Kaktovic Numerals
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- Опубліковано 13 вер 2021
- In the 1990s, a group of middle school students in the small arctic town of Kaktovic Alaska created their own number system to better reflect their own Inupiaq language which uses Base 20 instead of Base 10. The result was was an elegant writing system that may even improve how we teach mathematics. In just two years Kaktovic test scores went from the 20th percentile to above the national average.
Kids really went: Wow your numbers are dumb, I'll take a bunch of squiggly lines and make my own
That's what I should have written in the video description
lol
This is proof that culture isn’t just stuff that appeared thousands of years ago. Culture is still evolving every day, even if you don’t notice.
Culture is definitely evolving all the time in subtle and dramatic ways, including language and food-culture as well.
This is simply mind blowing. I remember a few years ago how one of my friends basically teached me how to change bases. She was studying to become a teacher and how sometimes simply teaching children how to change base allowed them to practice a multitude of adjacent skills. Anyways, she was telling me how a few children were having far less difficulty by simply changing from base 10 to base 12. That, however, is some next level stuff
Yeah, changing bases is a good way to show that things we take for granted like counting can be done differently. It opens up the mind to many other possibilities.
Absolutely amazing video as always, Ferret! :]
Much appreciated!
The multiplication table for this is remarkably simplified! You don't need to memorize a 20x20 grid like a base-20 Arabic numeral system would require. Instead, the sub-base 5 system can be taken advantage of and reduce the table to just 37 total products to memorize: 4x4 for each base (vertical numbers), 4x3 for each base x sub-base (vertical * horizontal), and 3x3 for each sub-base x sub-base (horizontal numbers).
Multiplication requires you to break up the vertical and horizontal parts of each number and FOIL the now single digits.
6 * 3 would breakup into a 5 * 3 + 1 * 3 = 3 5's (horizontals) and 3 1's (verticals) which is visually the character for 18!
6 * 6 -> 5 * 5 + 5 * 1 + 1 * 5 + 1 * 1 = 1 vertical in the 20's place and 3 horizontals and 1 vertical in the 1's place = 1 (16) in base-20 = 36 in Arabic numerals!
@@dannyshahlestari6679 I can't type Kaktovik numerals, so this will be limited to Arabic numerals for simplicity (either write it yourself or visit the Wikipedia page for a better visual).
You only need to memorize 3 VERY small multiplication tables:
The ones x ones:
| 0 | 1 | 2 | 3 | 4 |
----------------------------
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
The fives x ones:
| 0 | 1 | 2 | 3 | 4 |
----------------------------------
| 5 | 5 | 10 | 15 | 20 |
| 10 | 10 | 20 | 30 | 40 |
| 15 | 15 | 30 | 45 | 60 |
The fives x fives:
| 0 | 5 | 10 | 15 |
--------------------------------
| 5 | 25 | 50 | 65 |
| 10 | 50 | 100 | 150 |
| 15 | 65 | 150 | 225 |
Again, these will be at most 2 digit numbers because Kaktovik is in base-20. The Wikipedia page at the end of the Computation section gives a MUCH better visual of these tables than what I can do in a UA-cam comment.
The reason this works is due to the sub-base of 5. It effectively acts as if the number has been rewritten in a factored form. An easy example to see this would be doing 6 * 6 yourself with the Kaktovik numerals. You'll use the FOIL method to compute (5 + 1) * (5 + 1) to build the value for 36 (in Kaktovik numerals obviously).
There's no actual need to memorize that 6 * 6 = 36 (or any other multiplication greater than 5!), and that advantage is only really useful for children still learning their first numeral system. It reduces the amount of rote memorization needed for multiplication, which is certainly easier for children. The fact that it's been reduced by 63% is amazing!
Memorizing more multiplication pairs to shortcut some work just becomes extracurricular at that point.
Keep making videos man! this was the best
Glad you liked it. I stumbled upon this topic totally by accident (like most stuff on here) and it blew me away!
absolutely ingenious. soon enough this will be taught in schools.
Everyone drop what you’re doing he’s back!!
Tune in now before he disappears into the mist like The Flying Dutchman
Great content as always!
Thanks as always for watching and commenting!
Impressive video!
