Multiplying by 2, Getting Closer to 0 | p-adic numbers (p=2)
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- Опубліковано 17 гру 2024
- [Note] p-adic numbers and base-n numeral systems are distinct concepts.
The base-n numeral system (such as binary, decimal, etc.) is merely a matter of notation, while p-adic numbers represent a number system fundamentally different from real numbers.
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#math
[Note] p-adic numbers and base-n numeral systems are distinct concepts.
The base-n numeral system (such as binary, decimal, etc.) is merely a matter of notation, while p-adic numbers represent a number system fundamentally different from real numbers.
I always liked thinking about them as mod infinity numbers.
do you think it's a valid analogy, I think it gives me a better intuition but I also feel I can be just wrong.
ok perhaps mod infinity is not what I wanted to say. maybe it's better to compare with the complement of 2 that we use to represent negative numbers in computers so p-adics are complement of infinity... that's also a second way I think about it.
@@BleachWizz No, you're right. The P-adics are the inverse limit of the intergers mod p^n.
You have the rings of integers mod p, mod p^2, mod p^3, etc. and you take the inverse limit of this sequence, and you get the p-adics. So yes, you can think of it that way, that you're taking the integers mod p and continuously expanding it until you reach the p-adics at the limit, like it's mod p^infinity.
The integers mod p^n are just the truncation of the p-adics to n digits (in base p). It's basically similar to how we only deal with terminating decimals to 2 places (when it comes to stuff like money), though not quite since that wouldn't be a ring. (it would be though if we took all the terminating decimals, but that's going off-topic)
2 adic numbers have strong relationship with collatz conjecture.
If you are curious about it please recomment me for more detail.
P-adics are very unique alternative number systems that makes some computations easier too represent and do than in normal Reals.
@@f5673-t1h Does it make easier to work with elliptic curves over finite fields (first that came to mind)?
Watching this instead of studying for my stats final
STATISTICS?!
*_GOD BLESS STATISTICS🦅🦅🦅_*
This channel probably has stats videos
This channel only has pure maths vids lmao
Literally in the exact same situation. my stats final is tomorrow so that's even worse lol. Still, worth it to get a break from statistics for some... math.
I watched interstellar instead of studying for my physics final :p
I maybe p-addicted to Zundamon's videos
is that Lancer deltarune?!?!!?!
@ak_the_gr8 Oh, I'm not Lancer! I'm just a sweet little boy!
This is one of my favourite math channels ever
same
-1/12 is approaching
Ramanujan...
That's actually my suggested video right now (specifically, "The Return of -1/12" from Numberphile).
@@MsGinko put some misleading title like Ranma 1/2
That’s 1+2+3+4+5+…
@ I found the numberphile vid but probs can't post link
WAKE UP CHAT, NEW ZUNDAMON'S THEOREM VIDEO‼️‼️‼️‼️‼️
How is it possible that I didn't know before that there is a version of this channel in English, I only knew it in Japanese,
I'm so glad I found the channel
12:02 Most notably, the two's compliment number system used by computers is a truncated 2-adic number system.
Better than the veritaserum video
Glad I'm not the only one who thought this
I'd say watching Veritasium is fine if you have some prior knowledge, while Zundamon's Theorem is more focused on slowly exploring stuff from zero, accompanied by anime girls 🤡
@@HuyTheKiller zundamon isn't an anime girl. Has the same art style but she's from vocaloid/neutrino/I think cevio now
@@m4rcyonstation93 Weeelllll, aaakkkttuuuaahhllyyyyy🤓☝
I love eric rowland’s video about p-adic numbers, veritasium is bad at explaining things in general so I’ve stopped watching. Its unfortunate cuz they used to make great vids but now they’re meh. This channel is great tho
Note to self:
Always bring Zundamon with you to a dungeon.
Mathematics may call this p-adic numbers but programmers may have seen something similar before known as unsigned integer overflow :)
This is such a great video coming from a recent math bachelor. What makes these videos so great is the fact that Zundamon shares the exact same thoughts I do while watching the video, questions about rigor and epiphanies all corresponds to my thoughts while watching the video. So it's almost as the video is reading my mind.
thank you as always for your hard work zundamon’s theorem en! absolute cinema from the goat 🐐🐐🐐
I really like the small story elements during the video, they're short, add lore and engagement to the video, I think they're great! Please continue doing them!
These videos scratch a very weird part of my brain, thank you guys for another great upload
I'm commenting to get a hi from the creator of this amazing content!
What's interesting to me is that the infinite geometric series formula seems to, in a sense, "diverge" if |p|infinity doesn't make sense in the p-adic numbers); so in this way, the p-adics really are the complete opposite of the real numbers.
Yet p-adics contain some real numbers, so they're not really the opposite 🤔.
i love zundaemon theorem
BABE WAKE UP, ZUNDAMON JUST DROPPED🗣🔥🗣🔥
"That's impossible"
She is just like me frfr
wtf , why this channel is so good?
