Only U.S. President to prove a theorem

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Only U.S. President to prove a theorem

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2 жыл бұрын

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
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Пікірлер: 794

@WestExplainsBest 2 жыл бұрын

Imagine if presidential elections were determined by math proofs.

@dbrx758 2 жыл бұрын

not sure about many things but our world will be a better place for sure

@salti6780 2 жыл бұрын

@@dbrx758 What does math have anything to do with government? James Garfield was a mid president and got assassinated

@WestExplainsBest 2 жыл бұрын

@@danielsmith4715-d2k Most would not, true, but imagine what kind of thinkers would be elected.

@cloroxbleach7554 2 жыл бұрын

Politicians are the last people I'd expect to do any math lol.

@forcelifeforce 2 жыл бұрын

*Keep politics out of the forums.*

@asifurrahman9950 2 жыл бұрын

Learned it back in 8th grade. Didn't know a US president proved it

@MuhammadAshraf-ke1ww 2 жыл бұрын

From Bangladesh?

@jimmykitty 2 жыл бұрын

@@MuhammadAshraf-ke1ww I'm from 🇧🇩

@MuhammadAshraf-ke1ww 2 жыл бұрын

@@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।

@jimmykitty 2 жыл бұрын

@@MuhammadAshraf-ke1ww Of course it's true. Once Presh Talwalker had solved a problem sent from Bangladesh. Did you see that video?

@MuhammadAshraf-ke1ww 2 жыл бұрын

@@jimmykitty I think you are talking about this one. kzfaq.info/get/bejne/l5-lZNOb2MyudKc.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.

@giacomomosele2221 2 жыл бұрын

wow, that's actually a really cool way to prove pythagorean theorem

@smilya4664 2 жыл бұрын

IDK why

@peanutnerd 2 жыл бұрын

@@skull_crusher7416 Huh?

@davidkuten 2 жыл бұрын

@@skull_crusher7416 Tf u talking about. Thats what the equation is called

@sebastianfors4491 2 жыл бұрын

@@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.

@broannoying8765 2 жыл бұрын

@@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.

@sadeekmuhammadryan4894 2 жыл бұрын

*"I used the proof to prove the proof" - James A. Garfield* 😎

@andrwba 2 жыл бұрын

No?

@bilbot.baggins9019 2 жыл бұрын

He never used the Pythagorean theorem though?

@sadeekmuhammadryan4894 2 жыл бұрын

Yeah, just got it now, thanks

@jimmykitty 2 жыл бұрын

@@sadeekmuhammadryan4894 Ayyy hello! 😊

@sadeekmuhammadryan4894 2 жыл бұрын

@@jimmykitty Hi! 😁

@labzioui1 2 жыл бұрын

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."

@jimmykitty 2 жыл бұрын

Geometry has two great treasures; one of them is Theorem of Pythagoras! ❤

@lucabricardknipp 2 жыл бұрын

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!

@ajety 2 жыл бұрын

What's the other one

@jimmykitty 2 жыл бұрын

@@ajety Golden Ratio!

@jimmykitty 2 жыл бұрын

@@mustafizrahman2822 It's euler's identity!

@macicoinc9363 2 жыл бұрын

@@jimmykitty Golden Ratio is kind of overrated, Euler's formula is would argue is the second.

@ffggddss 2 жыл бұрын

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

@EccentricTuber 2 жыл бұрын

And it's a pretty elegant proof

@robertsharp67 2 жыл бұрын

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.

@justinmacarrhur1924 7 ай бұрын

Doesn t the other formulas derive from Pythagoras s Theorem though ?

@JiminatorPV 7 ай бұрын

@@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.

@justinmacarrhur1924 7 ай бұрын

@@JiminatorPV didn t say you need, but I think those formulas were found via Pythagora

@JiminatorPV 7 ай бұрын

@@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.

@fakeit6339 2 жыл бұрын

just half of the shape used in the original proof

@ivarangquist9184 2 жыл бұрын

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.

