Only U.S. President to prove a theorem
Вставка
- Опубліковано 14 кві 2022
- In 1880, James Garfield contributed a new proof of geometry's most famous right triangle theorem. #shorts #math #maths #mathematics
Mathematical treasure: Garfield's proof
www.maa.org/press/periodicals...
Engraved portrait
en.wikipedia.org/wiki/File:GA...
Gizmodo post
gizmodo.com/james-garfield-wa...
Subscribe: ua-cam.com/users/MindYour...
Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!
If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.
Book ratings are from January 2022.
My Books (worldwide links)
mindyourdecisions.com/blog/my...
My Books (US links)
Mind Your Decisions: Five Book Compilation
amzn.to/2pbJ4wR
A collection of 5 books:
"The Joy of Game Theory" rated 4.2/5 stars on 224 reviews
amzn.to/1uQvA20
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4/5 stars on 24 reviews
amzn.to/1o3FaAg
"40 Paradoxes in Logic, Probability, and Game Theory" rated 4.1/5 stars on 38 reviews
amzn.to/1LOCI4U
"The Best Mental Math Tricks" rated 4.2/5 stars on 76 reviews
amzn.to/18maAdo
"Multiply Numbers By Drawing Lines" rated 4.3/5 stars on 30 reviews
amzn.to/XRm7M4
Mind Your Puzzles: Collection Of Volumes 1 To 3
amzn.to/2mMdrJr
A collection of 3 books:
"Math Puzzles Volume 1" rated 4.4/5 stars on 87 reviews
amzn.to/1GhUUSH
"Math Puzzles Volume 2" rated 4.1/5 stars on 24 reviews
amzn.to/1NKbyCs
"Math Puzzles Volume 3" rated 4.2/5 stars on 22 reviews
amzn.to/1NKbGlp
2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.
My Blog
mindyourdecisions.com/blog/
Twitter
/ preshtalwalkar
Merch
teespring.com/stores/mind-you...
Patreon
/ mindyourdecisions
Press
mindyourdecisions.com/blog/press - Наука та технологія
Imagine if presidential elections were determined by math proofs.
not sure about many things but our world will be a better place for sure
@@dbrx758 What does math have anything to do with government? James Garfield was a mid president and got assassinated
@@alekstanton4715 Most would not, true, but imagine what kind of thinkers would be elected.
Politicians are the last people I'd expect to do any math lol.
*Keep politics out of the forums.*
Learned it back in 8th grade. Didn't know a US president proved it
From Bangladesh?
@@MuhammadAshraf-ke1ww I'm from 🇧🇩
@@jimmykitty
শুনে খুব ভালো লাগলো ভাই/আপু।
অনলাইনে কোনো বিদেশি ফোরামে নিজের দেশের পতাকা দেখলে এক অন্যরকম দেশপ্রেমের অনুভূতি হয়।
ক্লাস ৮ এ পিথাগোরাসের এই উপপাদ্যের এই প্রামাণটি পাঠ্য ছিলো, ক্লাস ৯-১০ এ এর প্রমাণটি একটু কঠিন অবশ্য।
@@MuhammadAshraf-ke1ww Of course it's true. Once Presh Talwalker had solved a problem sent from Bangladesh. Did you see that video?
@@jimmykitty I think you are talking about this one.
ua-cam.com/video/a9u1pjsfJDs/v-deo.html
Although I had seen the problem in a problem book of Math Olympiad questions.
The problem is interesting but I had no idea about the Chords formula.
Lame me I guess.
wow, that's actually a really cool way to prove pythagorean theorem
IDK why
@@skull_crusher7416 Huh?
@@skull_crusher7416 Tf u talking about. Thats what the equation is called
@@davidkuten he thinks the word ”theorem” was used solely because it sounds ”fancy” and not because of what the video is about, proving thus that he himself finds the Pythagorean theorem a momentous one to understand.
@@sebastianfors4491 yeah and I think he's a lower than average IQ person hence gets intimidated when even a small amount of intellect is radiated upon him.
*"I used the proof to prove the proof" - James A. Garfield* 😎
No?
He never used the Pythagorean theorem though?
