Can any Number be a Base?
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- Опубліковано 2 чер 2024
- There are many different ways to express numbers. The most popular is definitely the decimal system, or in other words base 10. Base 2 and base 16 are also used in computers. But did you know that we can make number bases not only from integers?
Chapters:
00:00 Introduction
02:22 Base 1
03:12 Negative bases
04:34 Fractional bases
10:06 Irrational bases
15:10 Imaginary bases
16:27 Small error here. You say "21 + 2i", but it is written "21i + 2".
*16:26
similar issue at 18:28 where he calls -8i "negative real"
Agree.
@@wham_sandwitch !?!?!?
@@wham_sandwitch,
My objective here is to point out a "minor" error that appeared in the video, with the aim of potentially correcting it to avoid confusion, especially when the content involves mathematics. I'm doing this in a constructive manner.
So, stop taking offense on behalf of others.
That's "based"
Base 10. 37568
Base 6/4. 4200020200002002004402
badoosh
Based on what?
thank you based god
Damnit i was gonna say that 😂@@Tartarus4567
and before this i didn’t think my number universe could get any bigger…. thanks!
look up "apeirology" and "googology", thank me later
Well. Have you heard of j and k numbers?
if you mean quaternions, i remember having a similar experience!
Actually it haven't gotten bigger it's just a new way to write old things...
have u heard of p-adic numbers
Interestingly, tally marks or even just counting with your fingers are an example of base 1, and probably the oldest number system we have. Roman numerals were also derived from tally marks, and they could be considered an example of a system with multiple bases, where the auxillary bases are 5 and 10
You tecnically can count to 1024 on fingers because it is possible to interpret finger position as binary.
And if you assume intermediate states, then even tertiary is possible which allows to count up to 59'049
@@DimkaTsv , yeah, the problem with that is that it is too much to be able to realistically keep track of and definitely too much to be able to recognize. Even with your fingers moving up and down to help, trying to keep track of what you are counting while concentrating on intricate finger movements will be virtually impossible as you continue for several hundred or even thousands. Even making tally marks, which is a far easier task, can make you lose track at such high numbers. But even worse would be trying to recognize what number is being represented. Say you asked me how many people I counted coming into the stadium for an event. I hold up my hands with my left pinky halfway up, my left index and thumb fully extended, my right index and middle halfway up, my right ring fully extended, and my right pinky, due to how my hands work, is potentially halfway up or trying to stay down. What number would that be? Before you even start to work it out, you have to ask how I was counting. Did I start from the right so it looked left-to-right readable for me, or did I start from the left so it would be left-to-right readable to other people? And, because this is positional, how were my hands in relation to each other? Did I have my hands facing away from me (to start and end with pinkies), facing towards me (to start and end with thumbs), or one facing toward while the other faced away (to make the smallest on each hand consistent with either pinky or thumb)?
All that to say, if you really need to count *that* high, there are far better methods than using fingers.
--addendum: Now that I think about it, you could use those states of your hands to encode even more numbers (using those four possibilities I listed as a leading 0, 1, 2, and 10 to get all the way up to 236,196), but, seriously, why would anyone want to?
@@SgtSupaman that is why i said "technically". It doesn't mean that counting in such way is efficient or practical at all.
think of the hour:minute:second format, where every digit space has a different base.
well,other than the minute and second marker, but the hour and millisecond markers have different bases.
I can see base pi being useful for trig. imagine cos(10)=-1 and sin(10/2)=1, etc. Also sum of reciprocal squares would be 100/(whatever 6 would be)
I mean we do basically use base pi for trig already. We just do it in a way where we can still use base 10 but also make it obvious we are counting in increments of pi. Ie sin(n pi).
base tau (2pi) could be better
measurement in radians comes close to what you are saying imho.
@@emremokokoand better. Because the pi oftentimes cancels out during calculations
Radians
my favorite type of number system that wasn't brought up here is factoradic, where instead of having one radix that you keep squaring, you take each digit as the next factorial, so each position can range from 0 up to the position number. to give you a feel for the system, here's some numbers counting from 0 to 24:
0, 10, 100, 110, 200, 210, 1000, 1010, 1100, 1110, 1200, 1210, 2000, 2010, 2100, 2110, 2200, 2210, 3000, 3010, 3100, 3110, 3200, 3210, 10000, and so on
to go back and forth it's very similar to a normal base; for example to render 5835241010(!) into base 10 you would do:
5*9! + 8*8! + 3*7! + 5*6! + 2*5! + 4*4! + 1*3! + 0*2! + 1*1! + 0*0!
