Let me know if you really wanna learn about multiplication and division in negative bases, I prepared notes on it but in the end I didn't think it was interesting enough to be worth the time. Join Wrath of Math to get exclusive videos, lecture notes, and more: ua-cam.com/channels/yEKvaxi8mt9FMc62MHcliw.htmljoin More math chats: ua-cam.com/play/PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO.html
@@veganwater381 To convert the decimal component to negadecimal, every other place value starting with the tenths place increases the number to the left of it by 1 (if it is 9, it goes to 0 and subtracts 1 from the place to the left of that) and then is replaced by 10 minus itself. Therefore, 0.999... becomes... 1. Just 1.
Your theme song for this video is obviously "Too Much Time on My Hands" by Styx. Also, you edited in the same audio snippet of the word "negadecimal" everywhere it appeared in the video, didn't you? That's effin' hilarious and proves that the song applies _hard._ Also also, you introduced an _i_ at 20:38. That's just _mean!_ Finally, you said early on that we can use _any_ number for a base. Oh yeah? Show me base _i,_ smart guy. 😁 P.S. I had commented two of the above quips separately in replies here but they disappeared. Let's see if this reply survives.
Or base Phi - looks like binary but there can be no adjacent 1s, and IIRC all integers are palindromes with the units place at the centre. If you don't like the inconvenient integers, you can approximate it using the Fibonacci numbers instead of powers of Phi. Even a matrix can be used as a base. All you need for your base is exponentiation (or similar function) by an integer, multiplication of that by an integer, then adding up load of those.
Or -1+/-i? Combo Class reveals that this (that being either Gaussian integer with a magnitude of sqrt(2) and real part -1) is almost like "Gaussian binary!" ("Gaussian integers," in case you didn't know, is a name for numbers a+bi where a, b are in the ring Z)
I love the fact that he used the same copy pasted audio clip to say “negadecimal” every single time; I’m just imagining him saying it repeatedly into the mic and then strenuously listening to each to find which one sounds the least like a slur 😂😂
It's probably something that he didn't account for while recording, so when youtube started detecting it as a slur, he had to use their quick editor to fix it post factum
honestly the whole "nega- sounds like a slur" thing is kinda dumb. Like, is anybody getting offended by people saying a word that's not even related to the slur? Why do we cared about people saying something like "negadecimal" but "negative" is fine? It's still different enough to be able to tell the difference, and I don't think I've ever seen anyone actually get upset about someone saying "nega-", just people pointing at it and saying "haha look it sounds like a slur"
@@_pitakoWow some people are really sensitive, imagine getting so worked up over something as small as someone using a robot voice to cover up "negadecimal" as a running gag through a video
Isn't this what normally happens? "I've spent months creating this new branch of maths." "What applications does it have?" "None yet, I just thought it was cool."
Problem with negative bases is that carry is 2 digits. John Colson's Negativo-Affirmativo Arithmetik is better. It uses digits from -4 to +5. Edit: Carry can be one digit if it is negative.
When I first learned about it I really started enjoying playing with quaterimaginary which is 'base 2i' and lets you also fold the imaginary part of a complex number into a single 'base-4' series of numbers without negative signs or the awkward +(number)i part. When adding and subtracting you have the 'carries become borrows' flip flop and you have to skip over a digit since every other digit flip flops between real and imaginary. Remarkably multiplication works pretty much like regular multiplication except you have to deal with the oddities of 'carrying' and 'borrowing'. Division, on the other hand, is rather tricky but it has the one advantage of not having to compute a complex conjugate.
I immediately visualize base +b as taking progressively finer steps in the same direction, being careful not to walk too far as to overstep. While base -b overshoots what it represents and corrects that by bouncing back and forth around it. It's a little fun to think of this dance on a number line.
Yeah I kinda scratched my head on that one because I know 13 in base 10 is 111 in base 3. He probably just forgot to multiply the number to the power of the base
@@TheRandomPersonOnUA-cam i think what he forgot is the fact that he was talking about 221, as it seems like he just decided to add them up for no reason
@@2wr633I guess it can be true. Forgetting the numbers in their respective digits and just thinking about the sum of the exponents is a common mistake especially for me.
The prefix "nega-" is pronounced the same as the N-word (no hard r) for some English speakers, and even for those that don't do that, the words are so similar sounding that it's better to be safe than sorry.
I found a use of this, we can use it as an example that there are equally as many naturals as integers. Because every integer in base 10 maps to a unique natural in base -10. And every integer in base -10 maps to a unique integer in base 10. Thus we have a bijection between the sets!
Fun fact: -1 in base 10 is 19 in base -10, so in the example of 34 + 89 in base -10, you can also think of that -1 carry into the hundreds place as sticking -1 in negadecimal to the front.
I thought about this idea before! My solution was have negative numbers every other number. so 100101 is 10^2 -10^1 + 10^0 This makes us use twice the space to represent all numbers, though. Major downside, but addition still works the exact same. And multiplication / division you need to carry across 2 places instead of 1, which is a very simple/ intuitive change.
