Math and different number system

[Hary R]

Hi guys,

This was bugging me after the last metric vs imperial thread. What if we use a non 10 based number system? 

Ok let's just take the one that all imperial system based know, the 12 based but instead of going from 1 to 12 it goes from 1 to 10 (e.g. 1-2-3-4-5-6-7-8-9-a-b-10) a and be replace 11 and 12. it result in 10/3 to be equal to 4 instead of 1,3333 (or 1/3) in the 10 based system, 3*5 = 13 instead of 15, 5*2 is a instead of 10 and so on (i guess you see what i meant).

How much will that affect all the calculus made in history and can someone with the 10 based system understand the result of those calculus if you explain them how your system work? How long will it take for him to understand? 

12 base is pretty easy but imagine it's a little more complicated, let say 14 based one or 22 based one.

This is a part of what if we meet an intelligent life form, can we understand each other can, can we understand their math?

Edited March 8, 2016 by Hary R

[mikegarrison]

The math doesn't change. Only the numbers do.

[Elrond Cupboard]

Being a computer programmer I work in base 16 a lot of the time, you get used to it.

[Hary R]

1 hour ago, mikegarrison said:

The math doesn't change. Only the numbers do.

I know and that was not my question.

1 hour ago, Elrond Cupboard said:

Being a computer programmer I work in base 16 a lot of the time, you get used to it.

This is the question, you work with it, you use the system in your everyday life (it's your job), but how long did it takes you to adapt to it, to adapt your mind to the result it gives you. (I should have gone for the hexadecimal as an example )  if i ask you to remake all the calculus in science and give result in hexadecimal only to someone who never use it, how long do you think will it take to that person to understand what your are talking about? (let's just say that that person is a scientist that never use hexadecimal.)

I can say, as a financial adviser, i'm used to do calculus but if someone give me the result of my return of investment ration in hexadecimal, the number i will see will make no sense to me, and I can't say how long will take for me to have the figures have a sense again.( if I were to switch to hexadecimal).

Edited March 8, 2016 by Hary R
missing words.....

[mikegarrison]

1 minute ago, Hary R said:

I know and that was not my question.

Really? I could have sworn you wrote: "How much our maths will be affected." Let me check.

Yup, you did. If that wasn't your question, you had a funny way of asking it.

[Hary R]

9 minutes ago, mikegarrison said:

Really? I could have sworn you wrote: "How much our maths will be affected." Let me check.

Yup, you did. If that wasn't your question, you had a funny way of asking it.

Sorry i'm not an english native so sometime i get confuse by how should i say something.

edit: ok i prefer to delete that part so my inability to say something to complicated in a foreign language will not affect the question.

Edited March 8, 2016 by Hary R

[mikegarrison]

34 minutes ago, Hary R said:

Sorry i'm not an english native so sometime i get confuse by how should i say something.

edit: ok i prefer to delete that part so my inability to say something to complicated in a foreign language will not affect the question.

OK. Thanks for the explanation.

[K^2]

1 hour ago, Elrond Cupboard said:

Being a computer programmer I work in base 16 a lot of the time, you get used to it.

Yup. Some bases are easier to get used to than others, but you can get used to pretty much anything.

[Hannu2]

I can not say how long adaptation takes. But it will never be perfect. Typically people learn basic arithmetics during childhood when brains are easier to develop. At least I have programmed over 25 years (from I was 14) and can not use hexadecimal numbers without extra work. Of course I understand values needed typically in programming, and many thing are easier in hexadecimal notation, but if government would change banknotes and price tags to hexadecimal I had to convert everything into decimal system before decisions.

[Green Baron]

2 hours ago, Hary R said:

Hi guys,

This was bugging me after the last metric vs imperial thread. What if we use a non 10 based number system? 

Ok let's just take the one that all imperial system based know, the 12 based but instead of going from 1 to 12 it goes from 1 to 10 (e.g. 1-2-3-4-5-6-7-8-9-a-b-10) a and be replace 11 and 12. it result in 10/3 to be equal to 4 instead of 1,3333 (or 1/3) in the 10 based system, 3*5 = 13 instead of 15, 5*2 is a instead of 10 and so on (i guess you see what i meant).

