Mathematics: figuring how to figure (...or not)

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Posted by Hervé

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https://www.sott.net/image/s20/41125...prikoly_31.jpg

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It makes perfect common core sense...

Source: https://youtube.com/watch?v=6blD-cfko1A

:)

But more seriously ... I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.


BTW that "A+ for creativity exam" -- that was certainly me in school. Yes I was a "smartass". :) (referring to this post: https://projectavalon.net/forum4/show...=1#post1215887)

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Posted by DeDukshyn

I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.

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Depends where the brackets are! In other words, what you do first, and what you do next.

• (6÷2)x(1+2) = 9.
• 6÷(2x(1+2)) = 1.

:focus:

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Posted by Bill Ryan

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Posted by DeDukshyn

I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.

---

Depends where the brackets are! In other words, what you do first, and what you do next.

• (6÷2)x(1+2) = 9.
• 6÷(2x(1+2)) = 1.

:focus:

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wow i feel old, I learned the 100 years old method Aye...

It might be the same for all my expectation, including knowing how to write and read properly....

As for patience, see the thread on narcissism, which has subtantially increased since year 2000 in young people, which means that patience and educating becomes a must

https://projectavalon.net/forum4/show...=1#post1216193

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Posted by Flash

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Posted by Bill Ryan

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Posted by DeDukshyn

I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.

---

Depends where the brackets are! In other words, what you do first, and what you do next.

• (6÷2)x(1+2) = 9.
• 6÷(2x(1+2)) = 1.

:focus:

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wow i feel old, I learned the 100 years old method Aye...

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Its actually wrong. 9 is the correct answer because when it comes to multiplication and division you have to move from left to right - whatever comes first; in BEDMAS for example, D and M are actually equals. Most people answer this as 1 - so you are not alone. :) But ... it made me realize that math is not an absolute -- it is bound by an agreed upon set of rules, as opposed to an unchanging fundamental.

:focus:

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Posted by DeDukshyn

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Posted by Flash

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Posted by Bill Ryan

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Posted by DeDukshyn

I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.

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As far as I am concerned the answer will always be 1

Parenthesis are always done first

Depends where the brackets are! In other words, what you do first, and what you do next.

• (6÷2)x(1+2) = 9.
• 6÷(2x(1+2)) = 1.

:focus:

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wow i feel old, I learned the 100 years old method Aye...

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Its actually wrong. 9 is the correct answer because when it comes to multiplication and division you have to move from left to right - whatever comes first; in BEDMAS for example, D and M are actually equals. Most people answer this as 1 - so you are not alone. :) But ... it made me realize that math is not an absolute -- it is bound by an agreed upon set of rules, as opposed to an unchanging fundamental.

:focus:

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The answer seems obvious to me.
You must work out the parenthesis first
the order of operation is , in this context:
parenthesis, multiply, divide
you don't work out the parenthesis and then put a temporary parenthesis around it so you can divide the 2 and multiply the three

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Posted by Ernie Nemeth

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Posted by DeDukshyn

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Posted by Flash

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Posted by Bill Ryan

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Posted by DeDukshyn

I ran into this one recently and it made me really think ... and I began to lose a little faith in the consistency that mathematics can always be relied upon for. Apparently, at one point in time this mathematical equation had a different answer than it does today ...

6 ÷2(1 + 2)

The correct answer in this day, is 9.
At one point in time (almost 100 years ago, I believe) the correct answer would have been accepted as 1.

---

As far as I am concerned the answer will always be 1

Parenthesis are always done first

Depends where the brackets are! In other words, what you do first, and what you do next.

• (6÷2)x(1+2) = 9.
• 6÷(2x(1+2)) = 1.

:focus:

---

wow i feel old, I learned the 100 years old method Aye...

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Its actually wrong. 9 is the correct answer because when it comes to multiplication and division you have to move from left to right - whatever comes first; in BEDMAS for example, D and M are actually equals. Most people answer this as 1 - so you are not alone. :) But ... it made me realize that math is not an absolute -- it is bound by an agreed upon set of rules, as opposed to an unchanging fundamental.