=1814400 + 322560 + 15120 + 3600 + 240 + 96 + 6 + 0 + 1 = 2156023
Is there a number system where you represent numbers as a sum of sqyare numbers?
The final 0 of each number seems to be redundant.
Btw: What about fractions in this system? I.e. what would be the meaning of digits to the right of the "decimal" point?
@@user-gd9vc3wq2h yes, the first digit is in 'unary' so it can only take one value, 0.
the most logical way to do fractions would be to count the other way, but factorial is undefined for negative numbers so mathematicians came up with a smarter way. if you have each digit after the decimal be 1/n! then you can represent any fraction with a number of digits equal to the denominator+1 or less. (on this end, 1/0! and 1/1! both evaluate to 1 and so neither can be anything but 0. like that ending 0 you mentioned before, they are sometimes just omitted but I will include them for completeness' sake). examples:
1/2 = 0.001
1/3 = 0.0002
1/4 = 0.00012
1/5 = 0.000104
1/6 = 0.0001
1/7 = 0.00003206
1/8 = 0.00003
1/9 = 0.0000232
1/10=0.000022
1/11=0.00002053140a
1/12=0.00002
any multiples are multiples of those just like any other system. to get an idea of what's happening here in your head: each number 1/n starts at the 1/n! position, and the number that goes at that position is (n-1)!. so a third starts at the 1/6s place and 1/3 is 2/6; or a fourth starts at the 1/24s place and 1/4 is 6/24(since 6 > 3, the biggest digit at this place value, you carry over to the next place and subtract 4, like how in addition if you add 8+8 you carry a 1 to the 10s place and put 16-10 in the ones place). the numbers work the same way going the other way; that is the 3rd digit can be 0 or 1, the 4th digit can be 0, 1, or 2, the 5th digit can be 0, 1, 2, or 3, etc.
@@user-gd9vc3wq2h also if you use this system for fractions a handful of transcendental numbers have fun decimal expanions:
e = 10.0011111111111111111...
sin(1) = 0.00120056009A00DE00HI00... (each group is +4)
cos(1)= 0.0010045008900CD00GH00...
sinh(1) = 1.0001010101010101010101...
cosh(1)= 1.0010101010101010101010...
it is called factorialadicpoint man 😂
For a number base a/b, you use digits 0..a-1, but that allows numbers to be written many different ways. In fact, you only need 0..floor(a/b).
Let's say for your example of 265, converted to base 7/3. You write it as 64366. But you could also write it as 1110020.001...
Similar for other non-integer bases.
According to this logic, 1 can also be written as 0.999999999… in base 10
(which is correct as they have been proven to be the same number)
Even that can create problems with, for example, having more than one representaion for 1.5 in base 1.5.
@@lox7182 Yes, I am aware that there are always some numbers that can be written in many ways. But I don't see a reason why one should use MORE different digits than required, right?
The video is like using decimal, but also allowing B to write eleven. No use for that.
@@supernt7852You can write every non-repeating rational number in three ways.
For example:
1, 0.999..., 1.000...
This man put so much work effort to show us the beauty of math. I’m highly appreciating your videos dude. I hope u get a good job and good life mate
There's also base φ, with digits 0 and 1, no two 1s in a row. 2 is represented as 10.01 in this base.
You can use base 2-i with the digits 0, 1, i, -i, and -1. Similarly, you can use base 2.5-√-0.75 with digits 0, 1, -.5+√.75, .5+√.75, -.5-√.75, .5-√.75, and -1.
this video is absolutely brilliant. i have been trying to figure out the implications of non-natural bases but i have never been able to figure it out myself. This video is exactly what i have been looking for for years, subscribed!
You make videos about topics I really wanted to know, but you can't really find them on the internet.