@gabriel-et3gy 2 жыл бұрын

@@ivarangquist9184 half of the shape used in the most famous proof, then.

@rjtimmerman2861 2 жыл бұрын

It is literally the same idea as with a full square, I wouldn't really call this a distinct proof

@ninja8flash742 2 жыл бұрын

makes you about what makes a proof distinct maybe all proofs are logically equivalent

@WahranRai 2 жыл бұрын

in 1880 in the USA, the first to solve this problem became president : James Garfield had won !

@jesselapides4390 2 жыл бұрын

we should bring these back

@ffggddss 2 жыл бұрын

And then, the following year, unfortunately became the second U.S. president to be assassinated. Fred

@HenrikMyrhaug 2 жыл бұрын

Damn, that was surprisingly simple!

@arnoldbissen9921 2 жыл бұрын

Instead of making a trapezoid, why not make a square (by linking four abc triangles). That would be easier to understand..

@AHBelt 2 жыл бұрын

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.

@rjtimmerman2861 2 жыл бұрын

@@AHBelt yeah, new and clever proofs are, but I would argue this is not really distinct from the well-known square Arnold referenced

@AHBelt 2 жыл бұрын

@@rjtimmerman2861 Sure, and I've actually read a book by someone who agrees with you on that.

@somedudes6455 2 жыл бұрын

It would.... But then it wouldn't be an original way to prove it. There isn't one single way to prove mathematical theorems.

@somedudes6455 2 жыл бұрын

@@rjtimmerman2861 um yes it is. It literally is a different way of proving it.

@anonymousanonymous1338 2 жыл бұрын

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).

@GenKoe6917 Жыл бұрын

Oh silly, silly Charles Guiteau. The definition of a Walking L

@karpholmes6942 Жыл бұрын

Thank you Sam O’Nella

@harshitgupta7987 2 жыл бұрын

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

@jilow 2 жыл бұрын

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.

@tomasskraban7899 2 жыл бұрын

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.

@user-lh5hl4sv8z 2 жыл бұрын

(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

@malaysarker6721 2 жыл бұрын

@@tomasskraban7899 which class math is this? Will I have to know some formula to do it?

@tomasskraban7899 2 жыл бұрын

@@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.

@nathanderhake839 2 жыл бұрын

This has been proven before by Euclid way back in ancient times. It is a nice proof tho.

@benjamincruz6633 Жыл бұрын

Yeah but there a bunch of cool ways to prove it, like this one

@ARKGAMING 2 жыл бұрын

I've never seen a proof to that with a trapezoid. It's pretty interesting

@nikko7345 2 жыл бұрын

Bro was named after a cat that ate lasanga 💀

@carultch Жыл бұрын

Other way around.

@asheep7797 15 күн бұрын

@@carultchnope. the _other_ other way around. the lasagna was named after the president who ate a cat.

@Ed19601 2 жыл бұрын

Pythagoras already proved that some 550 years BC. though it was known by the babylonians a millennium earlier already

@stj1203 2 жыл бұрын

Indeed incredible🤩

@suponjubobu5536 2 жыл бұрын

He was not the first to prove this, but he came up with this proof himself.

@rakhuramai 2 жыл бұрын

Wow this is a much more simple proof the Pythagorean Theorem than what I learnt back in high school!

@ScalarYoutube 2 жыл бұрын

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

@MrNicePotato 2 жыл бұрын

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.

@babyboy5553 2 жыл бұрын

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

@KataisTrash Жыл бұрын

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.

@charlessands6933 Жыл бұрын

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.

@jscb87 Жыл бұрын

And Richard Garfield, his great-great-grandson is also a mathematician and the creator of Magic The Gathering.

@darkexcel 2 жыл бұрын

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

@claudreindl7275 2 жыл бұрын

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.

@donsena2013 Жыл бұрын

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)

@Sdakouls3 2 жыл бұрын

Took me a second to spot that the area of the third triangle was (c^2)/2. Thanks for the quick mental workout!