"Lets stop using (-)" "Now we carry the -1" Since -1≈19 in negadecimals the carying becomes simpler. 4+7=190+1≈100-90+1=11 This also allows us to understand the non-unique representations in a different way. Any two non-unique representations must be separated only infinitesimally. In decimals, the infinitesimal can be represented by 0.00...001. In negadecimals, the positive infinitesimal is 0.00...001 in a positive place value and 0.00...019 in a negative place value, and vice versa is true for the negative infinitesimal. In the same way that 0.99...99+0.00...01 cascades in place value 1.90...909=0.09...09+0.0...019 Does as well
This makes me think of 2's complement, used for representing negative numbers in computers. The top-most place has a negative value, but the rest are positive. For example, for 8-bit numbers, the top bit has a value of -128, and the rest are +64 down to +1.
You can improve that slightly by having the leftmost digit imply an endless sea of 0s or 9s to the left depending on whether that digit is 0...4 or 5...9. (This requires prepending an extra 0 or 9 for half the numbers.)
In two's compliment, the leftmost bit is basically the minus sign. Computers don't use an actual minus sign because you can have +0 and -0 and that just messes up math.
I know the tenp in Canada probably goes below 0° F pretty frequently, but youd have to use a lot fewer negative numbers if you werent using Celcius. Or just use Kelvin, or, dare I say, Rankine, and you wouldnt have to use negatives at all for temperature.
I think it's really interesting that while in decimal, terminating expansions always have a non-terminating equivalent, in negadecimal, some non-terminating expansions also have a second non-terminating equivalent. Makes me curious if this is always the case or if 1/11 is a special case.
Just as I was looking into bases with negative numbers this video pops up! Please do balanced number systems next (like balanced ternary), I find them fascinating!
You've no way of knowing this upfront, but this video came along at just the right time. As one of my favourite fictional people would say, "Decidedly, I am unwell." This is a fun topic and I enjoyed your easy to follow exposition of it. When I go back to bed, in the spirit of one of Lewis Carroll's pillow problems, I'm going to bring a notepad and pencil with me and play with this concept. Cheers!
To remove the need of a minus sign to represent negative numbers, by far I prefer the balanced number systems. They still use a positive base, but have positive and negative digits, ideally the same number of positive and negative digits.
Couldn't we just use a nega-posi decimal system where we have a doubling of size but avoid using minus signs - let's say the numbers in odd places are positive and the numbers in even places are negative for example: 01 is equal to 1 but 10 is equal to -1 then if we wanna write out a negative number like -124 we simply write 102040 and then if we multiply 102040 by 10 we get 010204. It would be a lot simpler to get used to. Maybe you could showcase that in the next video??
i know it's not perfectly compact because you can write the same number multiple ways (for example 11 = 00 = 0 = 11111111 = 0011001100 but I think it's a fair trade taking that we don't have to use minus signs anymore - further more it would make it easier to write subtraction operations because rather then writing the whole boring sentence: 102 - 45 😴😴😴🥱🥱 you could just write 14052 🔥🔥🔥🥵🥵 and it's obviously the same as the number 0507
A more straightforward way to convert from negadecimal to decimal is to take advantage of the fact that having a 0 in every other place makes thr number equivalent to yhe decimal notation. a numbe rin the form EDCBA can be written as E0C0A - 0D0B0, and then do the suntraction as per normal decimal subtraction. For instance, 17492 = 10402-7090 = 9312.
I totally thought this video was gonna be about two’s complement and variations. It accomplishes the same goal, is widespread in computing, and is easier to convert too - still a fun video tho
Next you need to figure out how to remove that pesky . Symbol used to represent the negative powers of your base. When considering the powers of negative 10, those exponents are also written in negadecimal. Since -1 = 19(nd), 0.1 = 1×(10^-1) = 1x(10^19(nd)) = 10000000000000000000(nd'). Almost no thought has been put in here, so I'm sure that there are holes left to fill in to make this system work.
I'm smoking weed and the "negadecimal" TTS was tripping me out but i did hear one time you forgot to switch the audio and said "decimal" rather than "negadecimal" - However you're presenting this very well and unless you weren't paying attention to what was happening you would know what you meant anyways... so rather than time stamping it I simply kept watching until hearing the TTS so much drove me to point this out. lmao! Great videos tho, I watch a lot of math creators here and you somehow stand out a lot of the time and also the videos have good quality!
I tried making a binary version of this "negabinary" since i think binary is the best number system i = 0 l = 1 | ( using different symbols for a more digestible look ) the patterns you have to really remember for positive numbers is li, ll, ii, il and for negative il, ii, ll, li the order of positive numbers from one ( spaced out in pairs so you can see the pattern ) l l li l ll l ii l il l li li l li ll l li ii l li il l ll li l ll ll ... the order of negative numbers ( the pattern skips first two numbers since they start with 0 ) ll li l l il l l ii l l ll l l li l i il l i ii l i ll l i li l l il il l l il ii ... addition works the same as in the video, it’s just easier to "subtract the one" from the next digit, because there can only be two outcomes when you do that for example, 13 ( lllil ) -14 ( llilli ) l ll il+ ll il li ii ii ll ll ( -1 )
@@Why553-k5b_1the base number is always written as '10' in any positional numeral system using Arabic numerals. The word 'ten' and the numeral '10' only represent the same thing in decimal.