Thank you sooo so much! ^w^
Mind = blown
Respect for explaining such far out concepts in a way that is so easy to follow
Funnily enough, base 1 has a fun application where you can represent a string of numbers by having the “length” of the number represent a number in some other base, like base 9 for example, using 9 as a separator. This allows you to write any number of numbers in a string in base 1. Fun thought experiment.
I have a hard time following this explanation. Care to give an example?
@@GustvandeWal for sure! I made a terrible job of explaining but here we go with a “real world example”:
Imagine you have a typewriter with number keys and a spacebar and are tasked with writing down a string of numbers given to you. The string of numbers can be of any length and the numbers themselves belong anywhere in the set of natural numbers. If you were to find that, one day, the type writer was modified so that you no longer had a space bar, you would still be able to write down strings of numbers by converting those given to you to base 9, and using 9 as the separator. To further extend this, if you found that your typewriter now only had one key remaining, by using our base 9 rule established earlier, we can write any string of numbers as a string of numbers in base 9 using 9 as a separator, and using THAT number to represent the list using tally marks.
Example:
1, 10, 18, 27
Can be written as so in base 9 using 9 to indicate separation:
1911920930
And this number above is itself an integer that we can represent in base 1 with tally marks. That way, we can decode the original string of numbers!
@@sobhansyed4482 This just seems like the explanation of unary counting (tallying). Where is the "base 9; use 9 as a separator" part of the thought experiment?
Great content. I never thought bases could be something else than integers, but it actually makes sense.
I just spot a very little mistake at 16:25 it shows 21i+2 (which is absolutely correct) but the voice says "21+2i".
i saw that too!
what a great video, I enjoyed the insights and the production quality - thank you very much
Very well expressed and executed video. I never thought of this before. Thank you.
this is a beautiful video. the topic is so absurd but you explained it in the most understandable way possible
I came up with a numbering system that was "like" base-phi in an esoteric programing system. You could represent integers with strings of two commands: '+' to add one and '@' to redo part of the substring. It was like base phi, because it took about n log_phi commands to represent a particular integer n, similar to how it takes n log_b digits in normal base b.
May you clarify what you mean by 'redo[ing] part of the substring'?
I really liked this video! You explained in very well and the animations were fitting and eass to understand. I am really looking forward to watch more videos! Good job!
really cool video but i dont think you covered about the golden ratio base?
whats interesting about this is that if the base is the golden ratio, you get an interesting phenomenon.
(btw base golden ratio only needs 2 digits, 0 and 1)
let the golden ratio = phi
we know that phi = 1 + 1/phi
multiply both sides by phi.
we get phi^2 = phi + 1, (a(b+c) = ab + ac)
rewrite this as phi^x because we are in base phi
phi^2 = phi^1 + phi^0. (x^0 = 1)
remember that we can always multiply both sides by phi to increment all of the exponents.
its really cool cause we get 100 = 11 in base golden ratio. just something to note.
if you found this comment interesting, consider checking the combo class, another channel covering this topic and is the source of all these equations.
Oh yes combo class, the annoying guy who's constantly dropping shit and yelling, great
@@vampire_catgirlhow old are you?
Very good video. I kinda always wondered this. Good to see. I would like to see a tetration video of different group of numbers, that is a very difficult operation can be made by hand.
I remember reading several research papers in the 70's about unusual number bases. It was a long time ago, so my memory has faded, but I do remember being intrigued by negative number bases. Their main attraction is that no sign is needed to handle negative numbers. Nowadays, of course, two's complement arithmetic is so entrenched in all computers that nobody ever uses anything else.
This is so mind blowing and really well explained. I barely can believe what I see
i’m not gonna watch this video but, fractional bases… mind blown. haven’t thought about something that crazy in a while. thanks. ❤
great vid, and coolest end transition i've ever seen
Yes, I've thought about this question from time to time. And here is answer. Thank you!
This is getting really deep into number theory, had a hard time keeping up with the transformations, and still don't understand the utility if imaginary numbers or imaginary base number systems.
Also why I don't think I'll cut it as a mathematician, that being said great video, thanks for the breakdown.
Mind blown. What a great explanation!
im high af and have absolutely no business watching this but for some reason im here anyway lmao
12:55 But this problem should also happen for some Algebraic numbers. There are Algebraic numbers that you can't write in terms of radicals, for example a solution to some general quintic equation.