Yeah, just got it now, thanks
@@sadeekmuhammadryan4894 Ayyy hello! 😊
@@jimmykitty Hi! 😁
Geometry has two great treasures; one of them is Theorem of Pythagoras! ❤
Fun fact: Pythagoras was not the mathematician that discovered the theorem. It was in ancient Babylon about a thousand years before gim that someone found out this property of right angled triangles!
What's the other one
@@ajety Golden Ratio!
@@mustafizrahman2822 It's euler's identity!
@@jimmykitty Golden Ratio is kind of overrated, Euler's formula is would argue is the second.
And it's a pretty elegant proof
It uses a mixture of algebra and geometry. I don't think that level of algebra existed at the time of the ancient Greeks. A nice original proof, though.
Doesn t the other formulas derive from Pythagoras s Theorem though ?
@@justinmacarrhur1924 no, he only uses the areas of the shapes, and to deduce the expressions for those areas you don't need the Pythagorean theorem.
@@JiminatorPV didn t say you need, but I think those formulas were found via Pythagora
@@justinmacarrhur1924 I don't think they were found via Pythagoras either. And even if they were, as long as the Pythagorean theorem is not needed, it is a proof.
Yes ! Yes !
In 1876, Garfield demonstrated his talents as a mathematician by providing a proof of the Pythagorean theorem. His work was published in the New England Journal of Education. Mathematical historian William Dunham argued that Garfield's proof was "really a very elegant proof."
just half of the shape used in the original proof
You cannot say that there is a "original proof". This theorem has been proved in hundreds of ways all over the globe dating back to the babyloneans and ancient Egypt. There is no known first proof of this fact.
@@ivarangquist9184 half of the shape used in the most famous proof, then.
It is literally the same idea as with a full square, I wouldn't really call this a distinct proof
makes you about what makes a proof distinct maybe all proofs are logically equivalent
Thank you for posting this! I recall seeing, some decades ago, that he had been a schoolteacher, and had come up with a novel proof of the Pythagorean Theorem.
But the diagram for that, was twice this one. Namely, there was one big square, with another one (here ½ a square that's a rt. isosc. ∆ in white; green in your thumbnail), side=c, inscribed in it, tilted, so that there were 4 congruent right triangles (here in blue), with legs a & b.
Then, areas were equated:
BIG Square = 4 right ∆s + little square
(a + b)² = 4(½ab) + c²
a² + 2ab + b² = 2ab + c²
a² + b² = c²
QED
When done this way, it is perhaps more obvious that the four ∆s are right ∆s, and all congruent; and there's no need to use, or even know, the area of a trapezoid.
Was your version his original, and someone later turned it into what I've described here?
Anyway, I always thought that this was far superior to, and more elegant than, the tangled mess of a proof we were taught in high school geometry class, which may have been straight out of Euclid's _Elements,_ idk.
EDIT: From your link, I see that your trapezoid version was Garfield's original, and that it was published in 1876, not 1880, which was the year he was elected president.
Fred
This has been proven before by Euclid way back in ancient times. It is a nice proof tho.
Yeah but there a bunch of cool ways to prove it, like this one
Garfield was actually crazy smart. It was said he could write Latin with one hand and Ancient Greek with the other at the same time. He was head of a university in Ohio, and managed to win a surprise victory as president. Then he got shot by a crazy person who joined a sex cult and didn’t get laid (not a joke).
Oh silly, silly Charles Guiteau. The definition of a Walking L
Thank you Sam O’Nella
Bro was named after a cat that ate lasanga 💀
Other way around.
And now the government doesn't want a smart president, it wants an empty talking head
And an emptiness-shaking hand.
*Keep politics out of the forums.*
@@forcelifeforce What makes you think you can tell me what to do?
@@DmitDmit1 he's a republican 💀💀
Im sure that's what people thought of all the presidents before Biden and them. Im sorry the political radicalism has changed your thoughts but presidents have always been a controversial figure
in 1880 in the USA, the first to solve this problem became president : James Garfield had won !
we should bring these back
And then, the following year, unfortunately became the second U.S. president to be assassinated.
Fred
Damn, that was surprisingly simple!
Pythagoras already proved that some 550 years BC. though it was known by the babylonians a millennium earlier already
Instead of making a trapezoid, why not make a square (by linking four abc triangles). That would be easier to understand..