@@Why553-k5b_1in base 2, 2 is written as 10. Therefore in base 2, you can say "this is base 10". This works for any number because n^1=n, so every base will be represented by one copy of the base to the power of one (a 1 in what we call the "tens digit" in base ten) and zero copies of the base to the power of zero (a 0 in the ones digit), or 10
I'd never encountered *cue wav file* "NEGADECIMAL" or other negative base number systems before, so this was a view into a new way of thinking for me. I'd love to see more about the esoteric multiplication and division and you as a song-singing frog, especially now that we've already invested the time and effort in learning the absolute essentials like Canada (the country of my people) and amphibian body dysmorphia.
There might be some utility here for representing very large integers in numerical analysis applications, as this system eliminates the need for sign bits or any other system of tracking whether the stored integer is positive or negative; it's just a matter of whether the most significant digit is an odd-column digit or an even-column digit.
Balanced Ternary! (Or just binary with two's complement. Might need to do something about needing infinite ones going leftward. Maybe a hyphen could represent that?)
In the 2nd example of negative bases you should have mentioned balanced trinary instead of negatrinary as balanced trinary is more simpler to understand and also is possibly the most used negative base.
I'm done with complex numbers and that ugly i. I introduce quaterimaginary or base 2i. Using 4 numbers we can represent any conplex number. 20.2_2i = 2*(2i^1) + 2*(2i^-1) = 2*2i + 2*-0.5i = 4i + -i = 4i - i = 3i
Interesting, Then all of the negative numbers become interleaved with the positive ones, logarithmically I wonder what the number line (or, more likely "number space") looks like
Balanced ternary is my favourite. Rounding a number is nothing else than cutting the excess decimals. You can start an addition of many numbers without knowing whether the result will be positive or negative. Ideal to represent quantities that have no natural zero point. In particular every scale that is logarithmic. A drawback is that one half has no finite representation, but two infinite sequences of digits: zero, decimal point, followed by infinitely many 1s, or one, decimal point, followed by infinitely many (-1)s.
Some benefits of Balanced Ternary: Extremely good radix economy No - sign needed. Only 2/9 carries in addition table, even better than the 1/4 of binary. Negation is just flipping all the (+1) symbols to (-1) and vice versa.
This look interesting. I guess the arrows represent different combinations of -1 (left), +1(right), i(up), and -i(down). The question is then that base do you use for this? Or does it work in a different way?
For a number system to be useful every quantity has to be represented. It seems like certain quantities would not be representable in a pure negadecimal system. It's a worse problem to have than the Roman numeral system which stops at 3899. You'll have quantities which will be forbidden to be expressed.
Use of subscript '10' to indicate base ten feels circular. If you're really using other bases often enough to leave doubt then spell out 'ten' in the subscript.
I am very pleased to have seen your video as another numbering system, I called it FrEd, seems wonderfully perplexing and probably quite useless. Consider 4528 in decimal. Replace powers of ten in (assumed) decimal system with powers of ascending primes ordered so as to increase from right to left. Unlike decimal system it is a closed numbering system in sense that any prime to power zero adds a "1" and that can be awkward. Damn, bloody awkward really. So maybe we might need a symbol such as ¥ to denote "No power here" Anyway, in that awkward, closed numbering system, call it FrEd, 4528 in decimal returns 5792 in FrEd numbering system. FrEd seems to have advantage of turning p/q rationals into plain old p rationals as negative powers of primes can be helpful. Example rational 1/2 in decimal 0.5 is (-1) in FrEd and 1/3 becomes 0.3r becomes (-1)0 and see! It is awkward, bloody damned awkward 🙂 Further suggesting a limitation that 1 in decimal is for ever 0 in FrEd or ¥0 to disambiguate as 0¥¥¥0 would be very very bothersome
Can you differentiate (-n)^m and -n^m where n is a real positive number? 5:01 Idea: Mirror digits to make them negative and add a tilt to the symbol „8“ and it'll definitely cause no problems for any people.
I'm triggered in how you read the base -10 numbers as if they were base 10 numbers. The name "twenty-one" does not refer to the digit string "21" but to the number of vertical lines in "|||||||||||||||||||||" (I don't think you used this specific number, but I didn't want to type - and count - too many vertical bars).
Don't (b-1)s' complement and bs' complement (for base b) have advantages over negative bases? I'd rather write -(229) as 770 than having to use 190 just to represent ten.
Addition in negadecimal is wild. You have to exchange borrow and carry. 9+9 is 8 borrow 1. 1 borrowed from 0 is 9, carry 1. So 9+9 = 198. That's decimal 9 + 9 = 18. 190 + 190. 0+0 is 0. 9+9 is 8 borrow 1. 1+1-1 is 1. 190+190=180. That's decimal 10 + 10 = 20. Imagine teaching that to grade schoolers. I'll keep the minus sign.
Given the unfortunate pronounciation of negadecimal, I propose two alternatives: antidecimal: literally the opposite of decimal debtimal: debts can be viewed as having negative money
I wonder what negaunary is (base -1.) Unary is just another name for tally. They're all base one digits. Negaunary can only the have two states presence or absence. And then you got nullary digits (base zero) which is existential math. It doesn't matter what you do: nothing matters. It's just art.