Although it was not explained in the video, the non-radicals work beautifully. If you have a quintic equation and some root γ, then by the fact that it is a root of a quintic polynomial p(γ)=0 you can move the 5th term to the other side and obtain γ⁵ = q(γ) where q has a lower (4th) degree. In other words, γ⁵ can be written as number with digits being the coefficients of q. So to write any number in this base, you will need maximum of coefficients of q digits.
@@mzg147 Doesn't this assume that all coefficients are integers though?
@@yanntal954 Yeah, the integer case is easier. I still think it works in the general case too, but then you have those pesky digits reversals just like with the rational bases in the video.
@@mzg147 I am not fully convinced yet 🥺
3:00 it is, however, possible to use infinitesimals as a basis. They aren't gonna be good for covering R (you could still do it if you allow infinite ordinals) but they can have neat properties such as a ε³ < b ε² for any real a, b.
This can actually be useful. I used it to calculate a sequence dependent on low probability events in the limit where the probability is 0. (This is one particular way to get the Thue-Morse sequence, and using this "infinitesimal basis" number system you can extend that to more than 2 separate states)
Wait wait wait! I want to point out a lot of things:
First we want to state what a "generalized base b" should be. L'll start with just real numbers.
I think we should ask that, with a finite set of digits (natural numbers), we want to be able to write any real number as a sequence like: ±an[...]a0.b1[...]bn[...] (where we have a finite number of "a" digits on the left of the dot and an unlimited sequence of "b" digits on the right).
This is what the "traditional" base b allows ud to do.
We trivially notice that base 1 des not work, since we are not alloed to represent 1/2 in any way (just like any other fraction).
Moreover base -1 does not work even to represent just integers with just one kind of digit...
Using a negative integer just like you said causes no problem and could be done just fine (except for -1, of course).
Up to now we should point out that each number can be written in base b in just one possible way (with the exception of "b-1" periodic, where for exmple, in base 10 we can write 3.000000... and 2.999999... and they are the same number).
With non interger numebers we have to renounce to this property, but we'll be ok with that.
Now think how to write 1/3 in base 2, it should be 0,0101010101... (and that's the only way to write it).
Now how can we write 3 in base 1/2?
It should be the reversed of the previous writing, namely: ...0101010.000...
Here we have an unlimited sequence of digits in the left of the dot, and this cotraddicts our defiinition (see above).
We could stretch that definition to include unliited sequence of digits on the left. Quite strange but ok, let's do it.
We could now prove that, using any real number b (besides 0, 1 and -1) the amount of digit we need will be the maximum between b and 1/b, rounded up. Witch is much better than your proposal, since fof 3/7 we will just need 3 digit instead of 7.
Talking now about complex numebers: it's not clear which digits are you allowed to use in case of complex numbers like 2+2i.
Since it's a fourth root of -64 I suppose you will want to allow us to use all the integer digits between 0 and 63.
If so 1 can obviously be written as 1, while i will be -0.08 (you can easily check it).
Since you can wrote any number in base -64 using those digits and you can represent those numbers in base 2+2i by using just fourth powers (like 3000500020003.00040007... instead of 3523.47... in base -64), you can write any complex number like a+ib doing the operations: x-0.08y.
I'm not sure about which complex numbers cannot be used as a base b, but I'm pretty sure that the n-th roots of 1 cannot be use, regardless of the amount of digits we will allow us to use.
I don't know if other numbers with modulus 1 can be used (and, if I have to guess, I think they could work, with the proper, finite, amount of digits) nor I can extimate the amount of necessay digits for generic complex numbers, like e+iπ.
I hope I was understandable (I'm sorry but I'm not a native English speaker) and please answer me noticing my possible mistakes.
This is a great overview!
Very interesting video.
In positional numbering systems each base has particular characteristics to them, as for example the divisibility criteria vary from one base to another. In the decimal system, for example an integer is divisible by 5 if it ends in zero or five. In a base n, n ∈ ℕ, numbers ending in zero are multiples of the base. In base π, the sine function reaches zeros in integer positions of that base: .... -2, -1, 0, 1, 2, ...
