volts) will apply each "participation" (the new "unit" / the new "one") relative to the total voltage of the circuit and give us the correct value on that particular point of the system. I have NO IDEA if what I wrote makes any sense, as I said I am far from a math geek just the everyday nerd on youtube. It would be great if anyone could share some insights and/or send me some reading material... It took me forever to learn what are reciprocals, reciprocal rules and see how it's used beyond "copy the formula and replace the variables". Somehow I have a feeling that learning about this thought process might help me a lot with understanding how things actually work. My education never focused on reciprocals, we would learn to multiply fractions by putting one on top of the other and multiply TOP x BOTTOM / BOTTOM x TOP ... again, just another formula we would memorize and repeat (before year 2000)... Cheers from Brazil! Keep up the cool videos!!!
Hello and thank you for liking the channel. I will admit that I am not familiar with the formula on reciprocals as it relates to circuits. I hope you find an intuitive understanding of the formula!
Sounds like a good way of thinking about it. Essentially, you are taking resistances (which don’t add in parallel), converting them to conductances (through taking the reciprocal), adding the conductances (to get the combined parallel conductance), and then converting back to resistance. One motivation for why conductances add is parallel is the following: the conductance of a path times the voltage across that path yields the current that would flow. Two paths in parallel have the same voltage, so the sum of the currents through each path would equal the voltage times the sum of the conductances, so it would be reasonable to say the conductance of a parallel combination is just the sum of the conductances on each branch. Hope this makes sense.
@@matthankins6206 THANK YOU! That was a great explanation and I had not been able to think about it in the "conductance" light. I believe I now understand why reciprocals / inverse numbers are used in the formula. It is because the inverse of the resistance IS the conductance. Beautiful as it all came together! I confess I thought it would open up a new way of looking at this derivation, but it was "so simple" (with your help) that it completely makes sense why it was done this way, incredible thought process. I thought reciprocals would be more of a wildcard in derivations (due to my lack of understanding), but understanding its use in this formula has put my mind at ease. Once again, thank you mate! It's the kind of stuff almost impossible to find the right google search. Cheers!
Well actually, that’s debatable. The title could go to Archimedes who also calculated pi, but did so during the times of early Ancient Rome. Or it could go to Pythagoras, who developed the Pythagorean Theorem, allowing the discover of pi. The title could go to Nikola Tesla, or Benjamin Franklin, or Schrodinger, or Galileo, or Aristotle, or Socrates, or Plato, or Marcus Pollio. There are millions of candidates for this position, I don’t think we can declare just one person for it.
I've always wondered how pi was originally calculated. Plus the visual really helps! It's interesting how the quarter sections of the circle match up to the radians of the angle. Just another way to show that math always comes back to prove itself
Basically, the binomial expansion is used to expand a binomial to any power, it uses the binomial coefficient (the combination symbol of ncr), I suggest googling to find out more.
Epic Math Time made a really good video about it. Edit: Did want to note that it was originally supposed to only work for positive integer n; this is why Newton's discovery was novel. For a more rigorous definition, you can use a MacLaurin series expansion.
For (a+b)^x where x isn't a positive integer, we can find an approximate using that formula. Since the square root is ^1/2, we can only approximate it. It only works if x is between 1 and -1 and if x is rational.
At newtons days, the power had to be a positive integer, newton tried negative integers and it worked, and then he tried the non-integers and it also worked, for further details check out veritasium's video on CZcams called "the discovery that transformed pi"
The first mathematician to calculate pi was a greek philosopher he had draw a regular sided polygon with n sides and calculated for n tends to infinity what is the premeter of the shape which tends to be a circle with radius 1 cm so the limit is 2 pi which is 6.28318530718
my no-nut expirience: Day 2: I jogged a mile. Day 4: I cracked an egg with one hand Day 10: my dog ran out of the house, but I caught up. Day 20: I read the odyssey in Braille Day 35: my sex appeal formed an aura around me Day 40: went to the gym, all the squat racks emptied. Day 60: I learned to speak in Hieroglyphics. Day 70: my phone held a charge for 10 days. Brightness at max. Day 90: I tickled her G spot with my voice Day 100: my wifi works wherever I go. Day 150: I resuscitate my grandfather. He died in 1994. Day 360: I mine Bitcoin with my subconscious. Year 2: I am energy
Newton made a new method which was efficient and easy when compared to Pythagoras's method...almost everyone used Pythagoras's method to calculate the value of pi. Newton was different... And that's expected from him
Isn't it simpler to just explain that π = circumference divided by diameter? Because no matter the size of a circle, its circumference divided by diameter will always be equal to π. This is how you get the formula circumference, C = 2πr. (C/d = π) => (C/2r = π) => (C = 2πr) Please correct me if I'm wrong :)
@@alexkarter5854 in simple terms, circumference can be measured by using a string, no? And there must be other tools that are more accurate to measure the circumference of a circle.