So even digit number are negative and odd digit numbers are positive, so in the end don't you have to write negative sign anyways defeating the purpose?
not very space efficient for certain numbers like -12. and it is also very difficult to read as there's no indication of whether the digit is positive or negative.
No, that it is a decimal system isn't what makes it special. Base 8 or base 12 would be much better. What makes it special is solely that it is positional. And, ironically, that's what you go on to describe. Not that it is decimal, but that it is positional. Words mean things. You should have been clearer about what you were lauding.
Let me know if you really wanna learn about multiplication and division in negative bases, I prepared notes on it but in the end I didn't think it was interesting enough to be worth the time.
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I would love to! And also, is there a way to represent 0.99... in negadecimal? Does it involve imaginary numbers? Like 0.90..I + 0.09..?
@@veganwater381 To convert the decimal component to negadecimal, every other place value starting with the tenths place increases the number to the left of it by 1 (if it is 9, it goes to 0 and subtracts 1 from the place to the left of that) and then is replaced by 10 minus itself. Therefore, 0.999... becomes... 1. Just 1.
Your theme song for this video is obviously "Too Much Time on My Hands" by Styx.
Also, you edited in the same audio snippet of the word "negadecimal" everywhere it appeared in the video, didn't you? That's effin' hilarious and proves that the song applies _hard._
Also also, you introduced an _i_ at 20:38. That's just _mean!_
Finally, you said early on that we can use _any_ number for a base. Oh yeah? Show me base _i,_ smart guy.
😁
P.S. I had commented two of the above quips separately in replies here but they disappeared. Let's see if this reply survives.
i have a feeling you were pronouncing nega like niga so you had to put on an ai voice
Why not make life more difficult and add imaginary numbers to replace negative numbers ?
Oh wow that's way easier than writing a small horizontal line
😂😂
Sarcasm detected.
True
Obviously 😂
Except you still have to write it in the subscript
“i realised there’s one big problem, and that is the country of canada and it’s ten provinces.”
greatest maths video of all time
Ah, Canada. That giant suburb of the Arctic Circle.
Absolute cinema.
Just use Kelvin scale.
its.
if y’all (the viewers) think this is crazy wait till you find out about base 1+i. Yes, it is literally a complex-valued base.
Or base Phi - looks like binary but there can be no adjacent 1s, and IIRC all integers are palindromes with the units place at the centre. If you don't like the inconvenient integers, you can approximate it using the Fibonacci numbers instead of powers of Phi.
Even a matrix can be used as a base. All you need for your base is exponentiation (or similar function) by an integer, multiplication of that by an integer, then adding up load of those.
Or -1+/-i? Combo Class reveals that this (that being either Gaussian integer with a magnitude of sqrt(2) and real part -1) is almost like "Gaussian binary!" ("Gaussian integers," in case you didn't know, is a name for numbers a+bi where a, b are in the ring Z)
Sounds based.
@ChessTopia1xi is just i
I remember imaginary bases being possible, but not complex bases
2:16 2:42 2:52 4:08 4:30 4:37 6:00 7:21 7:31 7:42 7:48 7:54 7:59 8:20 8:29 9:56 10:12 10:14 10:48 12:33 13:03 13:12 13:22 13:30 13:41 14:56 15:19 16:07 16:31 16:32 17:53 17:58 18:11 18:21 18:29 18:40 18:47 18:57 22:42 22:51 23:04 26:47 27:24 🤨📸
😬
it would have been way funnier if instead of using a stupid robot voice, you just asked a black friend to help you with this project 🤣🤣🤣
@@regulus8518 That's assuming he has one
holy thank you for the good wheeze
@@inutamer3658I am black
I love the fact that he used the same copy pasted audio clip to say “negadecimal” every single time; I’m just imagining him saying it repeatedly into the mic and then strenuously listening to each to find which one sounds the least like a slur 😂😂
nega decimal...
@@lailoutherandwhat did you just call me?!
Is that really why he was doing that? :/
That's so funny, bro. I can't even. 😐 let's not say viNegar out loud. That might upset the black people. 🙄
jorgefoyld8538George Droyd💀
if you're worried about accidentally saying a slur, negative decimal also works
It's probably something that he didn't account for while recording, so when youtube started detecting it as a slur, he had to use their quick editor to fix it post factum
@alexzhukovsky8361 idk, from my personal experience people are better at detecting slurs than youtube
honestly the whole "nega- sounds like a slur" thing is kinda dumb. Like, is anybody getting offended by people saying a word that's not even related to the slur? Why do we cared about people saying something like "negadecimal" but "negative" is fine? It's still different enough to be able to tell the difference, and I don't think I've ever seen anyone actually get upset about someone saying "nega-", just people pointing at it and saying "haha look it sounds like a slur"
@@_pitako it's america, nothing here makes sense
@@_pitakoWow some people are really sensitive, imagine getting so worked up over something as small as someone using a robot voice to cover up "negadecimal" as a running gag through a video
1:05 Just use Kelvin, problem solved.
Bro. U smart
why didn't they think of that?
my god that's genius
Or measure coldness instead of temperature.
Heat is just the absence of cold.
Bro, why can't I reply to u- anyways that's like debt
A truly wonderful solution! If only we could find a problem for it to solve!
🤣
you've done it! you've broken mathematics down to its bare essentials!