The Most Beautiful Equations:
Euler: Euler's Identity
Einstein: Mass-Energy Equivalence
Zundamon: Geometric Series Formula
What is so beautiful about them?
another amazing zundamons theorem upload
I actually love these videos. I'm a maths student myself, but some of these concepts have either not been taught yet or are considered irrelevant to the course. This is, somehow, the first I've learnt about how p-adic numbers work (despite _using_ them in group theory)
Anime girls and math, the two best creations of mankind
Adic number systems are beautiful. I remember my first time learning that ...999 + 1 = 0 it blew my mind
You know what, at 4:50 "In the middle of discussion, what is it this time?" and here came 2 ads
2:59 PI BACKWARDS? 😶🌫️
Well....a slice of pie, not the complete pi.
@ yeah, i know. thats where it stops
I just stumbled upon this channel and I'm glad I did. You guys create amazing videos and i can actually understand maths through these than from other videos.
Hope you guys will continue uploading amazing videos like this :)
I love ur vids and especially the lore so much pls never stop
What blows my mind is that some p-adic numbers can equal the imaginary unit!
Beautiful video - I keep hearing "p-adic" being name-dropped, but never actually learned about them. I'm curious: would I be reading too much into things if I suspect an intentional pun at 0:19? Namely, when "there's no choice", i.e. in ZF without countable choice, there's no way to disprove [assuming ZF's consistency] the existence of an infinite sequence of two-element sets whose Cartesian product is empty - matching the equation on the screen!
Please make this a fun channel that can provide context or understanding about math that is acceptable to the general public, in other words please teach people about math at an advanced level but still acceptable even for someone who failed math class
"We can't move until we solve this mystery"
What's this, 7th Guest and Professor Layton?
Babe wake up, new Zundamons Theorem video just dropped.
Zundamon and Metan want to break the Internet. Another great video!
My brain aint braining, help(;-;)
i love p-adic numbers, they give us an idea on how modular arithmetic can be used to solve diophantine equations
1:42
We're not taking this result seriously enough.
we are witnessing modern turning a sphere inside out
they feel somehow similar to modulo p arithmetic. For example that sum of 10... = -1 in modulo 2 arithmetic too
There is at least a connection.
One way is to compare 2adic numbers to signed integers with 2s complement. There, it is the same as numeric value modulo integer limit. For example, -1 mod 65536 = 65535 = 0xFFFF (all 1's).
The other way is to store "1/3" or similar in a unsigned integer. When you have a 16 bit unsigned integer with 0xAAAB and multiply it with 3 you get 1. This is just a padic that is cut after 16 binary digits. And 0xAAAB = 43691 which is the multiplicative inverse of 3 and 65536 (0x10000).
10:10 The fact that p needs to a prime number, is it somehow related to the fact that any interger can be formed by the product of primes ?
For normal numbers? No
For piadic nimbers? Yeah
watching this right now instead of studying for my calculus finals 💀😭🙏
wait, why am i here? i have a math test tomorrow, and i know these knowledge are not going to be useful in the test 💀
just saw title. ah yes, p-adic numbers
a series on exploring the multiverse so vast that there's always gotta be somewhere weird thing is somehow true
Just to clarify: The real numbers and the p-adics are completely separate things. Don't go into your calculus class saying stuff like "2^infnity equals zero", because you're working with reals in calculus, not p-adics.
The two arise in the same way from the rationals: You define a way to measure distances between rational numbers, and then fill in the gaps. The only difference is how you defined distances.
From there, you get different definitions of convergence, and so series (that only have rational terms) that converge in one, may not in the other. (1+2+3+... and 1+1/2+1/3+... still diverge in both)
I specify "that only have rational terms" because the numbers that exist in one may not exist in the other. Like how the 5-adics have sqrt(-1) but not sqrt(2), the opposite of the reals, while the 3-adics have neither. Moreover, p-adics have their own transcendentals with no analoge in the reals (just like the p-adics don't have pi nor e).
The p-adics also have some strange (and more beautiful) behaviors. For example, series with terms that converge to zero, don't necessarily converge in the reals (e.g. 1+1/2+1/3+...). But in the p-adics, they DO necessarily converge (I'm not saying 1+1/2+1/3+... converges in the p-adics, read what I said carefully. The terms don't converge to anything under the p-adic distance).
So in the p-adics, a series converges if and only if its terms converge to zero. A statament that only holds in one directoin in the reals rather than both.
You also have some funky stuff, like how all triangles are isosceles in the p-adics. You take any three numbers in the p-adics, and two of the pairwise distances are equal. Like 1 5 6 in the 5-adics, the distances are 1/5 between 1 and 6, but 1 between the other two pairs. So they make an isoceles triangle.
When I saw the thumbnail I was thinking like this:
2^inf = 2 x 2 x 2…
(2^inf-1) 2 x 2 x 2… / 2
(inf - inf) = inf
2^inf = 0
Who is watching this before 16 hours of final exam 😂
Wow, what a cool channel!