:focus:

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The answer seems obvious to me.
You must work out the parenthesis first
the order of operation is , in this context:
parenthesis, multiply, divide
you don't work out the parenthesis and then put a temporary parenthesis around it so you can divide the 2 and multiply the three

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Ah! but many people, including myself were taught to use BEDMAS - brackets, exponents, divide, multiply, add, subtract, for order of operations. If you follow this strictly, you'll get 1 as the answer. If you know that divide and multiply are equal and need to follow order from left to right, it matters not which of these is first. Same goes with add and subtract. but since people tend to have learned by BEDMAS as the order, they will end up with the wrong answer if following it specifically. The oddest part I found, was that at one point 1 would have been accepted as the correct answer, signifying that math has agreed upon boundaries that define "correct" answers.

I was going to apologize for the off topic and suggest a new thread ... lol then I realized that already happened. (didn't notice at first responding to a quote notification)


EDIT:

Actually, I'm wrong on my order theory of why people get wrong ... If people followed BEDMAS they would get it right ... don't know why most people get this wrong then then ... weird ..

So...

The answer is ... 7!

Okay! So now I've moved all these posts to a new thread, I can respond to DeDuksyhn without going wildly off-topic on the Silly Season thread where they came from.

:)

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Posted by DeDukshyn

it made me realize that math is not an absolute -- it is bound by an agreed upon set of rules, as opposed to an unchanging fundamental.

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Nope. Math is absolute.. it's the notation that may not be. (Or that may be confusing, or misunderstood, or even possibly ambiguous.)

Math notation is just a set of instructions, like computer code, to tell the reader what to do. And that has a language. Language, like a spoken language, has to be agreed. If I say 'table', and you think I mean 'elephant', we've got problems. :)

That math language has certainly changed a LOT over the centuries, of course. One reason why English math was in the doldrums for a century after Isaac Newton's groundbreaking invention of methodical calculus was that his German contemporary Leibniz, who also invented the same thing at almost exactly the same time, used notation that was MUCH more user-friendly and easier for everyone else to understand.

(That's because as all this was new stuff, they each had to invent the notation as well.)

The underlying truths were identical. But it was the agreed user-friendly notation that really made the difference.

I used the example above that ETs would have the same math – of course. But then when they're writing down what 6 divided by 2 times 1 plus 2 is, we have NO idea what that would look like or precisely what they meant.

If you really want to scratch your head and stare at the wall, watch this short video: a proof that 1+2+3+4... (to infinity) = -1/12.

https://youtube.com/watch?v=w-I6XTVZXww

My favorite YouTube comment:

Dear God,
I'd like to file a bug report. (see attached video)
Amen.

:bigsmile:

Yup, we're all taught infix notation and precedence rules. In postfix notation, "6 ÷2(1 + 2") looks like this: "6 2 1 2 + * /" No ambiguity, no precedence rules, each operator requires exactly 2 operands and the answer is 1, but it's not user friendly, because it's not how we're taught and we're not used to it.

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Posted by Bill Ryan

Okay! So now I've moved all these posts to a new thread, I can respond to DeDuksyhn without going wildly off-topic on the Silly Season thread where they came from.

:)

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Posted by DeDukshyn

it made me realize that math is not an absolute -- it is bound by an agreed upon set of rules, as opposed to an unchanging fundamental.

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Nope. Math is absolute.. it's the notation that may not be. (Or that may be confusing, or misunderstood, or even possibly ambiguous.)

Math notation is just a set of instructions, like computer code, to tell the reader what to do. And that has a language. Language, like a spoken language, has to be agreed. If I say 'table', and you think I mean 'elephant', we've got problems. :)

That math language has certainly changed a LOT over the centuries, of course. One reason why English math was in the doldrums for a century after Isaac Newton's groundbreaking invention of methodical calculus was that his German contemporary Leibniz, who also invented the same thing at almost exactly the same time, used notation that was MUCH more user-friendly and easier for everyone else to understand.