(....hey, our work usually comes into vogue a century or two late)
Isn't this what normally happens?
"I've spent months creating this new branch of maths."
"What applications does it have?"
"None yet, I just thought it was cool."
the problem is the country of canada and its 10 provinces. duh.
You used an awful lot of minus signs in your attempt to eliminate the use of the minus sign! 😂
Yeah I feel like he didn't quiiite give it a fair shake. To do a negative carry in negadecimal addition, just add 19 instead of subtracting 1.
You can also use a 10-adic number system
still need to work on this plan 😂
@@merhaba3621 and deal with x*y=0, x=/=0^y=/=0
This is the 100th reply, and I'm the 10000000th liker. (If you get what I'm saying, good)
Bro heard Mr. Crocker say "NegaChin" and said "nah" 💀
Did UA-cam not like using "nega-" as a prefix and it tried to interpret it as another word?
what other word are you thinking of. Please say it so we all know exactly which word you are talking about.
Nega, please.
@@KingGisInDaHouse [redacted]
Would nega-racism be hatred of your own race? 🤔
Could be more than just a YT issue - for some American (meaning the American continent, not the USA) speakers, they literally sound the same
> There's one big problem, and that is the country of Canada
I've been telling people that for years and nobody believes me!
How can it be a problem when it doesn't exist
*“Blame Canada” starts playing*
Problem with negative bases is that carry is 2 digits.
John Colson's Negativo-Affirmativo Arithmetik is better. It uses digits from -4 to +5.
Edit: Carry can be one digit if it is negative.
ah yeah, i usually call it signed decimal or balanced decimal
Balanced numbers benefit from an odd base. 10 is not really a good base. With an odd base, rounding a number is merely removing the excess digits.
Dude... I know April Fool's Day is amazing, but can you at least hold off until the actual day? I swear, they're making holidays closer and closer...
It's already April first, in the right base.
When I first learned about it I really started enjoying playing with quaterimaginary which is 'base 2i' and lets you also fold the imaginary part of a complex number into a single 'base-4' series of numbers without negative signs or the awkward +(number)i part.
When adding and subtracting you have the 'carries become borrows' flip flop and you have to skip over a digit since every other digit flip flops between real and imaginary.
Remarkably multiplication works pretty much like regular multiplication except you have to deal with the oddities of 'carrying' and 'borrowing'. Division, on the other hand, is rather tricky but it has the one advantage of not having to compute a complex conjugate.
At the addition, at 10:39 you used negative sign. I giggled.
I immediately visualize base +b as taking progressively finer steps in the same direction, being careful not to walk too far as to overstep. While base -b overshoots what it represents and corrects that by bouncing back and forth around it. It's a little fun to think of this dance on a number line.
1:45 nope it's 18 + 6 + 1 :)
Yeah I kinda scratched my head on that one because I know 13 in base 10 is 111 in base 3. He probably just forgot to multiply the number to the power of the base
thank god im not the only one who noticed
@@TheRandomPersonOnUA-cam i think what he forgot is the fact that he was talking about 221, as it seems like he just decided to add them up for no reason
@@2wr633 I guess.
@@2wr633I guess it can be true. Forgetting the numbers in their respective digits and just thinking about the sum of the exponents is a common mistake especially for me.
Why are you unable to say the word negadecimal?
What do you mean? He said it like 100 times. That's 100 times in *NEGADECIMAL*.
He reused the audio for some reason
I suspect he pronounced the first e and an i... Let's, um, not go there.
The prefix "nega-" is pronounced the same as the N-word (no hard r) for some English speakers, and even for those that don't do that, the words are so similar sounding that it's better to be safe than sorry.
@@AstaryuuGaming i'm pretty sure he was just doing it for the funny
16:07 HE SAID IT
rats, knew I missed one!
@@WrathofMathwhy was every mention of negadecimal being replaced with a robot voice?
@@Mozartminecraft🤔
@@Mozartminecraft because if you say "nega" wrong it sounds like a slur and he is not risking that
@@MSBenwastaken by that logic every mention of ‘negative’ should be a robot voice as well no?
0:43 uncalled for
Real
Oh go apologize for something... :P
I found a use of this, we can use it as an example that there are equally as many naturals as integers. Because every integer in base 10 maps to a unique natural in base -10. And every integer in base -10 maps to a unique integer in base 10. Thus we have a bijection between the sets!
Fun fact: -1 in base 10 is 19 in base -10, so in the example of 34 + 89 in base -10, you can also think of that -1 carry into the hundreds place as sticking -1 in negadecimal to the front.
- Hmm i wonder what comes after 9 in negadecimal
- 190
- What.
I thought about this idea before!
My solution was have negative numbers every other number.
so 100101 is 10^2 -10^1 + 10^0
This makes us use twice the space to represent all numbers, though. Major downside, but addition still works the exact same. And multiplication / division you need to carry across 2 places instead of 1, which is a very simple/ intuitive change.
Hmm if you take this idea but represent it in base 20, it would take up no more space than in decimal... i think?
@@Bengalnoestimido Interesting idea, I can't figure out how to make it work.
For part two, to avoid using multiplication and division symbols, try doing logarithmic arithmetic in Neg-Base 10!