After Eric Rowland and Veritasium, now Zundamon presents p-adic numbers 👀.
(I still think that Eric Rowland's video is the best to explain the p-adic as he goes into much more details 😅…)
2^(+inf.)=+inf., but 2^(-inf.)=0.
huh... the hallway and numbers have swapped sides from the JP iteration of this video. wonder what's up with that?
It's probably related to the fact that Japanese is written right to left whereas English is written left to right. Although google says that left to right is becoming more common in Japanese writing.
@sponk13Japanese is written top to bottom (vertically), right to left. And left to right is not becoming more common. We right left to right when we write horizontally and right to left when we write vertically.
It’s written vertically in most non-STEM books. It’s almost never written vertically in the STEM fields because it’s hard to write mathematical formulas vertically. Other than that, in handwriting, it’s up to preference.
@@vonneumann6161 I stand corrected. I’m glad someone who knows more could clarify. I had now idea it was so specific to context.
thank you both for clearing things up for me!
Lets wait till zundamon starts to teach the korean language
Ando bien pedo viendo esto xd, pura calidad
Math and anime girl... The youtube algorithm really gets me
Do they have more similar properties to the real and hyperreal numbers if you use an omegadic (ω-adic) number system?
Also, is a googolminex (10^-(10^100)) considered an extremely large number in polyadic systems? And, can there be a system where digits can extend infinitely on both sides? And, are there hyperreal polyadic systems where you can have digits at ω or more places on the left? And, is there an equivalent for the real numbers?
BTW, an unexplored topic in this video is how you can generate fractions as well as negative numbers, like 1/2=1̅2₃ (...11112)₃. (In the real number system we have 1/2=0·1̅₃=(0·1111...)₃.) Can you also generate irrational numbers by having a non-repetitive digital sequence on the left side?
Thanks but now I feel more confused 😭
There were some numbers where ab=0 and a,b≠0 but i forgot its name
But what are the practical applications for this?
In the end she is saying about quantum mechanics
I didn't get even a single thing in the vid 😢. Time to google
I might be wrong. But we are just using a different distance function innit?. In 2-diac numbers, the powers of 2 converges to zero since it becomes close to the 0 according to some weird distance function. That's why we get the wierd infinite digits at the right side.
[I am just pulling it out of hat. I am just guessing]
I am guessing for 1/(2^n) it grows without bound
nice story!
So, we just assign unique numbers in p-based numeral system to all real values from 0 to 1?
Im bad at math so idk why im watching this
Wonderful channel. Underrated one definitely
Really simpler?? Exponential function in p-adic is not even continuous!!!
If this is true then
infinity + negative infinity= 0
Why is 10000000...(2) equal to 0(2)? Why did 1 on the left side disappear?
Because it goes infinitely to the left
Because we define it to be this way in this context. To understand how things are defined, and thus how they work after that, I encourage you to research how they are mathematically defined. This video's goal is to explore intuitively how to manipulate such (man-invented) concepts, I think it does a great job at it. For the full story, it's about how to define these notions mathematically and what more we can do with them.
because we define it as the limit of a sequence of numbers that, under a certain notion of "distance" (the p-adic norm), gets arbitrarily close to 0. much like how in the real numbers, .0000...0001 is treated as being 0, because the sequence that defines it gets arbitrarily close to 0 under the traditional absolute value notion of distance
rgp maker ahh music hehehe
Math bros why we are here just to suffer?
Correct me if I am wrong, but saying 1.... 0000 .... 000_(2) ---> 0 in the limit does not seems correct to me
em -1.
3
How.
yippe new vid
I... uh
3 i
Aleph-1 😎😎😎😎
Formula: 2^(n+1)=2ⁿ×2
If 2^∞ is 0, than 2^∞-1 is also 0, since 0/2 is 0, and it continues to 2¹=0 which is not true since 2^1 is 2.
So 2^∞=0 is a fake statement. Nice try.
" 2^∞ is 0, than 2^∞-1 is also 0 " wrong. If you has defined 2^∞ = 0 then 2^∞ -1 = -1, not zero. Fail analysis.
You can’t “continue to 2^1” because you have to do it infinitely from 2^inf
We work in a, self-imposed arbitrarily defined space of numbers. In the real numbers "2^infty" obviously does converge to infinity and not 0. Then it's about how things work in the way we have defined them to be, which might be different from usuals. So "2^infty = 0" is a fake statement in the usual real numbers, but we specifically work in a different way here.
Also, your argument is wrong in the real numbers too. 2^-(n+1) = 2^-n * 1/2 and "2^-(infty)" does indeed converge to 0 but it doesn't mean we can 'continue' to 2^-1 = 0 which would be false in the reals. The argument that we can 'continue' is not working anyway.
Zundamon is so goated ong 🧌