(That's because as all this was new stuff, they each had to invent the notation as well.)

The underlying truths were identical. But it was the agreed user-friendly notation that really made the difference.

I used the example above that ETs would have the same math – of course. But then when they're writing down what 6 divided by 2 times 1 plus 2 is, we have NO idea what that would look like or precisely what they meant.

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6 ÷ 2(1 + 2) = 6 ÷ 2 + 4 = 3 + 4 = 7

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Posted by Bill Ryan

If you really want to scratch your head and stare at the wall, watch this short video: a proof that 1+2+3+4... (to infinity) = -1/12.

https://youtube.com/watch?v=w-I6XTVZXww

My favorite YouTube comment:

Dear God,
I'd like to file a bug report. (see attached video)
Amen.

:bigsmile:

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Hi Bill.

I thought we are here to make sense of things. Regardless of what these scientists claim and despite what numbers continue to appear in physics, the logic employed is wrong.

The first reduction of S is an average. An average is not an answer. It is not even an approximation. It is merely acknowledging the limits of time-bound intelligence. We cannot count to infinity. If a universe is populated by only a 1 then no matter the operations there will always be the one. This short-circuit of reasoning employed in the video and hoisted on our most brilliant minds is the definitive proof that our understandings are purposely bound to contain our thinking.

Just because we do not understand the concept of infinity does not mean little games can be played with numbers to make it more palatable.

So, we know 1+ 2 = 3. That answer and a bit of logic proves that their answer cannot be right because the answer must be larger than the first sum.

And now we can see that the first average is completely wrong leading to a mess by the end of the calculations. Much like our world where morality gets modified by ethical considerations. A false premise at the start leads to increasingly larger and larger mistakes.

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Posted by Ernie Nemeth

A false premise at the start leads to increasingly larger and larger mistakes.

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Exactly. :) It was all in fun, though done semi-seriously — because there are some really interesting mathematical issues in play here.

Such chaos ensued among the comments — and many of them were extremely funny — that the guys at Numberphile, who made the video, made another much longer one, sort of in apology, to discuss the ins and outs of convergent or divergent infinite series.

Basically, to risk summary in a sentence, if an infinite series is NOT convergent (i.e. doesn't add up to a finite number), it has to be handled with a great deal of care since many 'rules' for finite entities cease to apply.

But all this is at the very heart of advanced math, so it's all legitimate discussion: and if it makes intelligent YouTubers think hard for a little while, that surely has to be no terrible thing. :)

Just ran across this in one of Jayke's posts...."today scientists have substituted mathematics for experiments, ....and eventually build a structure which has no relation to reality." Tesla:bigsmile:

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Posted by Bill Ryan

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Posted by Ernie Nemeth

A false premise at the start leads to increasingly larger and larger mistakes.

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Exactly. :) It was all in fun, though done semi-seriously — because there are some really interesting mathematical issues in play here.

Such chaos ensued among the comments — and many of them were extremely funny — that the guys at Numberphile, who made the video, made another much longer one, sort of in apology, to discuss the ins and outs of convergent or divergent infinite series.

Basically, to risk summary in a sentence, if an infinite series is NOT convergent (i.e. doesn't add up to a finite number), it has to be handled with a great deal of care since many 'rules' for finite entities cease to apply.

But all this is at the very heart of advanced math, so it's all legitimate discussion: and if it makes intelligent YouTubers think hard for a little while, that surely has to be no terrible thing. :)

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At least I get a hearty hail before I must admit that I am incorrect (Yes, I know, but it's true!). To my chagrin I made an error in that simple math equation. Multiplying through by 2 does not solve the parenthesis. The parenthesis are there to denote that the addition is to be done first - even if you multiply through by 2.