"Lets stop using (-)"
"Now we carry the -1"
Since -1≈19 in negadecimals the carying becomes simpler. 4+7=190+1≈100-90+1=11
This also allows us to understand the non-unique representations in a different way. Any two non-unique representations must be separated only infinitesimally. In decimals, the infinitesimal can be represented by 0.00...001.
In negadecimals, the positive infinitesimal is
0.00...001 in a positive place value and
0.00...019 in a negative place value, and vice versa is true for the negative infinitesimal.
In the same way that 0.99...99+0.00...01 cascades in place value
1.90...909=0.09...09+0.0...019
Does as well
This makes me think of 2's complement, used for representing negative numbers in computers. The top-most place has a negative value, but the rest are positive.
For example, for 8-bit numbers, the top bit has a value of -128, and the rest are +64 down to +1.
Just stop being logical. Spoils a good fantasy story!
cool, don't think I've ever heard of that
Yes, but using the tens complement for negative numbers is far too logical. Negadecimal does not have this problem.
You can improve that slightly by having the leftmost digit imply an endless sea of 0s or 9s to the left depending on whether that digit is 0...4 or 5...9. (This requires prepending an extra 0 or 9 for half the numbers.)
In two's compliment, the leftmost bit is basically the minus sign. Computers don't use an actual minus sign because you can have +0 and -0 and that just messes up math.
I know the tenp in Canada probably goes below 0° F pretty frequently, but youd have to use a lot fewer negative numbers if you werent using Celcius. Or just use Kelvin, or, dare I say, Rankine, and you wouldnt have to use negatives at all for temperature.
go all in and use a complex base like -1+i
too many minuses for me
I think it's really interesting that while in decimal, terminating expansions always have a non-terminating equivalent, in negadecimal, some non-terminating expansions also have a second non-terminating equivalent. Makes me curious if this is always the case or if 1/11 is a special case.
🥷🏿 decimal
😭💀
Just as I was looking into bases with negative numbers this video pops up! Please do balanced number systems next (like balanced ternary), I find them fascinating!
Perfect timing, I will!
Me when I am bored after classes when I stay in school and they don't allow phones:
Wait till he realises there are numbers between numbers...
*cough* 3.141 *cough*
You've no way of knowing this upfront, but this video came along at just the right time. As one of my favourite fictional people would say, "Decidedly, I am unwell." This is a fun topic and I enjoyed your easy to follow exposition of it. When I go back to bed, in the spirit of one of Lewis Carroll's pillow problems, I'm going to bring a notepad and pencil with me and play with this concept. Cheers!
This is why Canada needs to use Kelvin for its temperature...
Nigadecimal 💀 🤫👉🏻🧏🏻
To remove the need of a minus sign to represent negative numbers, by far I prefer the balanced number systems. They still use a positive base, but have positive and negative digits, ideally the same number of positive and negative digits.
I'm a fan of Knuth's quarter-imaginary base (2i); using just the numbers 0-3 it maps all complex numbers to real positive numbers.
Couldn't we just use a nega-posi decimal system where we have a doubling of size but avoid using minus signs - let's say the numbers in odd places are positive and the numbers in even places are negative for example: 01 is equal to 1 but 10 is equal to -1 then if we wanna write out a negative number like -124 we simply write 102040 and then if we multiply 102040 by 10 we get 010204. It would be a lot simpler to get used to. Maybe you could showcase that in the next video??
i know it's not perfectly compact because you can write the same number multiple ways (for example 11 = 00 = 0 = 11111111 = 0011001100 but I think it's a fair trade taking that we don't have to use minus signs anymore - further more it would make it easier to write subtraction operations because rather then writing the whole boring sentence: 102 - 45 😴😴😴🥱🥱 you could just write 14052 🔥🔥🔥🥵🥵 and it's obviously the same as the number 0507
"glorified hypen" - plus the idea that Canada invented negative numbers because it was always too cold. Great stuff.
*WHAT* decimals?!
Bro really didn't want to be cancelled
A more straightforward way to convert from negadecimal to decimal is to take advantage of the fact that having a 0 in every other place makes thr number equivalent to yhe decimal notation.
a numbe rin the form EDCBA can be written as E0C0A - 0D0B0, and then do the suntraction as per normal decimal subtraction.
For instance, 17492 = 10402-7090 = 9312.
Bro can say eggman nega right
I totally thought this video was gonna be about two’s complement and variations. It accomplishes the same goal, is widespread in computing, and is easier to convert too - still a fun video tho
Next you need to figure out how to remove that pesky . Symbol used to represent the negative powers of your base.
When considering the powers of negative 10, those exponents are also written in negadecimal.
Since -1 = 19(nd),
0.1 = 1×(10^-1) = 1x(10^19(nd)) = 10000000000000000000(nd').
Almost no thought has been put in here, so I'm sure that there are holes left to fill in to make this system work.
When you wrote for the first time 2i-1 i freaked out how you get imaginary here
I'm smoking weed and the "negadecimal" TTS was tripping me out but i did hear one time you forgot to switch the audio and said "decimal" rather than "negadecimal" - However you're presenting this very well and unless you weren't paying attention to what was happening you would know what you meant anyways... so rather than time stamping it I simply kept watching until hearing the TTS so much drove me to point this out. lmao!