Yes, but that had already been done. I've read that he was working as a teacher at the time. Maybe he found this by accident and found it interesting. I've read there's a book by Elisha Loomis that contains 367 proofs of the Pythagorean theorem. Finding new and clever proofs of known things seems to be fun for some.
@@AHBelt yeah, new and clever proofs are, but I would argue this is not really distinct from the well-known square Arnold referenced
@@rjtimmerman2861 Sure, and I've actually read a book by someone who agrees with you on that.
It would.... But then it wouldn't be an original way to prove it. There isn't one single way to prove mathematical theorems.
@@rjtimmerman2861 um yes it is. It literally is a different way of proving it.
He was not the first to prove this, but he came up with this proof himself.
Hey presh i want you to take a look at this interesting problem-
"Gold is 19 times as heavy as water, and copper is 9 times as heavy as water, the ratio in which these two metals be mixed so that the mixture is 15 times as heavy as water"
A)1:2
B)2:3
C)3:2
D)19:135
The correct answer is C) 3:2
Will you please solve this
With out loss of generality and For simplicity let's say 1cm ^3 of gold is 19lbs and 1cm^3 of copper is 9lbs. And let say water is 1lb for 1cm^3.
If we did an equal amount of gold and copper. Say 1cm^3 of each and we melt it, mix it and cool. Then we'd have 2 cm^3 weight 28lbs but that's for 2 cm^3. So 14lbs cm^3.
Not quite right but close.
Basically we need total weight divided by total volume and we want that that to be 15cm^3.
We want to solve this:
Let x be the number cm^3 of gold and y be the number of cm^3 for copper.
#1 (19x + 9y)÷(x+y) = 15lbs for 1cm^3.
The 19x + 19y is the total weight.
And x + y is the total volume.
#2 x+y = 1cm^3
We can scale the total volume to anything we want so I am choosing 1cm^3 to be the total as it makes the numbers easier.
Now what do we do with these two equations.
A. Using the eq. #2 we can simplify the denominator of eq. #1 to be 1.
B. Also we can arrange equation #2 to be y = 1-x.
Substituting both A and B on eq #1 we can write:
(19x + 9(1-x))/1 = 15 cm^3
19x + 9 -9x = 15
19x - 9x = 6
10x = 6
X = .6
Substituting X= .6 Into eq . #2
.6 + y = 1
Therefore y = .4
Ratio is .6 : .4
Same as 6 : 4
Same as 3 : 2
Done.
Easy.
To make 1 litre (or any other unit of volume, mass or whatever) of something with value 19 and something of value 9 to have value 15 (here it is density relative to water, but can be anything) we simply use this equation:
19*x + 9*(1-x) = 15, where x is amount of gold in one litre.
By solving you get x = 3/5, which means you have 3/5 of liter of gold ans 2/5 liter of copper in one liter of mixture.
Therefore 3:2. Easy peasy.
(19x+9y)/(x+y)=15
(19x is gold with x added mixture, 9y is copper with y added mixture, (x+y) is to find the average)
15x+15y=19x+9y
15x+6y=19x
6y=4x
3y=2x
@@tomasskraban7899 which class math is this?
Will I have to know some formula to do it?
@@malaysarker6721 it's just weighted average. 19 and 9 are averaged values, x and 1-x are weights. Sum of weights is 1, so we don't have to divide by it. It's like someone mentioned (19*x + 9*y)/(x+y) = 15, but I siplified it so that x+y =1 and substituted for y. It's without loss of generality, no problemo. Hope it helps.
And I don't think it's too advanced. It's just a neat trick with weighted average. You just need to know equations.
I've never seen a proof to that with a trapezoid. It's pretty interesting
Wow this is a much more simple proof the Pythagorean Theorem than what I learnt back in high school!
Pythagore : am i a joke to you?
I thought that Garfield was only good at eating lasagna...
That’s a really neat proof
US Presidents in 1880s: smart as heck
US Presidents in 2020s: **snoring**
Indeed incredible🤩
Maths and lasagne, Garfield’s two favourite things
And Richard Garfield, his great-great-grandson is also a mathematician and the creator of Magic The Gathering.