@@LMNOP12 this isnt the whole truth every mathematician were trying to get most significant value of pi aryabhatta find 5 significant value but there was one guy who spend his whole life and find 39 significant value and it would be very much easier to do it with calculus
I have had all the prerequisite knowledge for this proof for years and I never ever thought of this. That's the difference between a rote-learned engineer and Sir Issac Newton.
@@itsiwhatitsi Yes it is. Isn't it? (BPRP reference). The reason is that this variation of the binomial formula is derived from the Mclaurin expansion of (1+x)^n.
@@haziqridzwan5199 we don’t know much, but based on some of his letters, he was romantically interested in his niece (I believe she was a step-niece but IDK). He lived with/near her for several years.
Remember: This was done by hand.
bruh computing is never the hard part
@@callmetpm2586 but definitely not the easiest part
This apple provided miracles to him
@@PizzaMan137 except, in this case, it is
I mean , how do you even consider newton a human being? At this point I convinced that he was a fking alien
I may be bad at mathematics, it remains so fascinating to me, thank you for existing and for educating us
wow, thanks!
If you're a bad at math then how do you get educated from this, isn't math sequential learning.
@@clashoclan3371 why must there always be a reply to a comment which is just unnecessarily toxic and annoying?
Same
@@alpaqa Law of the Internet
The amount of things Newton discovered makes me believe that he may lived for more than 200 years... I mean so much discoveries in a lifetime... Great
yeah, from what I've read he was a very hard working person. He closed his door and worked so hard that he forgot to eat dinner sometimes.
@@normanmai52 Well in their era there wasn't that much to do compared to us now so I'm guessing his just discovering new methods the entire day and the day after that.
@@wr245g9 Tf are you saying?
@@alephnull6691 hes saying thats an activity, not a task or job to them.
@@alephnull6691 hes saying that there were less activities to do so newton just sat and did this for fun cause he has time this wasn't his job
This man also just CASUALLY INVENTED CALCULUS
Along with Leibnitz,independently
Wow, pure magic. I didn't even notice the moment when x^2+y^2=1 turned into a wall of formula and I have no idea where it came from. Thanks for explaining it so well.
So clever. Also, remember the guy literally came up with the process of integration all by himself, so it's not only that he was the first to do this, he was one of the first to BE ABLE to do this, and because of his own work. Say what you will about Newton, but this is the work of a genius.
me : i hate math
also me : hmm this video interesting
Welcome to analysis. Enjoy your stay.
Depends on how you are taught
"God said, Let Newton be!...And all was light." - Alexander Pope
“Nature and nature's laws lay hid in night: God said, 'Let Newton be! ' and all was light.” The great English poet Alexander Pope (1688-1744) wrote this epitaph to be inscribed on the tomb of Sir Isaac Newton (1643-1727), and captured in two lines the immense importance of Newton to the history of modern science.
Whats also interesting is that he further expanded on the method to use a smaller region because it would converge faster (i think 1/3 of the region?). It totally makes sense to not include that in a short but I think this point adds to the ingenuity of the method.
hmm.. that would mean ud have to do 2x as much arithmetic per term, so taking 1/3 of the region mustve caused it to converge quite a bit faster
1/3 is bigger than 1/4
-Straight to the point
-Less than one minute
-Simple explanation
-Good representation
This video is what I was looking for
in order to get expand (1-x^2)^1/2 Newton broke the previous rule of only using the binomial expansion formula for powers that were counting numbers (positive integers)
Newton was the first to realise that the binomial expansion formula applies to all real numbers, if you're comfortable with infinite series.
Newton took it a step further: He integrated the circle fromula between x=0 and x=1/2, which gave him the area of a sector of angle Pi/12 radians + a triangle with base 0.5 and height (root 3)/2
root 3 is equal to the binomial expansion 2(1-1/4)^1/2
Therefore Pi = 12 x(integral ((1-x^2)^1/2) - (root3)/8)
This value converges quite quickly
This is incredible. Thank you for creating thos
pi was invented by Indian Hindu scientist Aryabhatt
i never feel so stupid today
Wow I never knew that π could be this interesting. Thanks for being here to teach me this. Please do keep up your hardwork. Great work and thanks❤
awesome content, keep on going my dude!