6 ÷ 2(1 + 2) = 6 ÷ (2 + 4) = 6 ÷ 6 = 1

edit to add: syntax is everything in math

because if the equation had been:
6 ÷ 2 x (1 + 2) the answer would be:

6 ÷ 2 x 3 = 9

but since there is no operator before the parenthesis the 2 is the first operation, then the parenthesis - sheesh, made me think, though

I bow to the master.
And sneer at Dedukshyn :facepalm:

if one has not yet seen the British film "The Man Who Knew Infinity" about a young Indian man (no, not native American; from the subcontinent), an autodidactic with no formal higher education but entered Oxford and who completely revolutionized mathematics; it's truly worth viewing; if/IF info in this film is 100% the truth- I normally don't put much worth in films but this is a good view and I hope the info contained herein is true; besides it stars Jeremy Irons (one of my favorite actors) as the professor who believes in/befriends the young Indian mathematician-

Larry

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Posted by Cardillac

if one has not yet seen the British film "The Man Who Knew Infinity" about a young Indian man (no, not native American; from the subcontinent), an autodidactic with no formal higher education but entered Oxford and who completely revolutionized mathematics; it's truly worth viewing; if/IF info in this film is 100% the truth- I normally don't put much worth in films but this is a good view and I hope the info contained herein is true; besides it stars Jeremy Irons (one of my favorite actors) as the professor who believes in/befriends the young Indian mathematician-

Larry

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Many thanks, agreed, and very highly recommended. :thumbsup: :thumbsup:

Yes, the film is totally accurate. It's the extraordinary true story of Srinivasa Ramanujan, a self-taught Indian genius who was a two-time college dropout, worked as a lowly office clerk, and died at the age of 32 — having left behind dense, scribbled notes that are still keeping mathematicians busy to this day, 100 years later.

He was a devout Brahmin, and had an astonishing mathematical intuition. He said that many of his insights were revealed to him by the Goddess Namagiri. He would often do his math while listening to the drums in his local Hindu temple.

He wrote to the famous English mathematician G H Hardy, who was stunned by his work, and arranged for him to travel to Cambridge. This very unusual partnership (Hardy and Ramanujan, who could scarcely be more different from one another) is the storyboard of most of the film.

The book The Man Who Knew Infinity is here:

• http://avalonlibrary.net/ebooks/Robe...0Ramanujan.pdf

... and the film is here:

• http://avalonlibrary.net/The_Man_Who..._Ramanujan.mp4 (850Mb)

:star:

@Bill

sorry, my "misteak," Bill: yes, it was Cambridge, not Oxford!

There's nothing wrong with the video for this sum at all, it's just poking beyond the scope of the fabrication of our reality, which is based on certain types of infinities. There are many types of infinities and it can drive a mathematician mad studying them too deeply (Georg Cantor).

If this were not a simulated reality, these kinds of anomalies would not show up. But for the very reason they show up in the physics hints at how ongoing study of physics will indeed begin to reveal the source code of reality. This one is not alone, it shares its position with Euler's identity and many others, some discovered already, many not. Uncanny simplicities hinting towards a kind of order that we are still largely unaware of.

Factors like the 1/12th hint towards the divisions of the musical octave as being more important than one might think, as also the inversion into negative numerical space adds another clue.

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Posted by Bill Ryan

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Posted by Cardillac

if one has not yet seen the British film "The Man Who Knew Infinity" about a young Indian man (no, not native American; from the subcontinent), an autodidactic with no formal higher education but entered Oxford and who completely revolutionized mathematics; it's truly worth viewing; if/IF info in this film is 100% the truth- I normally don't put much worth in films but this is a good view and I hope the info contained herein is true; besides it stars Jeremy Irons (one of my favorite actors) as the professor who believes in/befriends the young Indian mathematician-

Larry

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Many thanks, agreed, and very highly recommended. :thumbsup: :thumbsup:

Yes, the film is totally accurate. It's the extraordinary true story of Srinivasa Ramanujan, a self-taught Indian genius who was a two-time college dropout, worked as a lowly office clerk, and died at the age of 32 — having left behind dense, scribbled notes that are still keeping mathematicians busy to this day, 100 years later.