The study of mathematics today has a bias to base 10. There are many things related to this particular base. Developing mathematics using other numerical bases as a center could lead to interesting discoveries within mathematics and beyond.
Thank you very much for the video.
Your video and animation are incredible! I hope you will continue to post other video.
You're way of explaining is so great
Sres. Digital Genius, reciban un cordial saludo, gracias por ampliar los conocimientos en este tema apasionante. Éxitos.
What a remarkably beautiful system
Between delight and eye opener 😊 thnks. A meaningfullless universe lues ahead.
Wow, it took a lot of pause-replay and pencil work, but I sorta-kinda-got-it. What fun! TY.
That was a great video!
One of the craziest videos I've ever watched in my life, period. I knew how to calculate base 2 and stuff, but never even cared to think about other numbers as base. I'm absolutely mind blown, you deserve all the subs and views in the world.
For 10:00 you could just say for bases a/b, We use the digits 0, 1, …, max(a, b) - 1
The most interesting video I have seen in a long time! Now I want to know what every number is in every base! 😂
Gotta admit that your videos are so well-made! I've managed to understand each one of them so far!
Wondering what your next video will be about, but your content is very interesting! and it can make anyone learn something new! ^^
Keep up the great content Digital Genius! ;)
Already knew about the topic but the visuals are great
Your videos are useful!
You make me like the bases
Thank you !
Very much enjoyed this :]
My favorites are base Fibonacci and base factorial, but I’m not good enough to succinctly explain those here. Look them up if you’re interested! Micheal Penn did a good video on it
If we used the base 1 you suggested (basically just tally marks) we can only write natural numbers, and there would be no (functional) decimal point.
I certainly wouldn’t call that a counting base, it seems much easier to just put it with 0 as “bases you can’t count in”.
Exactly what I was thinking
If we look at any base to represent number adding 0 to the left of the number and to the right after decimal point shouldn't change the number but since 0 is the only number we can use this rule breaks here
Also if we convert any base 1 number we are multiplying every number with 0 so the answer is just 0
I just started the video and already hear about transcendental number.
I don't think I have attained such a realm of understanding yet 😅
Nice summary in the video 😃👍
woah, cool video!
this is the first time I've seen someone actually talk about base 1, I always wondered about it since everyone I know doesn't understand how bases work so they just think, "Base 1, 1 digit, boom, tally marks."
It's gonna be useful in achieving accuracy for (deep) space navigation/exploration.
This reminds me of that age-old question that has vexed many an erudite academic: how many angels can fit on the head of a pin? Not saying this of you, but this is a good example of the quasi-mystical nature of cultic mathematics. Numeromancy and Pythagoreanism are very much still alive. I did enjoy the video, though. Cheers!
Upto this point I found, You are the second person on this planet who is seriously working on base system representation. Well, In my work, I'm trying to extend this for polynomials to represent polynomials with base of other polynomials. Very nice vedio. All the best for your reasurch and future. 😊
You should look up Combo Class, they have a number very similar to this that looks at negative, square root and transcendental bases too. Fascinating stuff!
Second person? Then may I present you to imaginarybinary, an extremely underrated channel that created a very unique way of using 2i as a base.
Combo Class with Domotro is also a fun channel that often works with maths theory and number base systems
Please check out my latest video about the most difficult puzzle in the world.
@@felipevasconcelos6736Base-2i was first proposed by the legendary Donald Knuth.
it`s best video for learn english, thanks
LOL I've been making jokes about "base π" for few years and I never knew that base π was some serious stuff! Thanks for your good work❤
Math is so fucking cool. Only as an adult I’ve come to really appreciate this science and just in awe of its ability to be complex and simple at the same time.
Congratulation for you. ( x 1000 ) . Super visualization. Can I ask you, what software do you use to make these amazing mathematical visualization.
I use Python, Manim library
To represent all real numbers, the largest digit must at least be b-1. Hence, the digits 0,1,2 are insufficient for base π. For instance, 3 has the representation "3" in base π.
Note that d/b + d/b² + d/b³ + ... = d/(b - 1) Is the largest number with digits at most d and zeroes before the decimal point, but 1 is the smallest number with nonzero digits before the decimal point. If d < b - 1 is the largest digit, the numbers between d/(b-1) and 1 have no representation, in fact all numbers x where dbⁿ/(b-1) < x < bⁿ have no representation (we just shift the argument by n digits).