@oloyt6844 2 жыл бұрын

What??

@destructchronos3125 2 жыл бұрын

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)

@clevercat1218 2 жыл бұрын

We can also prove it by drawing perpendicular from side has 90° on base then similar triangle 🔺 and then apply thales threom

@michaelsmyth3935 2 жыл бұрын

Most important formula in machine tool pipefitting and tool building. Used it evey day.

@Nikhil_P7 2 жыл бұрын

if f:[0,7)→[1,∞) and g:[6,∞)→[3,∞) be two functions. if 3x-y=17 and y-2=0 are tangents to the graph of f(x) and g(x) at x=5 and x=7 and h(x)=g²(x+f(x)), then h'(5) is equal to ?

@78anurag 2 жыл бұрын

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

@jeremykraenzlein5975 2 жыл бұрын

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.

@DavidAnimate21 8 ай бұрын

US Presidents in 1880s: smart as heck US Presidents in 2020s: **snoring**

@levistepanian5341 2 жыл бұрын

That’s a really neat proof

@DmitDmit1 2 жыл бұрын

And now the government doesn't want a smart president, it wants an empty talking head

@user-oh2kt8lf6g 2 жыл бұрын

And an emptiness-shaking hand.

@forcelifeforce 2 жыл бұрын

*Keep politics out of the forums.*

@DmitDmit1 2 жыл бұрын

@@forcelifeforce What makes you think you can tell me what to do?

@mikechad27 2 жыл бұрын

@@DmitDmit1 he's a republican 💀💀

@anonymousperson3023 2 жыл бұрын

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

@avidetroja2163 2 жыл бұрын

But what when we don't know the formula of area?🤔🤨

@justa9560 2 жыл бұрын

Prove it

@vhm0814 2 жыл бұрын

By dividing the trapezoid by 2 triangles, you can easily find that formula of area.

@draganandrei5356 2 жыл бұрын

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.

@matthewbell4273 Жыл бұрын

Maths and lasagne, Garfield’s two favourite things

@megamentebr7716 2 жыл бұрын

It's looks like A2 + B2 = C2 But with extra steps

@xjuhox Жыл бұрын

Imagine that Euler, Gauss, Newton, Leibniz or Riemann didn't know this proof!

@jaypaint4855 2 жыл бұрын

That was so simple but so genius

@roxiethecockapoo1138 2 жыл бұрын

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...

@dmi. 2 жыл бұрын

Pythagore : am i a joke to you?

@Awesome-ct7vr Жыл бұрын

Yesss algebra proof!! To a geometry theorem! These are the best kind. Where it all connects

@Mebasically 2 жыл бұрын

I still prefer the old proof with the four congruent right triangles arround a square but nice thing to know...

@MeDoMeer 2 жыл бұрын

This is such a nice proof as well!

@Aiharon Жыл бұрын

I thought that Garfield was only good at eating lasagna...

@conit4125 2 жыл бұрын

Incredible that makes a whole 2 notable about president Garfield.

@Pr0t4t0 2 жыл бұрын

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.

@MagicBoterham 2 жыл бұрын

Area_trapezoid = h*(parallel_side1+parallel_side2)/2

@siddharthagarwal5756 Жыл бұрын

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

@nathanmays7926 2 жыл бұрын

1876: “a^2 + b^2 = c^2” 2022: “The number one threat is the strength, and that strength that we’ve built is inflation.”

@tazguy371 Жыл бұрын

Nice one!

@pavlos712 2 жыл бұрын

Still avoiding to mention Pythagoras xD

@JJ-sd4kb Жыл бұрын

What your math teacher meant when they said you need to show your workings

@abuhuraira5581 2 жыл бұрын

Meanwhile Donald Trump: I am gonna solve one of the greatest mysteries in mathematics, The Reimann hypothesis.

@devkhair9d286 Жыл бұрын

It can easily be proved by the concept of similar triangles

@nickfas1429 2 жыл бұрын

1 this was proven like 2500 years ago from Pythagoras.2 you can just draw a square in each side of the triangle and compare the areas.