Appreciate it!
وجدت القناة عن طريق الصدفة وهي فعلا مدهشة و مذهلة في نفس الوقت شكرا جزيلا لصاحب القناة فماتعرضه رائع يارجل 👍
he estado buscando durante bastante tiempo como calcular Pi, era una de mis mas grandes dudas, el saber por que se llega a ese resultado tan preciso en los decimales, vi muchos videos y casi ninguno solucionó mi duda, todos dando explicaciones largas y se desviaban del tema, sin embargo, he aqui en un video de tan solo 53 segundos por fin sé como calcular Pi, ya puedo descansar en paz, muchas gracias amigo.
You could integrate (1-x^2)^0.5 with substitution x = sin(t)
@Dark Matter its good
Never saw this explained like this before! Fabulous!
I really love math and I hope to be a mathematician when I grow up. Your videos always explain things easily which we all really appreciate, thank you!
first time i actually understood this
Just using Pythagoreans theorem, I was able to calculate pi to ten decimal places... I could have gone further but I did it by hand. It took several days and I got bored.
It makes me wonder why it was so hard to calculate before Newton when the theorem has been known for centuries.
Another thing that blows my mind is that if we know the radius of a circle, we can never measure the circumference and vice versa. We can see the beginning and end but it can't be exactly measured... 🤯
an important thing is that he was the first to apply the binomial expansion to the circle because for that he needed to use a fractionary power, and no one before had tested the binomial expansion for a non-integer n, but newton did and proved he could, that's the beauty of what newton did, expanding a theorem, proving the expansion is possible, start playing with it and using the other knowledge he "created" to revolutionize pi approximations
I am pretty happy that I was able to understand the explanation
I really like math and so in class we were taking equations that require pi and so I memorised 20 decimal digits of it
I found legendary channel Today 🙂
So Newton used his own developed calculus to calculate pi. Genius!
gorgeous video! Please make more like this about history of math
Thanks mate! Just found your channel and I am loving it.. I am not an academic nor a math ninja but I love seeing these "behind the scenes"!
I was actually looking for something you might be able to help me (and many more, I hope) with...
I am trying to "understand" the thought process behind reciprocal and reciprocal formula. I have been learning about circuits and was faced with the Parallel Operator/Sum which strikes me as something very "deep" in derivation of formulas and algebra/math itself but 99% of people will only ever copy the formula and solve it without ever understanding why..
How I see it is that by using the reciprocals of parallel domains (in this topic, circuits with resistances) the "formula" allows us to "sample" a new kind of "unit" of that abstract domain where we can then calculate the participation (the % contribution of the total) of each "parallel resistance" in the overall system and the final step (multiply by the volts) will apply each "participation" (the new "unit" / the new "one") relative to the total voltage of the circuit and give us the correct value on that particular point of the system.
I have NO IDEA if what I wrote makes any sense, as I said I am far from a math geek just the everyday nerd on youtube. It would be great if anyone could share some insights and/or send me some reading material... It took me forever to learn what are reciprocals, reciprocal rules and see how it's used beyond "copy the formula and replace the variables".
Somehow I have a feeling that learning about this thought process might help me a lot with understanding how things actually work. My education never focused on reciprocals, we would learn to multiply fractions by putting one on top of the other and multiply TOP x BOTTOM / BOTTOM x TOP ... again, just another formula we would memorize and repeat (before year 2000)...
Cheers from Brazil! Keep up the cool videos!!!
Hello and thank you for liking the channel. I will admit that I am not familiar with the formula on reciprocals as it relates to circuits. I hope you find an intuitive understanding of the formula!
Sounds like a good way of thinking about it. Essentially, you are taking resistances (which don’t add in parallel), converting them to conductances (through taking the reciprocal), adding the conductances (to get the combined parallel conductance), and then converting back to resistance.
One motivation for why conductances add is parallel is the following: the conductance of a path times the voltage across that path yields the current that would flow. Two paths in parallel have the same voltage, so the sum of the currents through each path would equal the voltage times the sum of the conductances, so it would be reasonable to say the conductance of a parallel combination is just the sum of the conductances on each branch.
Hope this makes sense.
@@matthankins6206 THANK YOU! That was a great explanation and I had not been able to think about it in the "conductance" light. I believe I now understand why reciprocals / inverse numbers are used in the formula. It is because the inverse of the resistance IS the conductance. Beautiful as it all came together!