In particular, the number 3 has no representation in base π with digits 0,1,2, as 2π/(π-1) = 3 - (π - 3)/(π - 1) < 3 is the largest such number with 1 digit before the decimal point and π > 3 is the smallest such number with at least two digits before the decimal point.
Unless we allow digits that represent values below 0 . Such as in so-called _balanced_ representation systems.
For example, the _balanced ternary_ system is basically a base 3 representation system, but instead of digits {0, 1, 2} it uses digits that represent the values {0, 1, -1} . There is no convention for which symbol to use that represents -1 , but suppose I'll use the letter "h" for that.
So
0₃ = 0
1₃ = 1
1h₃ = 2
10₃ = 3
11₃ = 4
1hh₃ = 5
1h0₃ = 6
1h1₃ = 7
10h₃ = 8
100₃ = 9
101₃ = 10
11h₃ = 11
110₃ = 12
111₃ = 13
1hhh₃ = 14
1hh0₃ = 15
1hh1₃ = 16
1h0h₃ = 17
1h00₃ = 18
1h01₃ = 19
1h1h₃ = 20
1h10₃ = 21
1h11₃ = 22
10hh₃ = 23
10h0₃ = 24
10h1₃ = 25
100h₃ = 26
1000₃ = 27
1001₃ = 28
etcetera.
The negative of a number is then obtained by simply swapping 1's with h's and _vice versa_ :
h₃ = -1
h1₃ = -2
h0₃ = -3
hh₃ = -4
h11₃ = -5
h10₃ = -6
h1h₃ = -7
h01₃ = -8
h00₃ = -9
etcetera.
Great video! Super interesting. Where would come across this or apply it in mathematics?
These feels like changing the clef in music theory.
All your base are belong to us
jesus christ, the way you explain is beautiful
I like base prime.
1 = 1
10 = 2
100 = 3
200 = 6
120 = 210 = 60 = 12
In this base integers can have multiple representations based on the factorization chosen.
Also the next prime is easy to find 😂
One of the things that fascinates me about negative bases is that the negative sign is useless, because that would create two representations of each number (eg. 1011 and -101 in base -2)
There's a reason why there are "big groups" of powers with the same sign on the form of z=|x| + |y|i
(a similar point could be made for z=|x|+|y|i)
If we write z on the polar form, we have z = |z| cis(theta).... where theta is between 0 and pi/4(or 90º)... since we are on the first quadrant in the complex plane.
[cis(x) = cos(x) + i * sin(x) = exp(ix)]
And z^n in the polar form is z^n=|z|^n cis(n theta)
Note that the smaller the theta, the higher the n you need to change from quadrants(which happens when you change the signs of |x| or of |y|)... which means the higher the sequences of powers with the same sign.
Fascinating, thanks! Would you be willing to cover the p-adic numbers sometime?
please talk about non-constant bases as well, where digits can scale by different multiples
so e.g. factorial base, where n-th digit can be from 0 to n-1 and has value of n!
That was an entrance exam question for the Oxford MAT. They called the factorial base "flexadecimal".
Wouldn't you need infinite amount of number symbols if the scaling gets bigger?
@@Henrix1998yes, but you can express integers up to n!-1 if you use n different digits in base factorial
Really like the look of those negative integer bases. Those look rather compact and don't need a sign. Kewl.
part of me hopes we keep finding more types of numbers that branch outside of complex numbers so I can see what their base number system looks like
There's also *multi-base* positional numeral systems, which remind me of HashCat's "Mask Processor". They open up a literally infinite multiverse of possibilities, by using *arbitrary sequences* of numbers instead of the typical powers of N.
The Fibonacci numeral system uses the Fib sequence, which is essentially just rounded powers of the Golden Ratio. So, in theory, Fib System should have the same radix economy as base φ.
There's also one for Primes. And if your sequence is finite, you can simply repeat it using powers!
Imagine a *"Collatz3"* num-sys, it would be: _×3 + _×10 + _×5 + _×16 + _×8 + _×4 + _×2 + _×1... (then repeat using all squares, then cubes, etc...)