@pankajchavda6422 2 жыл бұрын

This proof is similar to Bhasharachary's proof

@arnavverma2461 2 жыл бұрын

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 :(

@JosephStalin-yk2hd 2 жыл бұрын

Right..

@VeteranVandal 2 жыл бұрын

@@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.

@arnavverma2461 2 жыл бұрын

@@VeteranVandal it simple bcoz you know how , thinking while knowing no one has ever done it , wouldn't be that simple...

@djt-lu8tw Ай бұрын

That signature is more impressive to me

@DerpyUniverse 2 жыл бұрын

He must’ve had a lot of lasagna that day

@DotDotEight Жыл бұрын

theres also the semicircle one

@srayes1001 Жыл бұрын

Next level thinking.

@shanoobs8383 2 жыл бұрын

Thanks Garfield

@brunoramey50 2 жыл бұрын

Next, on MindYourDecisions : "let's use the Garfield's theorem" 😉

@timelyseeker 2 жыл бұрын

I sent you an email about a proof I need help on. It's very interesting, I explained how everything works

@Gabrielkk_ 2 жыл бұрын

Everything you mean reality?

@timelyseeker 2 жыл бұрын

@@Gabrielkk_ no, I mean in email lol I'm talking about brane space, fluid mechanics, and moduli space

@Gabrielkk_ 2 жыл бұрын

@@timelyseeker lol, may i ask how u did it?

@timelyseeker 2 жыл бұрын

@@Gabrielkk_ complicatedly, I don't know - That's why I was asking for a proof

@Gabrielkk_ 2 жыл бұрын

@@timelyseeker Oh lol, now i understand. I tought u said like "i've a proof for everything"

@sarashepherd3264 2 жыл бұрын

What a complicated way of going about this,

@iamthesaltonyourwounds2313 Жыл бұрын

There is another way to prove it by making squares on the two smaller sides and adding their areas to see if it matches the area of the square made from the longer side

@alva72nashir3 9 ай бұрын

it's same proof of square (a+b)².. and he cut it then the squaee become trapezoid

@someguy7723 2 жыл бұрын

Its why he was killed, big equations couldn't have someone out there disrupting their marked

@Banzybanz 2 жыл бұрын

Beautiful proof. In school we did the proof using similarity.

@WilliametcCook 2 жыл бұрын

Another onto the list of Garfield's many accomplishments

@VarenyaMaheshwari 8 ай бұрын

It was first proofed by ancient Indian professor - Bodhyana in 800 bce . Just imagine 😊

@jcortese3300 Жыл бұрын

Damn, that's super clever.

@TimJSwan 2 жыл бұрын

sadly, people still don't understand that condorcet voting would actually select winning candidates better.

@goauld88 2 жыл бұрын

I feel like that was already known by then...

@kittyn5222 Жыл бұрын

James Garfield was very skilled

@NoName-lu5tg 2 жыл бұрын

You guys don't know, how Indian PM explains about the origin of extra 2ab in formula of (a+b)²

@chasethescientistsaturre5009 2 жыл бұрын

This is just perfect.

@HeadCannon19 Жыл бұрын

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

@TONIO-ru4iu 2 жыл бұрын

MR GARFIELD! YOU ARE A GENIUS!!!!

@charlesdbruce Жыл бұрын

I just posted a diagram I made for this proof on my facebook page cover photo a week or so ago! What a coincidence! I love sharing this proof with people. Garfield was president in a time where many presidents were learned men.

@JLvatron 2 жыл бұрын

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!

@aug3842 2 жыл бұрын

ye but that’s like calling someone who says “i bought 2 dozen eggs” a mathematician

@JLvatron 2 жыл бұрын

@@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!"

@JLvatron 2 жыл бұрын

@@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!

@user-lh5hl4sv8z 2 жыл бұрын

@@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.

@Dude408f 2 жыл бұрын

Not sure if that (4x20) would be considered verb "math"