Still avoiding to mention Pythagoras xD
My 9th grade geometry class mentioned that there had been many proofs of the Pythagorean Theorem, including one by a US president, but gave no further details. Glad to finally see the rest of the story.
If you do that two more times you would have a whole square… then that’d be the same the moment you removed the 1/2 from both sides.
1876: “a^2 + b^2 = c^2”
2022: “The number one threat is the strength, and that strength that we’ve built is inflation.”
He must’ve had a lot of lasagna that day
That was so simple but so genius
This is such a nice proof as well!
Abe Lincoln also used math in his famous Gettysburg Address.
“4 score and 7 years ago, …”
You have to math it to understand it’s 87 years ago!
ye but that’s like calling someone who says “i bought 2 dozen eggs” a mathematician
@@aug3842 If they say dozen as in a case, then no.
But if they refer to dozen as in actual 12, then yes, they're a mathematician!
"Just like You can be!"
@@creamwobbly Wow, I never heard of huitante before!
But if Abe's speech had it written as 2 separate words (4 score), then he was Mathing!
@@JLvatron It’s inconclusive to tell if the person is a mathematician or of a sample bias with high or low probability of being which; The given situation is with too much equivocation for there isn’t any elucidation whither; It’s only a man seemingly soever in a world that doesn’t exist. It also revolves around your ideology about mathematicians.
Not sure if that (4x20) would be considered verb "math"
Next, on MindYourDecisions : "let's use the Garfield's theorem" 😉
I have heard several amazing things about Garfield. He could speak several languages he was ambidextrous and he could write one language with one hand WHILE writing another language with the other hand. Garfield sounds like he was an amazing man.
Another cool trick, if you mirror the shape created at the end of the video, flip it upside down and attach it onto the existing shape. You will get a square with a smaller inner square. That smaller inner square that is slightly rotated is c^2
"What have your Government did in 5 years"
*We proved the Pythagoras theorem*
Took me a second to spot that the area of the third triangle was (c^2)/2. Thanks for the quick mental workout!
What??
Another onto the list of Garfield's many accomplishments
Meanwhile Donald Trump:
I am gonna solve one of the greatest mysteries in mathematics, The Reimann hypothesis.
Issac Newton: *Apple falls.* 'If the apple falls, does the world too fall...?"
*Thus the discovery of gravity*
Then I suppose....
James Garfield: *Flips Dorito chip*
Thus the discovery of the pythagorean theorem...
Is anyone gonna tell him this was proved a zillion different ways before that?
And that does what to minimize the fact that he came up with a new proof of it?
You take 2 squares, one inside the other, and rotate the inner one until it touches the sides of the larger. Calculate the areas inside, total equal to the area of the larger square. Same result.
Isn't this essentially the rearranging proof by rearranging the triangles into a square, except we take half the square (the trapezium) and instead of the whole square
Nice one!
Incredible that makes a whole 2 notable about president Garfield.
Damn, that's super clever.
Imagine that Euler, Gauss, Newton, Leibniz or Riemann didn't know this proof!
Thanks Garfield
Yesss algebra proof!! To a geometry theorem!
These are the best kind. Where it all connects
or do the thingy with 4 right angle triangle to form a square with 4 sides of c
and rearrange the 4 of the same triangles to form 2 squares, a² and b²
today's president : "i know the theorem better than anyone else..."
JB: I remember the day my nurse was coming to give me my bottle. I’d arranged my blocks as a proof of the theorem on the floor. So she just ignored the proof and put the blocks away. I have no idea how she became the vice President.
Most important formula in machine tool pipefitting and tool building. Used it evey day.
I can’t wait to use this one day
you sound like my math teacher
What software do you use to edit geometric movements and formulas?
DAMN YOU GARFIELD
This is just perfect.
What your math teacher meant when they said you need to show your workings
Wouldn't it be visually clearer and easier, if he uses 4 square triangles instead of two? That way, the result is a square, with c^2 in the center (so its even visually clear that it works).
The calculation would remain fairly simple too:
(a+b)^2 = 4ab/2 + c^2
a^2 + 2ab + b^2 = 2ab + c^2
a^2 + b^2 = c^2
Maybe its just me, but that strikes me as simpler, since I don't need to know how to calculate the area of a trapezoid, and its visually using squares the whole time too.