At some point this reminds me of vectors spaces.... imagine you are on a 2d space... but you can only walk along the diagonals(you can walk any real quantity of space...) (1,1) and (1,-1)
You can still go anywhere, in that space, but to go from (0,0) to (1,0)... you need to go to (1/2, 1/2)... then to (1,0)
You went +1/2 * (1,1) + 1/2 * (1,-1) = (1,0) in total
So... just like you can go anywhere in 2d moving along the cardinal directions a(1,0)+b(0,1)... you can also go anywhere moving in diagonals(1,1) (1,-1)... this means both are valid basis for R².
(In this case... every pair of vectors which aren't aligned can be used to form a basis R²... that's kinda obvious: if they were aligned, the second doesn't really allow you to go anywhere new... and R² is a *2d* space...)
In 3d... we need 3 vectors that aren't co-planar... as in, given the plane defined by 2 of those vectors... the third needs to be outside of it in order to reach any point in R³... else the third vector won't allow us to go anywhere we couldn't have gone with 2 of them.
And then, for something completely different, we have transforms.
Imagine we have a function, more than a infinite sequence of numbers... but we "decompose" that function in waves.
In different circumstances we have different transforms. For periodic functions (with real numbers as output) you can them with a (possibly infinite) sum of (possibly shifted and scaled) cosine waves[each with period being a fraction of the period original function]t... which can be described with 2 real numbers(the shifting and the scaling...)
So from a continuous function to a sequence of pairs of real numbers...
@@matheusjahnke8643 That's really interesting! You've got #SoME4 material right there. I wish I could collab with you, but I have no time nowadays
Base Grahams Number?
imagine
thanks for this.
please less ads, this is knowledge sharing and not entertainment
Super interesting. I'm a little confused about the selection of digits for a rational base. Do you always use the larger of the numerator and denominator?
Yes
Это интересно, это великолепно, надеюсь, это понадобится в моей жизни)
2:33 No, a tally system uses multiple instances of a single digit to represent numbers; you simply count the number of digits to get your value since they all have equal weight.
Can you write -2*Zeta(3)-Gamma'''(1) in base (EulerGamma + Pi/Sqrt(6))?
simple. this is really an extended modifiable diophantine equation (EMDE) which is simplifiable by the isomorphism M -> ą_0 \ {S_sum, 0, +} according to an addition-like operator in a field of 0 nondifferentiable manifolds, where there are actually _3_ nth riemann roots of unity, so the determinant can be approximated by the limit as x approaches [y : y(x) not in S_product *] and the rest of the solution has been reduced to a trivial kirimeta-vu 3-model partially integrated gödel system
great question!
Base 2+2i can certainly express of multiples of i, you just need a lot of symbols. For example, i itself is expressed as 0.00000(512) where you need to use the 512th symbol. This is because (2+2i)^-6=i/512.
Great content to think about the way we write numbers.
But I see some problems which limits the use of such bases: It includes counting and number comparison.
In a Natural Number base you can count easily and any carry will go to the next column on the left. With a base of √5 a carry will skip one column and comparison 4*√5 > 5 while in natural bases a non zero digit more left is always of greater value.
Base 1 does not make sense because all digits have the same value (always 1) and each digit can take only Zero. Incrementing to 1 would result in a 0 with a carry to the next digit on the left which is then incremented and leads to an infinite number of digits affected.
If there was a way to represent the ideas of percentages with one set of symbols , it'd be possible to represent each digit as a percentage of the digit, with none being 0 and full being equivalent to (1/A) in the next digit for base A
The logical conclusion, however, is that you'd encode all the information in the leftmost digit, which means you only need that one digit's fill-percentage plus whatever power it's raised to... so we've effectively rediscovered scientific notation
I feel like base 10i would be pretty interesting to try using.
Keep it up!
thank you for this xD ive asked this question before but couldnt find an answer
@Digital Genius
So in base -10 , square root of 35 = i*5 in Base 10
so you dont need complex here right to express (square)roots of negative numbers
Is there any advantage to use base -10 instead of base 10
How a graph would look like?
Would Imaginary Numbers disappear here?
This video would be more interesting if some applications for the various bases could be shown.