Underapreciated video
Around 5:30, you don't have to know what number is represented no matter the system(as long as it's a positional number system) to calculate with them, just know what are the digits and the table for all of the digits for each operation. I tried to understand better binary numbers and those were my conclusions, as long as you stay in one number system then you don't have to worry about what is this number in another, like do you think what 10231391 is? It's 10231391 but only in base 10, so as long as we stay in base 10 then it's just it, just 10231391 you know that 1 less is 10231390, and 1 more is 10231392, etc etc
One thing, it's best if you have a word system for naming those numbers, so if they are base 20, then naming them in base 10 is switching systems all the time which would be a problem, I used base 10 names in base 2 as going down doesn't make many problems, just is a little confusing
Your videos are really good
Thanks as always!
Can this system be used for a duodecimal and a sexagesimal system?
What is a sub base?
Can there be a sexagesimal system with a duodecimal sub base system?
In this case, the sub-base of 5 just means the horizontal strokes are increments of 5 rather than keeping vertical strokes going on the bottom. So you don't switch to a new register until you reach 20, but you can count 1's and 5's in different parts of the same character. In other cultures this was done with a dot.
I suppose if you did a duodecimal version of this, the sub-base would be 3?? So every group of 3 would be counted up at the top with a maximum of 3 horizontal strokes (equalling 9) and then up to 2 vertical strokes until you start over again at 12. If you wanted the sub base to be 6, you'd need up to 5 vertical strokes, so the balance between top and bottom would be sorta weird. A sub base of 3 seems better for aesthetics...
Could this work in base 10?
imma start doing math like this on paper to freak people out
@Rodrigo Ferreira well, i can understand it. struggling with multiplication and less than whole numbers.
dude you're in my mind
@@winkydinky1436 and walls
@@umbigbry what do you mean.i am confused
@@winkydinky1436 i am in your walls.
This is all really cool, but one question. How do you write pie in this numeral system. I wanted to write this to see what cool mathematical arcane scroll of Inuit magic I could make, but due to the number system eventually going right to left when going rather than how we go left to right and add another number to the end. I just want to know if they have solved this problem as well as how they tackle decimal numbers.
Pi in base 20 is: 3.2GCEG9GBHJ9D2...
So in Kaktovic it would be 𝋃.𝋂𝋐𝋌𝋎𝋐𝋉𝋐𝋋𝋑𝋓𝋉𝋍𝋂...
(You will need a font that can display Kaktovic unicode characters.)
Its basically you writing an Abacus numbers. Chinese Abacus have base 15 and Japanese Soroban is base 10.
Ferrets, eh?
I don’t understand why this is being given the impression for being revolutionary. Explain to me how this technique differs from say dividing 888 by 111 in the decimal system. Here you can easily see that each digit of 111 fits 8 times into the digits of 888. Similarly to the decimal system there are divisions, like 252/18, that can’t be solved by simply looking at the patterns, 6/3 in the kaktovic system as an example.
For 6/3 just expand the sub base and then divide
@@introprospector that doesn't make the kaktovic system anything more special than the decimal system. If you need to expand into the subbases, you are only substituting work rather than making it easier.
@@Jawis32 That's literally the same thing you do in decimal. This is like saying decimal doesn't work well because 5/2 requires extra steps. Expand the remainder into the next base below and then divide.
@@introprospector Could you explain how to expand the sub base? I am trying to learn how to use this number system and this is stumping me.
@@introprospector
"That's literally the same thing you do in decimal."
False.
0 1 7 2 8
____________
7 3 | 1 2 6 1 4 4
7 3
---------
5 3 1
5 1 1
---------
2 0 4
1 4 6
--------
5 8 4
5 8 4
--------
0
126144 / 73 = 1728
Long division, no expansion required like what you're talking about
For long division: Base 10 > Kaktovic visual tricks
Im stealing someone elses comment from another video and would genuinely love to know how this is solved in Kaktovic
"I tried 100 divided by 11 couldnt figure out how to do it.. unless I got something wrong.. the symbols for 11 just never appear in the symbols for a hundred..
heres what I did:
so the symbols for a hundred is 2 symbols. the symbol for 5 (one line on top) followed by the symbol for 0 because we are in base 20 so 5x20 + 0x1 = 100
the symbols for 11 is 1 symbol, 2 on top to make 10 and 1
now trying to fit the symbols for 11 into 100 and counting how many times it appears give you 0, it never matches..
it seems to me the examples in the video are cherry picked so they work.. or I messed up pretty badly..
heres another one:
6 divided by 3
6 is one on top + 1 on bottom (5+1)
3 is 3 on bottom
they also never match.. theres only 2 lines in 6 so you can never match 3 line in it.
you would have to break the 5 (top line) of 6 into bottom lines to make a match, something like: \/\/\/ divided by \/\ to make it work visually
it's like if someone showed you how easy it is to divide in base 10 saying you just remove zeros ! and they show you example : 20 / 10 = 2, 300 / 100 = 3, 36 000 / 100 = 360
like that's cool but it really only works for specific cases, again unless I messed up somewhere... (please point it out to me if I did)"
it seems whenever you have a zero in the dividend, you have to divide like normal
Yeah sometimes you have to mentally break one or more of the “five” lines in order to get something to appear.