Alternatively, develop the cosine law from a dot-product analysis and then notice that the cross-product term goes to zero when the included angle goes to PI/2:
C^2 = A^2 + B^2 - AxBxCos (Theta)
I assumed it would be Jefferson. No idea Garfield was a nerd.
...Today I learned how the Pythagorean Theorem formula is a true formula. I just plugged in the numbers into the formula just cause my teacher told me to without questions.
This proof is similar to Bhasharachary's proof
Indian mathematicians proved many theorems much before than western mathematicians even knew they exist , but bcoz they were Indian , not much credit is given to them :(
Right..
@@arnavverma2461 sure. But many of the proofs are also parallel, to be fair. I'm not sure this one is, but it's a very simple one.
@@VeteranVandal it simple bcoz you know how , thinking while knowing no one has ever done it , wouldn't be that simple...
James Garfield was very skilled
Next level thinking.
It's looks like
A2 + B2 = C2
But with extra steps
I still prefer the old proof with the four congruent right triangles arround a square but nice thing to know...
That is why a "cat" can only be the best President
you mean that all this time i could have called it "the garfield theorem"?
But what when we don't know the formula of area?🤔🤨
Prove it
By dividing the trapezoid by 2 triangles, you can easily find that formula of area.
Area of a right triangle is exactly half of the rectangle its sides form, which by definition is side multiplied by side. So area is a*b/2.
Or just build squares with a, b and c being the size of their sides and calculate the area. Take a triangle with a=3, b=4 and c=5
Meanwhile Pythogoras sitting in the corner crying
Beautiful proof. In school we did the proof using similarity.
I love this proof because it’s so simple. You could probably explain this to an elementary schooler and they’d be able to follow along with their math knowledge
We can also prove it by drawing perpendicular from side has 90° on base then similar triangle 🔺 and then apply thales threom
Al kashi and pythagores be like : bruh , i guess we are presidents now
Very nice proof of the Gaugou theorem.
I feel like that was already known by then...
Pythagoras proved his own theorem in Ancient Greece. Now, a new proof was found, many moons later. cool.
That was when presidents were smart.
If you combine four isosceles right triangles where both legs have a length of one, you can create a square. The area of each triangle is equal to one half, so the area of the square is equal to two, so each side is equal to the square root of two.
theres also the semicircle one
I sent you an email about a proof I need help on.
It's very interesting, I explained how everything works
Everything you mean reality?
@@Gabrielkk_ no, I mean in email lol
I'm talking about brane space, fluid mechanics, and moduli space
@@timelyseeker lol, may i ask how u did it?
@@Gabrielkk_ complicatedly, I don't know - That's why I was asking for a proof
@@timelyseeker Oh lol, now i understand. I tought u said like "i've a proof for everything"
He also got assassinated by a crazy guy…
I prefer using euclidean metriques. If a and b are ortogonals, then: ||c||^2 = ||a+b||^2 = g(a+b, a+b) = g(a,a) + 2g(a,b) + g(b,b) = ||a||^2 + ||b||^2 (g(a,b) = 0 because they are ortogonal)
It was good
And he loved Lasagna, but hated Mondays.
This is Pythagoras theorem. Why 😂
So how did Garfield know his proof would work out? How exactly did he know to make a trapezoid for some point in his explanation? If someone would answer, it'd clear a good lot for me.
What a complicated way of going about this,
I have another proof:
Take the length of c and draw a square with it. Find its area. Do the same with a and b. There c²=a²+b²
Bro I passed this stuff so long ago. Why the fridge am I in awe at this? Is it because I’m not forced to watch it happen?
Theoretically, he would have been an excellent president, he was a good speaker, well educated, and was popular with a lot of the people.
Unfortunately, we will never know as he died less than 3 months into his presidency.
Died/was murdered. Kind of an important distinction.
@@brianbarber5401multiple factors had a hand in his death. Guiteau’s bullet was only part of it.
Its why he was killed, big equations couldn't have someone out there disrupting their marked
i’m not surprised
Petition to make all world elections determined by logical or scientific skills.