Most humans use base 10 (decimal). Almost all computers use base 2 (binary). Base 16 (hexadecimal) is useful for displaying binary numbers in a form that is easier to remember and read. The same holds for 8 (octal). If somebody use base 5, that would make sense since we have 5 fingers per hand. For clocks we use base 12, base 24 and base 60. For weeks we use sort of base 7.
But what is the use for base -10, 3/7, pi, or 2i?
Wondering if base 2+sprt7 would work, is it possible to get integers or would it be similar to transcendental numbers
You can Get some Integers,
for Example, 2+sqrt(7) In base 2+sqrt(7) is 10.
11+4sqrt(7) is 100. ((2+sqrt(7))^2)
6+3sqrt(7) b10=3 b(2+sqrt(7))
bn meaning base n
Pretty sure We could Express in the Digits
0,1,2,3,4
As these Are 0r(Something), while 5 is 1r(something)
Not a Very useful Base tho, You can Only express A few Numbers.
Very nice video. The animations were nice as was the commentary. There were some spots where I thought you were going a little too fast, but maybe that’s just me being slow 😅
I had an existantial crisis while thinking about the contents of this exact video like 1-2 years ago. Had completely forgotten about it until I came across this video. Turns out it's much more intuitive if you get a pen and a paper rather than trying to have it make sense all in your mind.
17:00 Could i be represented in base 2+2i with the digits 1, 0, 0, 1, 61, and then a 4 after the decimal point? This would, in base 10, become -64 + 2+2i + 61 + 1 - i, equaling i, right?
Nice synopsis; good explanations. My feedback:
IMHO, "base 1" is strictly the digit 0, and no other. Allowing any other digit breaks the rule for other positive integer bases, and is "cheating."
Speaking of breaking that rule, though, here's a neat trick I came up with some years ago, but which I strongly suspect isn't original.
In base 3, instead of (0,1,2), use digits (-1,0,1), for which let's use the marks (\,0,/). Because then, all numbers, + and -, can be written without using algebraic signs.
So e.g., +1 is just /; -1 is just \; the integers (..., -5, ..., 0, ..., 5, ...) are (..., \//, \\, \0, \/, \, 0, /, /\, /0, //, /\\, ...). Fractions can be converted from base 3 in the obvious way.
The negative of any given number is represented by vertically flipping the number's representation.
Other odd positive integer bases, b, would be analogous, using ½(b-1) new (non-mirror-symmetric) symbols for the +ve digits and their mirror-images for the -ve digits.
Another crazy idea of mine - "factorial base." Your discussion here was all about fixed bases, but what if each successive digit is in a different base?
Specifically, make the least significant digit binary, the next ternary, then bases 4, 5, 6, ... Similarly, after the "point," or radix, the digits are in bases 2, 3, 4, ...
The non-ve integers would be 0, 1, 10, 11, 20, 21, 100, ... I call it "factorial base" because "1" followed by (n-1) zeros, n ≥ 1, represents (n!) in this scheme.
Carrying out additions and subtractions in factorial base isn't too hard, but multiplication is totally impractical.
A couple nice features are that every rational number terminates; and that e = 10.11111111... forever.
A drawback is that there have to be an unlimited number of digit symbols.
Fred
Yes, others have thought of it
en.wikipedia.org/wiki/Balanced_ternary
But it speaks well for you that you came up with it on your own as well :)
You should look into ternary computer systems then. I wondered about there being base 3 computers, and after some very shallow research that confused me, they do exist. I remember (-1,0,1), (0,1/2,1), and (0,1,2) being some possible representations. You might understand it better than did.
Yep this is called balanced ternary and it’s used by some computers :)
The factorial base is sometimes called flexadecimal
@@larsokkenhaug148 Interesting. But not the best name for it, as it doesn't have anything to do with the decimal system.
Maybe they should have named it, "flexanary"? ;-)
Not as catchy...
Wow what a great vid
My brain executed a full factory reset at multiple points in this video.
I love the fact that e has the highest radix efficiency
*e*
Something that the usual radix economy calculation doesn't account for is the absence of a symbol being able to act as another symbol, so the radix economy graph gets sorted slightly. This makes base 2 become the most efficient base.
Nice 👍🎉
Underrated!