I confess I thought it would open up a new way of looking at this derivation, but it was "so simple" (with your help) that it completely makes sense why it was done this way, incredible thought process.
I thought reciprocals would be more of a wildcard in derivations (due to my lack of understanding), but understanding its use in this formula has put my mind at ease.
Once again, thank you mate! It's the kind of stuff almost impossible to find the right google search.
Cheers!
Newton was the most incredible human in history.
Well actually, that’s debatable. The title could go to Archimedes who also calculated pi, but did so during the times of early Ancient Rome. Or it could go to Pythagoras, who developed the Pythagorean Theorem, allowing the discover of pi. The title could go to Nikola Tesla, or Benjamin Franklin, or Schrodinger, or Galileo, or Aristotle, or Socrates, or Plato, or Marcus Pollio. There are millions of candidates for this position, I don’t think we can declare just one person for it.
Newton wrote more on theology than science
Lol you can always count on people to chime in with their own opinions as to why your comment is wrong
@@JM-md4ri "as to why your OPINION is wrong", i think you meant
@@swordz828 and it can also be given to chanakya, aryabhatta they are indians, they found pi before 2000 years ago
Brilliant and genius.
Complicated as Collision of Blocks with Pi numbers
“The closer we get to pi” ha…good one
I've always wondered how pi was originally calculated. Plus the visual really helps! It's interesting how the quarter sections of the circle match up to the radians of the angle. Just another way to show that math always comes back to prove itself
Can someone explain the binomial expansion theorem?
Basically, the binomial expansion is used to expand a binomial to any power, it uses the binomial coefficient (the combination symbol of ncr), I suggest googling to find out more.
Epic Math Time made a really good video about it.
Edit: Did want to note that it was originally supposed to only work for positive integer n; this is why Newton's discovery was novel. For a more rigorous definition, you can use a MacLaurin series expansion.
For (a+b)^x where x isn't a positive integer, we can find an approximate using that formula. Since the square root is ^1/2, we can only approximate it. It only works if x is between 1 and -1 and if x is rational.
“Binomial expansion”
Looks suspiciously like a Taylor series expansion
the true question is how did newton think of that ?
Because he is Newton
@@prasadbhalerao8556 this makes sense !
@@prasadbhalerao8556 lmao
Look up leibniz formula for pi
@@prasadbhalerao8556 unsatisfactory
amazingly done. THanks!
Thank you too!
the number complete off pi is insane
How is equation for a circle (x²+y²=1)?
I feel like this whole short is an excuse to say "this is a great slice of pi"
These guys back then we're on whole different level
Indian mathematicians Madhava and Aryabhata made very significant contributions in finding the exact value of π (pi).
Agreed
It's nice to see things like this, but I dont explicitly want to take the upper college level math courses to use these ideas
integration isnt too crazy man dont worry. DEs is where its at
@@hugoanimal8273 what does DE stand for?
@@AifakhYormum differential equations probably
I didnt know that the exponent of the binomal could be a rational, great video.
At newtons days, the power had to be a positive integer, newton tried negative integers and it worked, and then he tried the non-integers and it also worked, for further details check out veritasium's video on CZcams called "the discovery that transformed pi"
@@okabekun844 Thank you!
The binomial series is a sort of analytic continuation if you could say that
@@oni8337 exactly
You can literally put the "Mr. Incredible being uncanny" meme here.
Ma brain bouta bust open
Samjha nhi aaya per accha lga😁
alr, imma open this when my dad come to my room
Found this vid in recommendations, i will sub
wow thanks newton, for making my school life so hard
Thank him that we have rovers in mars or electronics.
@@clashoclan3371 it's called satire Mr clash o clan
That would do a great Tiktok video. DO u have a profile in there? It would be very interesting as it is very popular, specially among kids/teens.
Btw newton actually calculated Pi/12+ _[some number I forgot]_ because he integrated from 0 to 1/2 though at first he integrated from 0 to 1
Ppl who made 69420-sided polygon to estimate pi: "WTF?!!"
He actually did another thing. To make the expression converge faster, he integraded from 0 to 1/2.
The first mathematician to calculate pi was a greek philosopher he had draw a regular sided polygon with n sides and calculated for n tends to infinity what is the premeter of the shape which tends to be a circle with radius 1 cm so the limit is 2 pi which is 6.28318530718
The man was surely 🅿️
gee I thought he just took some spaghetti and, boom! pi!