How much will that affect all the calculus made in history and can someone with the 10 based system understand the result of those calculus if you explain them how your system work? How long will it take for him to understand? 

12 base is pretty easy but imagine it's a little more complicated, let say 14 based one or 22 based one.

This is a part of what if we meet an intelligent life form, can we understand each other can, can we understand their math?

If you want to get behind it, get a feeling trying base 2. Convert your birthdate to base 2, cookbook recipe, anything. Might be a good idea to buy a basic programming book.

The representation of numbers doesn't change the rules to handle them, only the functions that do the operationsif you don't want to convert from one base to another for doing an operation and then convert back. It's as easy as that. To get an idea: a multiplaction/division with the base is a shift operation :-) How long it takes to "understand" ? Depends ... someone with computer science background has advantages.

Examples: - coding 26 letters into 7 bit (base 2) characters. Coding decimal (base 10), octal (base 8) or hexadecimal (16) numbers into binaries (2). Coding already coded letters from one standard (codetable) to another (country specific, operating system specific, ...). It's only the representation, an addition remains an addition.

For every-day life i'd say that in history and culture there have been different systems, when switching from one to another people had to rethink a shorter or longer while :-)

 

Edited March 8, 2016 by kemde

[FancyMouse]

Only unary would change things. It's not defined exactly the same way, though - so I don't know if it counts here.

[Hary R]

Yep math is math no matter what.

The interesting part is (and guys I knew it from the beginning), the result is in fact the same, the number may have change due to the system but what it represent is the same whether you use decimal, octal, hexadecimal or any other system.

[GoSlash27]

I think I'll let Tom Lehrer explain how it works in different base numbering systems.

The mechanics are the same regardless of the numbering system.

Best,
-Slashy

[musicpenguin]

Kinda off topic (I think?), But Binary is pretty cool/interesting. Base two number system.

Almost like a completely different language that you can make numbers or words out of! 

[K^2]

Eh. Ternary is far more efficient. It's a shame we didn't go with it for the computers back in the day. Setting aside a huge boost in memory capacity and performance, even simply splitting data is more convenient. Instead of nibbles, we'd have sets of three trits, which is sufficient to encode 0 + 26 letters of alphabet, which would make for more convenient representation than hex. Alphanumerics would be sent with pairs of these allowing for larger alphabet and more error correction data, while a data stream would probably run in threes. A 9 trit instruction set would already put us close to 16bit machines with only a slight overhead over 8bit architecture. We'd probably have powerful personal computers half a decade sooner.

The only downside is that powers of 3 don't fall close to powers of 10, and the base-3 logic is slightly less intuitive than base-2 logic. Although, tritwise-Z₃ is always an option. Kind of like the bitwise XOR/AND make up a Z₂ ring. In fact, calling the tritwise operators AND and XOR would work just fine.

[Kerbart]

19 hours ago, Hary R said:

Hi guys,

This was bugging me after the last metric vs imperial thread. What if we use a non 10 based number system? 

You mean like roman numerals? Because any ordinal system is based on the concept of "10" -- it just has different values depending on the chosen base (2 for binary, 8 for octal, 16 for hex, etc)

 

[cubinator]

1 hour ago, Kerbart said:

You mean like roman numerals? Because any ordinal system is based on the concept of "10" -- it just has different values depending on the chosen base (2 for binary, 8 for octal, 16 for hex, etc)

 

We did manage to figure out the Mayan number system through archaeological studies, and they used a base-20 system a bit like Roman numerals. Numeric systems are not hard to figure out, unlike language, and to be honest, the first thing I would do if aliens landed in my back yard is write a list of numbers, then maybe some simple arithmetic like 4+5=9, etc.

I use binary to count with my fingers, it's way more efficient if you're willing to put up with numbers like 100, 101, and 10000100.

