Kaktovic Numerals 

That's what I should have written in the video description

lol

Culture is definitely evolving all the time in subtle and dramatic ways, including language and food-culture as well.

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.

@@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 RU-vid 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.

Much appreciated!

Glad you liked it. I stumbled upon this topic totally by accident (like most stuff on here) and it blew me away!

Tune in now before he disappears into the mist like The Flying Dutchman

Thanks as always for watching and commenting!

Thanks as always!

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...

I've never used an abacus but it does seem similar in principle.

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 😊

@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.

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

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.)

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.

That's always seemed so strange to me, since the numbers used in the West are called "hindu-arabic" numerals!

At least for very basic operations, yes!

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?