Everything went above my head except the yummy pie
That is even way simpler than reading a huge book full of flushiness
There is an error on 14:18. The integral of (-1/8x^4)dx is (-1/8)(x^5)/5
Don't even ask
- Sonic
But value of π was already calculated in Ancient India before Newton 👍👍🇮🇳
New subscriber for good content! 😘
The moment I saw the first frame of the video my dumb brain knew what would happen
my no-nut expirience:
Day 2: I jogged a mile.
Day 4: I cracked an egg with one hand
Day 10: my dog ran out of the house, but I caught up.
Day 20: I read the odyssey in Braille
Day 35: my sex appeal formed an aura around me
Day 40: went to the gym, all the squat racks emptied.
Day 60: I learned to speak in Hieroglyphics.
Day 70: my phone held a charge for 10 days. Brightness at max.
Day 90: I tickled her G spot with my voice
Day 100: my wifi works wherever I go.
Day 150: I resuscitate my grandfather. He died in 1994.
Day 360: I mine Bitcoin with my subconscious.
Year 2: I am energy
my brain at 3 am
calc is cool as hell
Keep up the good work
Thanks, will do!
Yes I totally followed that..
I totally understood everything u said
The grade 5 lesson i learned became more and more complex
Pi value was given by ramanujan sir
Hats off to the sacred genius. Newton is a God. He gifted the world, with his legendary inventions.
Newton made a new method which was efficient and easy when compared to Pythagoras's method...almost everyone used Pythagoras's method to calculate the value of pi. Newton was different... And that's expected from him
And hence mathematicians stopped trying to flex on each other by bisecting higher sided polygons
Isn't it simpler to just explain that π = circumference divided by diameter? Because no matter the size of a circle, its circumference divided by diameter will always be equal to π. This is how you get the formula circumference, C = 2πr.
(C/d = π) => (C/2r = π) => (C = 2πr)
Please correct me if I'm wrong :)
The thing is, how would you know the circumference in the first place? You can't really "measure" it.
@@alexkarter5854 in simple terms, circumference can be measured by using a string, no?
And there must be other tools that are more accurate to measure the circumference of a circle.
- So if *(4pi R²=A)* what is the formula of the surface area *(A)* of a 4 spacial dimensional sphere?
i just pity those mathematicians who spend years to find an accurate value of pi
pi was invented by Indian Hindu scientist Aryabhatt
@@LMNOP12 this isnt the whole truth every mathematician were trying to get most significant value of pi aryabhatta find 5 significant value but there was one guy who spend his whole life and find 39 significant value and it would be very much easier to do it with calculus
@@Ujjwalseth2412 basic is the most important...a building won't survive without basic and initial knowledge...keep in mind
Understandable,have a great day
Like Ludolph?
My brain exploded
Taking cal one this semester and this just blew my mind 🤯🥧
rip our forgotten hero "ramanujan"
What about when mathematicians 200 years later tried to redefine it into 3.2... ?
And that's because I selected foreign language as carreer
pls looking at my math grades is already a math problem
he actually integrated from 0 to 1/2 since the higher power fractional terms contribute less and therefore converges quicker and more accurately to pi
I have had all the prerequisite knowledge for this proof for years and I never ever thought of this. That's the difference between a rote-learned engineer and Sir Issac Newton.
So the Binomial formula with exponent 1/2 has infinite many elements 😝
all binomial expansions where n isn’t positive integer do
@@ricoseb I didn’t know that. That is interesting
@@itsiwhatitsi Yes it is. Isn't it? (BPRP reference). The reason is that this variation of the binomial formula is derived from the Mclaurin expansion of (1+x)^n.
Well, π is transcendental after all, not rational.
Bruh! Wut u guys are talking ?😂👌
I mean what u guys are doing currently 12th??
This video made pi make more sense to me.
Me: Watching the full video.
Also me: *Doesn't know wtf is going on*
Related able comment
Me also tink tis is related able
dude my head hurts
Interesting, nice vids
Side note: Newton was the first person to apply the binomial expansion to a fractional exponent like this.
Uh... numbers go brrrrr
I like your words magic man
Shift+EXP
and he did all of this while succeeding No Nut Lifetime lol
Unless you count his niece
what...
@@cara-seyun wha-
@@haziqridzwan5199 we don’t know much, but based on some of his letters, he was romantically interested in his niece (I believe she was a step-niece but IDK). He lived with/near her for several years.
@@cara-seyun doesnt make it any better
You…You……made
My brain hurt
Newton: Binomial Expansion *fwoosh*
Damn when i do math , i'm just staring at the paper not thinking about anything , just want every answers to appear automatically in front of me.