He was a devout Brahmin, and had an astonishing mathematical intuition. He said that many of his insights were revealed to him by the Goddess Namagiri. He would often do his math while listening to the drums in his local Hindu temple.

He wrote to the famous English mathematician G H Hardy, who was stunned by his work, and arranged for him to travel to Cambridge. This very unusual partnership (Hardy and Ramanujan, who could scarcely be more different from one another) is the storyboard of most of the film.

The book The Man Who Knew Infinity is here:

• http://avalonlibrary.net/ebooks/Robe...0Ramanujan.pdf

... and the film is here:

• http://avalonlibrary.net/The_Man_Who..._Ramanujan.mp4 (850Mb)

:star:

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One of my favourite anecdotes, occurred while Ramanujan was ill. GH Hardy happened to mention the number of his taxi (1729) that brought him to Ramanujan's residence as being quite a boring number. To which Ramanujan replied that 1729 was not a boring number at all: it was a very interesting one. He explained that it was the smallest number that could be expressed by the sum of two cubes in two different ways: 12 cubed + 1 cubed; or 9 cubed + 10 cubed both equalling 1729.

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Posted by happyuk

One of my favourite anecdotes, occurred while Ramanujan was ill. GH Hardy happened to mention the number of his taxi (1729) that brought him to Ramanujan's residence as being quite a boring number. To which Ramanujan replied that 1729 was not a boring number at all: it was a very interesting one. He explained that it was the smallest number that could be expressed by the sum of two cubes in two different ways: 12 cubed + 1 cubed; or 9 cubed + 10 cubed both equalling 1729.

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Yes, a delightful true story, also in the film. Hardy marveled that anyone could just happen to know such a thing. His Cambridge colleague John Littlewood, who's in the film as well, famously remarked prior to the incident that "every positive integer was one of Ramanujan's personal friends."

:)

Thanks for posting the film, Bill......truly amazing!! :flower:

I'm more of a words person than numbers, but I thought this was fascinating. I've never before seen the arithmetic technique of the Chinese girl on the right. They didn't teach this method when I was at school, and that's a shame. It's execution is both simple and elegant.

To understand how she did it, see under tweet.

https://x.com/gunsnrosesgirl3/status...48139720561149

Method:
100-97=3
100-94=6
97-6=91
3x6=18
9118

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Posted by Mark (Star Mariner)

Method:
100-97=3
100-94=6
97-6=91
3x6=18
9118

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When Carl Gauss was a tiny kid at school in Germany back in the 1780s, his teacher gave the class an assignment which he figured would keep them occupied for half an hour or more. He told them to add up all the numbers from 1 to 100. Like: 1+2+3....+98+99+100.

Instantly, little Carl wrote the answer on his slate: 5050. His teacher scolded him for his insolence — until he realized that 5050 was correct.

What Carl had done was add up 1+100 (=101), 2+99 (=101), and so on, all in pairs, then multiplied 101 x 50, because there were 50 pairs like that.

Very simple — but Carl had worked it out on his own. With an IQ in the 200 range, Gauss would grow up to become one of the greatest mathematicians to have ever lived.

This fun conversation was had between scientists Alexander von Humboldt and Pierre Laplace, both Gauss's contemporaries.

Humboldt: “Who is the greatest mathematician in Germany?
Laplace: “Johann Pfaff.”
Humboldt: “But what about Gauss?”
Laplace: “Oh, Gauss is the greatest mathematician in the world."

:focus:

Amazing.

Context:
Aaryan Shukla, a 14-year-old math prodigy from Maharashtra, India, has amazed the world by setting a Guinness World Record for mentally adding 100 four-digit numbers in just 30.9 seconds. Known as the "Human Calculator," this teenage genius showcased his extraordinary mental agility during a competition in Dubai, where he broke six records in a single day. His lightning-fast calculations are a testament to years of practice and a natural talent that emerged when he was just six years old.

https://x.com/TheFigen_/status/1901691679171850578