Great videos tho, I watch a lot of math creators here and you somehow stand out a lot of the time and also the videos have good quality!
I too came here high wondering what the hell was up that. 😂
Kelvin >>> Celsius >>> Fahrenheit
Negasexary is the best base. 213 is a very NICE number. It is the Jens "Flammable Maths" Fehlau constant.
I tried making a binary version of this "negabinary" since i think binary is the best number system
i = 0
l = 1
|
( using different symbols for a more digestible look )
the patterns you have to really remember for positive numbers is
li, ll, ii, il
and for negative
il, ii, ll, li
the order of positive numbers from one ( spaced out in pairs so you can see the pattern )
l
l li
l ll
l ii
l il
l li li
l li ll
l li ii
l li il
l ll li
l ll ll
...
the order of negative numbers ( the pattern skips first two numbers since they start with 0 )
ll
li
l l il
l l ii
l l ll
l l li
l i il
l i ii
l i ll
l i li
l l il il
l l il ii
...
addition works the same as in the video, it’s just easier to "subtract the one" from the next digit, because there can only be two outcomes when you do that
for example, 13 ( lllil ) -14 ( llilli )
l ll il+
ll il li
ii ii ll
ll ( -1 )
every base is base 10
Base 2 and base -i×sqrt(e^2) are shocked by this basephobic comment
proof?
@@Why553-k5b_1the base number is always written as '10' in any positional numeral system using Arabic numerals. The word 'ten' and the numeral '10' only represent the same thing in decimal.
@@Why553-k5b_1in base 2, 2 is written as 10. Therefore in base 2, you can say "this is base 10".
This works for any number because n^1=n, so every base will be represented by one copy of the base to the power of one (a 1 in what we call the "tens digit" in base ten) and zero copies of the base to the power of zero (a 0 in the ones digit), or 10
which is base 1010 itself
I'd never encountered *cue wav file* "NEGADECIMAL" or other negative base number systems before, so this was a view into a new way of thinking for me. I'd love to see more about the esoteric multiplication and division and you as a song-singing frog, especially now that we've already invested the time and effort in learning the absolute essentials like Canada (the country of my people) and amphibian body dysmorphia.
2:14 uh ×_×
There might be some utility here for representing very large integers in numerical analysis applications, as this system eliminates the need for sign bits or any other system of tracking whether the stored integer is positive or negative; it's just a matter of whether the most significant digit is an odd-column digit or an even-column digit.
The WHAT decimal
Balanced Ternary!
(Or just binary with two's complement. Might need to do something about needing infinite ones going leftward. Maybe a hyphen could represent that?)
Bro did NOT wanna say negadecimal wrong
the duality of man in the comments
in combined ternary and negaternary, you could weight anything with one of each weight
guys i just thought of a really funny joke-
negadecimal
@@davidmella1174NO-
Balanced ternary (and in general, balanced odd basis) are much healthier alternatives to implement negative numbers
In the 2nd example of negative bases you should have mentioned balanced trinary instead of negatrinary as balanced trinary is more simpler to understand and also is possibly the most used negative base.
You made me ask what a nega is out loud 😵💫
I'm done with complex numbers and that ugly i. I introduce quaterimaginary or base 2i. Using 4 numbers we can represent any conplex number.
20.2_2i = 2*(2i^1) + 2*(2i^-1) = 2*2i + 2*-0.5i = 4i + -i = 4i - i = 3i
Interesting,
Then all of the negative numbers become interleaved with the positive ones, logarithmically
I wonder what the number line (or, more likely "number space") looks like
Balanced Ternary is a fun and overpowered system.
Definitely got that one on my list, it's very cute and I didn't want to bring it up here because it's too fun to just mention and not dig into
Balanced ternary is my favourite. Rounding a number is nothing else than cutting the excess decimals. You can start an addition of many numbers without knowing whether the result will be positive or negative. Ideal to represent quantities that have no natural zero point. In particular every scale that is logarithmic. A drawback is that one half has no finite representation, but two infinite sequences of digits: zero, decimal point, followed by infinitely many 1s, or one, decimal point, followed by infinitely many (-1)s.
Some benefits of Balanced Ternary:
Extremely good radix economy
No - sign needed.
Only 2/9 carries in addition table, even better than the 1/4 of binary.
Negation is just flipping all the (+1) symbols to (-1) and vice versa.
Just do what a computer does and use the leftmost digit negative
kinda reminds me of a notation I developed years ago to display complex numbers using arrows:
for example: (19-16i) = ⇘⇖⇑⇘ and (11+4i) = ⇒⇗⇖
(19-16i)+(11+4i)=(30-12i) = ⇘⇖⇑⇘ + ⇒⇗⇖ = ⇒⇓⇘⭗
(19-16i)-(11+4i)=(8-20i) = ⇘⇖⇑⇘ - ⇒⇗⇖ = ⇓⇗⇓⇖
(19-16i)*(11+4i)=(273-100i) = ⇘⇖⇑⇘ * ⇒⇗⇖ = ⇒⇓⇘⇑⇒⇓
(19-16i)*(11+4i)=~(1.0584-1.8394i) = ⇘⇖⇑⇘ * ⇒⇗⇖ = ⇓⇗.⭗⇗⇖⇖⇐⇒⇐⇖⇓⇘
This look interesting. I guess the arrows represent different combinations of -1 (left), +1(right), i(up), and -i(down). The question is then that base do you use for this? Or does it work in a different way?