Edited March 9, 2016 by cubinator

[PB666]

22 hours ago, Hary R said:

Hi guys,

This was bugging me after the last metric vs imperial thread. What if we use a non 10 based number system? 

Ok let's just take the one that all imperial system based know, the 12 based but instead of going from 1 to 12 it goes from 1 to 10 (e.g. 1-2-3-4-5-6-7-8-9-a-b-10) a and be replace 11 and 12. it result in 10/3 to be equal to 4 instead of 1,3333 (or 1/3) in the 10 based system, 3*5 = 13 instead of 15, 5*2 is a instead of 10 and so on (i guess you see what i meant).

How much will that affect all the calculus made in history and can someone with the 10 based system understand the result of those calculus if you explain them how your system work? How long will it take for him to understand? 

12 base is pretty easy but imagine it's a little more complicated, let say 14 based one or 22 based one.

This is a part of what if we meet an intelligent life form, can we understand each other can, can we understand their math?

Silly. take out all the middlemen and go base two. Base 10 is 5 x 2, base 12 is 3 x 2 x 2, base 30 is 2 x 3 x 5 - all of these bases have in common is 2.

 

[RizzoTheRat]

14 hours ago, K^2 said:

Eh. Ternary is far more efficient. It's a shame we didn't go with it for the computers back in the day

Better capacity than binary but presumably harder to implement.  On/Off is nice simple system for an electronic computer.  there were some early ternary computers, how did they work?  Positive and negative voltages, or high and low voltage would presumably both be pretty complicated to manage.

[LN400]

Just as a brain teaser: Imagine the concept of integration using Roman numerals. Can it even be done? How do you treat limits using Roman numerals?

Can you use any additive system, like the Roman numerals, at all for calculus or is a positional system required?

How much did the introduction of zero mean to our mathematics today? How much did the invention of the positional systems mean? Imaginary numbers?

As for base 12 over base 10, or using base 2 or any other such base, it doesn't matter as far as expressing values are concerned. You will always have those values you can't express as a fraction. The values may not be the same for every base but for any base there will be irrational and transcendental numbers.

As for the basic operations you can perform, they are the same for each of these bases. Addition, subtraction, multiplication, division and powers. Sqrt(11001₂) = 101₂ just as sqrt(25) = 5.

Another brain teaser: How would Euclid cope with the idea that parallell lines may cross each other? It's hard to imagine by many even today, long after the introduction of geometry that allows that. Not all accept the idea at all. There are people who still insist all numbers are rational, that imaginary numbers are not even used by anyone serious about maths. And still we haven't left planet Earth.

As for maths being One Universal Truth, that is not really the case. What you perhaps could rather say is any maths out of a vast space of possible maths is universally true. Problem occurs when we say, meet some extraterrestrial using maths we are not aware of exists and they are unaware of the existence of our maths. What would such an alien's idea of a number be? We can not say that our understanding of numbers is universal. Our understanding and our numbers are results of how our brains evolved as our sensory organs evolved. There is no reason to believe a brain evolved on another planet would work the same way as our own. We can't be so anthropocentric as to think evolution strives to produce "humans" elsewhere. We are quite unique and so is, possibly, our maths.

EDIT: As a side note: A Japanese mathematician of some caliber, Shinichi Mochizuki has possibly proved the abc conjecture. Problem is, no other mathematician is capable of telling whether or not the proof is valid. Workshops where the experts in the relevant fields came together were organised. Not to see if the proof was valid, but simply to understand what on god's green earth the possible proof was saying. They still haven't succeeded. The human understanding can not be pushed beyond all limits.

Edited March 10, 2016 by LN400
typo

[Hary R]

2 hours ago, LN400 said:

your post 

This is interesting. In the tread that inspire me the question here (and this is a simplified version of what i really had in mind) I said that alien measurement system will be nothing like what we have. It's just a speculation but the chance of it to be true is far greater then the opposite.