For 100/11… let’s see…
(Note: I’ll use - for 5, > for 10, ≤ for 15, and ȣ for 0, as well as \ for 1, V for 2, V\ for 3, and W for 4. Also, sorry for the wall of text.)
100/11 would be - ȣ ÷ >\ . Mentally breaking the -, we get (W ≤W + \) ÷ >\. Let’s split this into (W ≤W ÷ >\) + (\ ÷ >\).
Let’s look at the first part first. We can see >\ once in W ≤W, so (W ≤W ÷ >\) becomes ((W -V\ ÷ >\) + \).
Next, let’s mentally break the next \ ȣ into two copies of >. We then get ((V\ >>-V\ ÷ >\) + \) (yes, I know KI numerals don’t actually have more than 3 horizontal lines at a time, but please bear with me here). We can see >\ twice in the “ones” place, so we get ((V\ -\ ÷ >\) + V\).
Next, let’s break the -\ into VVV and one of the \ ȣ s into >>. We then get ((V >>VVV ÷ >\) + V\). Again, we can see >\ twice in the “ones” place, so we get ((V VV ÷ >\) + -).
Next, let’s break another one of the \ ȣ s into >>. We then get ((\ >>VV ÷ >\) + -). Again, we can see >\ twice in the “ones” place, so we get ((\ V ÷ >\) + -V).
Breaking the final \ ȣ into >> gives us ((>>V ÷ >\) + -V), which simplified, results in (-W).
Adding the second part, \ ÷ >\, and noting that this second part is less than one, we get -W R \.
So - ȣ ÷ >\ = -W R \.
Yes, it’s complicated, but it can be done 😊
It's very similar to abacus.
I've never used an abacus but it does seem similar in principle.
Arabic numerals look a bit different.
3:50 2 + 3 in kaktovic numerals doesn't look not even a bit alike. That argument is just as convenient as it goes to the given example. The same would happen if you summed 10+ any other number (1, 5, 9, 7...) they would be alike, thus not being really a good point to give it as an advantage over the arabic numeral.
That's where the sub-base 5 comes in. \/ + \/\ = ~ (I don't have a higher horizontal line to type with) because there's 5 vertical lines you're combining from 2 other numbers. You'll have to write these out yourself to see it better, but 10 + 6 = 16 is still visually convenient because 10 has 2 horizontal lines and 6 has 1 horizontal and 1 vertical, together giving 16 with 3 horizontal and 1 vertical. Just like decimal, you still have to add from the right digits and carry, but you don't have to think as much to get the digits (more important for children still learning their first number system).
15 + 15 gives 6 5's which is equivalent to carrying 4 of them (which is 20 btw) and leaving 2 5's behind. That leaves us with a 1 in the 20's place and a 10 in the 1's place which is the correct answer but more visually convenient than Arabic numerals.
15 + 16 has the same convenience but you would see you can combine that 10 (left behind as seen in the 15 + 15 example) with the 1 from the 16 to still get an obvious 11 in this system.
17 + 18 likewise allows you to just count up how many 5's you have (6 from the horizontals and 1 from the verticals) and get an answer that has 3 5's in the 1's place left behind (which is correct, 1 in the 20's and 15 in the 1's for 35(base-10)).
A similar visual feature could certainly be applied to decimal as well, but the Arabic numerals don't do that.
Arabs use a different form of numerals than we do, even if they were both introduced by Arab traders
That's always seemed so strange to me, since the numbers used in the West are called "hindu-arabic" numerals!
So, a numbering system where you don't even have to understand math to get the correct answer to problems?
At least for very basic operations, yes!
Hindu-Arabic numerals ain't bad.
It's simply genius (but it looks really boring)
They do all look really similar, it's like if we counted using only tally-marks.
@@ferretsensei I just realised that it could be a disadvantage when you need to quickly distinguish them from afar, like on a digital clock or a traffic sign. Even though we also have similar symbols that can sometimes get mixed up (3 8; 1 7; 3 5...).
@@ferretsensei Why did youtube remove my reply?
Does anyone else notice how perfectly these Kaktovic numerals parallel the Mayan numerals? Fail, Inupiaq!
how do i do this: i.ibb.co/7N8gBbP/0.png