Nevermind, just noticed it's balanced ternary. Cool way to compress the two components into one
I love [[negadecimal]], and I understand why you'd use a TTS
For a number system to be useful every quantity has to be represented. It seems like certain quantities would not be representable in a pure negadecimal system. It's a worse problem to have than the Roman numeral system which stops at 3899. You'll have quantities which will be forbidden to be expressed.
Normally I just count up from negative infinity and I can reach any number I want without using the minus sign.
Use of subscript '10' to indicate base ten feels circular. If you're really using other bases often enough to leave doubt then spell out 'ten' in the subscript.
Wow! This sure was a whole lot of words I couldn't follow! 😄
I am very pleased to have seen your video as another numbering system, I called it FrEd, seems wonderfully perplexing and probably quite useless.
Consider 4528 in decimal. Replace powers of ten in (assumed) decimal system with powers of ascending primes ordered so as to increase from right to left.
Unlike decimal system it is a closed numbering system in sense that any prime to power zero adds a "1" and that can be awkward.
Damn, bloody awkward really. So maybe we might need a symbol such as ¥ to denote "No power here"
Anyway, in that awkward, closed numbering system, call it FrEd, 4528 in decimal returns 5792 in FrEd numbering system.
FrEd seems to have advantage of turning p/q rationals into plain old p rationals as negative powers of primes can be helpful.
Example rational 1/2 in decimal 0.5 is (-1) in FrEd and 1/3 becomes 0.3r becomes (-1)0 and see! It is awkward, bloody damned awkward 🙂
Further suggesting a limitation that 1 in decimal is for ever 0 in FrEd or ¥0 to disambiguate as 0¥¥¥0 would be very very bothersome
Robot voice hurts my brain.
Can you differentiate (-n)^m and -n^m where n is a real positive number?
5:01 Idea: Mirror digits to make them negative and add a tilt to the symbol „8“ and it'll definitely cause no problems for any people.
I'm triggered in how you read the base -10 numbers as if they were base 10 numbers. The name "twenty-one" does not refer to the digit string "21" but to the number of vertical lines in "|||||||||||||||||||||" (I don't think you used this specific number, but I didn't want to type - and count - too many vertical bars).
Don't (b-1)s' complement and bs' complement (for base b) have advantages over negative bases? I'd rather write -(229) as 770 than having to use 190 just to represent ten.
I’d rather read my negative numbers as (x * -1) than (y = a + b + c)
You can express negatives in binary without a negative sign using twos complement.
negawatt?
So Canada has 10 provinces... or is that -10 provinces... I'm so confused.
There's nothing positive about Canada, especially the temperatures.
Canada has 190 peovinces, problem solved
i always knew somehow sqrt(-1) was canada's fault....
Addition in negadecimal is wild. You have to exchange borrow and carry.
9+9 is 8 borrow 1. 1 borrowed from 0 is 9, carry 1. So 9+9 = 198. That's decimal 9 + 9 = 18.
190 + 190. 0+0 is 0. 9+9 is 8 borrow 1. 1+1-1 is 1. 190+190=180. That's decimal 10 + 10 = 20.
Imagine teaching that to grade schoolers. I'll keep the minus sign.
Given the unfortunate pronounciation of negadecimal, I propose two alternatives:
antidecimal: literally the opposite of decimal
debtimal: debts can be viewed as having negative money
I think antidecimal is good, though debtimal is pretty cute it only works well with decimal. Debternary isn't so smooth.
And yet he says negative perfectly fine, no robot voice required. It's almost like there's no problem with the pronunciation of the word
Because everyone knows what "negative" is and "nega" isn't a commonly used prefix?
@alexzhukovsky8361 Still doesn't sound bad, and it's obvious that it's just short for negative
I don' see any problems with that word
Okay, but what about the practicalities of just plain counting a number of objects. Would that go wonky in a negative base system?
I wonder what negaunary is (base -1.)
Unary is just another name for tally. They're all base one digits.
Negaunary can only the have two states presence or absence.
And then you got nullary digits (base zero) which is existential math. It doesn't matter what you do: nothing matters. It's just art.
Close enough, welcome back Vsauce
So even digit number are negative and odd digit numbers are positive, so in the end don't you have to write negative sign anyways defeating the purpose?
not very space efficient for certain numbers like -12.
and it is also very difficult to read as there's no indication of whether the digit is positive or negative.
twos complement naturally falls out of binary numbering... no need for negative anything to compute anything
No, that it is a decimal system isn't what makes it special.
Base 8 or base 12 would be much better.
What makes it special is solely that it is positional.
And, ironically, that's what you go on to describe.
Not that it is decimal, but that it is positional.
Words mean things.
You should have been clearer about what you were lauding.
Why the synthetic voice when saying 'negadecimal'? It's kinda annoying.
To get people to comment about it. Rather immature tbh
The negative situation just got crazy...
8:24 Wrong use of carry!!!
but if you don't have a minus sign how could you subtract numbers...