Here you add that even the concept of math might be different, that our understanding of what math is might be different of that of a different civilization, a different spices that evolve differently.

bottom line is we can't be sure until we meet another intelligent life form.

[LN400]

Here's a talk that raises some interesting points on maths including how 1+1 does not neccessarily equal 2. It's a long vid but the first half hour is interesting enough.

 

EDIT:

Hary R is right in that we just can't make any educated guess, only wild guesses, until the day we would meet these aliens, if that ever happens.

I'll be wildely guessing here so, consider this: We put a lot of emphasis on the Unit, 1 if you will but what exactly is a unit? I don't think many would disagree that by 1 we mean a single object of some kind be it a physical object or a non-physical one but is that really all "1" can be and does it even have to be just one 1, or could there be different 1s that are not the same but still share this quality of oneness? We use 1, the unit, to build up other numbers as collection of units and/or parts of a unit. The value 1 has it's own set of properties. Some or all properties in that set may be shared by other numbers but no other number have the same, full set of properties. With these numbers we can play around and construct other objects, like matrices, or a grid of values arranged to represent something. What if some other species elsewhere in the universe don't think in terms of units as we do but instead think of numbers as matrices of properties so that to them, the idea of a single digit value say 1, is just plain unthinkable. One can try to imagine a system of matrices where each element in one such matrix has their own set of properties, like our matrices but to these aliens, it's the matrix itself that is the number and they can have multiple matrices we have no other way of translating other than to 1. yet they are all different, and they build a consistent numeral system just as powerful as ours, or even more powerful allowing for operations we can't even imagine.

Edited March 10, 2016 by LN400

[SgtSomeone]

On 3/9/2016 at 5:07 PM, K^2 said:

Eh. Ternary is far more efficient. It's a shame we didn't go with it for the computers back in the day. Setting aside a huge boost in memory capacity and performance, even simply splitting data is more convenient. Instead of nibbles, we'd have sets of three trits, which is sufficient to encode 0 + 26 letters of alphabet, which would make for more convenient representation than hex. Alphanumerics would be sent with pairs of these allowing for larger alphabet and more error correction data, while a data stream would probably run in threes. A 9 trit instruction set would already put us close to 16bit machines with only a slight overhead over 8bit architecture. We'd probably have powerful personal computers half a decade sooner.

The only downside is that powers of 3 don't fall close to powers of 10, and the base-3 logic is slightly less intuitive than base-2 logic. Although, tritwise-Z₃ is always an option. Kind of like the bitwise XOR/AND make up a Z₂ ring. In fact, calling the tritwise operators AND and XOR would work just fine.

I thought the reason we went with binary is because it was easy to represent each digit: it's either on or off. 0 or 1. Having three possible states means you need transistors (or vacuum tubes!) to be half way... something. Either half charged, or half open, or half full. Which is a lot harder to design and build for, and squeezing all sorts of current through 'half open' gates would introduce all sorts of electrical friction and heat in a modern computer.

[FancyMouse]

11 hours ago, LN400 said:

Here's a talk that raises some interesting points on maths including how 1+1 does not neccessarily equal 2. It's a long vid but the first half hour is interesting enough.

1+1=0 in binary doesn't mean 1+1 <> 2 because in binary, 2=0. Note I'm not talking about binary number system, only just binary digit, or more precisely, Z/2Z or F_2 or GF(2).

I didn't hear the video for more than 1 minute. It's all people talking around without a board showing any symbol or digit makes me feel the whole thing isn't serious at all.

11 hours ago, LN400 said:

We put a lot of emphasis on the Unit, 1 if you will but what exactly is a unit? I don't think many would disagree that by 1 we mean a single object of some kind be it a physical object or a non-physical one but is that really all "1" can be and does it even have to be just one 1, or could there be different 1s that are not the same but still share this quality of oneness? We use 1, the unit, to build up other numbers as collection of units and/or parts of a unit. The value 1 has it's own set of properties. Some or all properties in that set may be shared by other numbers but no other number have the same, full set of properties. With these numbers we can play around and construct other objects, like matrices, or a grid of values arranged to represent something. What if some other species elsewhere in the universe don't think in terms of units as we do but instead think of numbers as matrices of properties so that to them, the idea of a single digit value say 1, is just plain unthinkable. One can try to imagine a system of matrices where each element in one such matrix has their own set of properties, like our matrices but to these aliens, it's the matrix itself that is the number and they can have multiple matrices we have no other way of translating other than to 1. yet they are all different, and they build a consistent numeral system just as powerful as ours, or even more powerful allowing for operations we can't even imagine.

(Mathematical) unit is defined by the property that anything multiplying it doesn't change the result. In a field, or more generally in a commutative ring with a unit, unit is unique because if we have 1 and 1', 1=1*1'=1' by the definition of unit. Anything else about 1 is just consequences which must be true given the axioms/definitions.

In the case of matrices where multiplication is not commutative, once a two-sided unit exists (identity matrix, in this case), then it can also be proven that there's no other unit, whether two-sided or one-sided. Same proof.

It's for sure out of the scope of number systems, but unfortunately what you are worrying is already taken care of by the more general, more abstract concepts of algebraic structures. And people are already dealing with a "number" system on matrices (I mean seriously, even defining metric, topology, talking about series, convergence, polynomials etc.) viewing a matrix as what normal people sees a number for a long time. It isn't alien stuff, either - you just probably didn't know.

[LN400]

7 hours ago, FancyMouse said:

1+1=0 in binary doesn't mean 1+1 <> 2 because in binary, 2=0. Note I'm not talking about binary number system, only just binary digit, or more precisely, Z/2Z or F_2 or GF(2).

I didn't hear the video for more than 1 minute. It's all people talking around without a board showing any symbol or digit makes me feel the whole thing isn't serious at all.

(Mathematical) unit is defined by the property that anything multiplying it doesn't change the result. In a field, or more generally in a commutative ring with a unit, unit is unique because if we have 1 and 1', 1=1*1'=1' by the definition of unit. Anything else about 1 is just consequences which must be true given the axioms/definitions.

In the case of matrices where multiplication is not commutative, once a two-sided unit exists (identity matrix, in this case), then it can also be proven that there's no other unit, whether two-sided or one-sided. Same proof.

It's for sure out of the scope of number systems, but unfortunately what you are worrying is already taken care of by the more general, more abstract concepts of algebraic structures. And people are already dealing with a "number" system on matrices (I mean seriously, even defining metric, topology, talking about series, convergence, polynomials etc.) viewing a matrix as what normal people sees a number for a long time. It isn't alien stuff, either - you just probably didn't know.

But now you are applying our understanding of it to all kinds of maths, you even say "defined as". Defined for sure but defined by us, the humans. Rings, fields, the unit, they are all human inventions or a result of the human understanding to bring order in our understanding of our maths. My point is, we can not assume that our inventions are applicable to all maths. As for the matrices, you just highlighted my point. We are limited in how much we can imagine. That goes for each and every one of us from dumbo to Hawking. We have different limits but that's irrelevant. We all have our limits. We can not imagine what's beyond our limit.

As for that vid, and 1 + 1 = 0, one thing they do mention is modular arithmetics (NOT binary as you presented it as) where 1 + 1 = 0 is as valid and as significant as 1 + 1 = 2. The same operation, the same value (1) and 2 distinctly different results. That was one of the points I was talkng about. Now if we can have 1 + 1 = 0 or 2 depending on the rules, then it's false to say that 1 + 1 is always 2. If that is false then who are we to say there aren't other ways to add those 1's to get even more different answers, IF we change the rules.

 

EDIT: As for the lack of board. Mathematics isn't only about scribbling down symbols, numbers, equations and so on. Our mathematics has roots going firmly into philosophy, logic and reasoning. Take philosophy and reasoning out of it and you won't have any maths as we know it. For that, you won't need a blackboard.

Edited March 11, 2016 by LN400