12 Digit Number System

[Kanad 351]

The decimal number system is the one that we all most commonly use. Its popularity is because of its simplicity and ‘easiness’, compared to the other number systems.

So, I was wondering - why does the decimal number system have 10 digits? Why exactly 10? My brother commented that that’s because its called the DECimal system. But still, why did we choose 10? Why not some other number? Some of you might say that it’s because ‘10′ is simple, if you are one of them, you fail to notice that the reason ‘10′ is ‘simple’ and ‘easier’ is because we took it as the ‘base’ and have been calculating on it since we were toddlers.

So, I decided to make up my own new number system - one with 12 digits. I called the 11th digit ‘zo’ [which would be equal to ‘10′ of the decimal system] and the 12th digit ‘ri’. At first, calculation was unbelievably difficult, but soon I got used to it, and it worked perfectly well. I didn’t see what the issue was with choosing ‘10′ as the base over every other digit. I was calculating on my fingers when it struck me - we have 10 fingers! Well, not exactly 10 ‘fingers’ but 8 fingers and two thumbs, which are collectively called ‘digits’. It was a moment of revelation for me. We have 10 digits, because nature and evolution gave us 10 ‘digits’!

When we run out of ‘digits’, we put add +1 to the ten’s place and return to 0 in the unit place and when we cross 9 ‘tens’, we put +1 in the hundred’s place and so on. This works perfectly well with any given number of digits [though I have only tried with even; I shall try a 7 digit [odd] number system soon].

This makes us think. When we come across ‘weird’ quantities like  299,792,458, it’s not that they are weird, unnatural or odd, it’s just that’s the way our number system represents those particular quantities.

I did the calculations on a piece of paper. All the arithmetic calculations - addition, subtraction, multiplication, division work perfectly well. Unlike the decimal system though, for the number to have a terminating decimal value in my 12 digit number system [e.g. 58.27], it would have to be divisible by “3^n × 2^2n”, unlike the decimal number system’s “2^n × 5^n” [because the factors of 12 are 2*2*3, whereas that of 10 are 5*2]. This would imply that numbers having [recurring?] non-terminating value in our 10-system would have a terminating value in some other system. At least for rational numbers. I am not sure about irrational numbers.

I think it would be better to make a video about this. Only if I had a camera…  

I am well aware that this is not a ‘discovery’. This is something very basic that some people and most mathematicians most probably would already know. Still, finding it for myself was quite a feat, one that I am proud of.

This brings me to a question: Is it possible for irrational numbers [numbers with non-recurring decimal value] to have a terminating value in some other n-digit decimal system? If yes, then what should be the value of n for the square root of 2 to have a terminating value? @Master Flap @Donald J Trump

[AJ Mysterio 1,494]

I swear to God that went completely over my head but I still continued reading it to the very end

[Conan Edogawa 837]

1 hour ago, Kanad said:

The decimal number system is the one that we all most commonly use. Its popularity is because of its simplicity and ‘easiness’, compared to the other number systems.

So, I was wondering - why does the decimal number system have 10 digits? Why exactly 10? My brother commented that that’s because its called the DECimal system. But still, why did we choose 10? Why not some other number? Some of you might say that it’s because ‘10′ is simple, if you are one of them, you fail to notice that the reason ‘10′ is ‘simple’ and ‘easier’ is because we took it as the ‘base’ and have been calculating on it since we were toddlers.

So, I decided to make up my own new number system - one with 12 digits. I called the 11th digit ‘zo’ [which would be equal to ‘10′ of the decimal system] and the 12th digit ‘ri’. At first, calculation was unbelievably difficult, but soon I got used to it, and it worked perfectly well. I didn’t see what the issue was with choosing ‘10′ as the base over every other digit. I was calculating on my fingers when it struck me - we have 10 fingers! Well, not exactly 10 ‘fingers’ but 8 fingers and two thumbs, which are collectively called ‘digits’. It was a moment of revelation for me. We have 10 digits, because nature and evolution gave us 10 ‘digits’!

When we run out of ‘digits’, we put add +1 to the ten’s place and return to 0 in the unit place and when we cross 9 ‘tens’, we put +1 in the hundred’s place and so on. This works perfectly well with any given number of digits [though I have only tried with even; I shall try a 7 digit [odd] number system soon].

This makes us think. When we come across ‘weird’ quantities like  299,792,458, it’s not that they are weird, unnatural or odd, it’s just that’s the way our number system represents those particular quantities.

I did the calculations on a piece of paper. All the arithmetic calculations - addition, subtraction, multiplication, division work perfectly well. Unlike the decimal system though, for the number to have a terminating decimal value in my 12 digit number system [e.g. 58.27], it would have to be divisible by “3^n × 2^2n”, unlike the decimal number system’s “2^n × 5^n” [because the factors of 12 are 2*2*3, whereas that of 10 are 5*2]. This would imply that numbers having [recurring?] non-terminating value in our 10-system would have a terminating value in some other system. At least for rational numbers. I am not sure about irrational numbers.

I think it would be better to make a video about this. Only if I had a camera…  

I am well aware that this is not a ‘discovery’. This is something very basic that some people and most mathematicians most probably would already know. Still, finding it for myself was quite a feat, one that I am proud of.

This brings me to a question: Is it possible for irrational numbers [numbers with non-recurring decimal value] to have a terminating value in some other n-digit decimal system? If yes, then what should be the value of n for the square root of 2 to have a terminating value? @Master Flap @Donald J Trump

Oh, goody!

First, you might have noticed that the base 10 system is more convenient and simplistic than other systems. Also, we have 10 fingers and toes.

It is notable that the finger made a base 10 system extremely obvious and intuitive. As Georges Ifrah expands...

 

Quote

 

Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and "ten" are respectively moro andmbouna: moro is actually the word for "hand" and mbouna is a contraction of moro ("five") and bouna, meaning "two" (thus "ten"="two hands").

It is therefore very probable that the Indo-European, Semitic and Mongolian words for the first ten numbers derive from expressions related to finger-counting. But this is an unverifiable hypothesis, since the original meanings of the names of the numbers have been lost.

 

 

 

Quote

...the hand makes the two complementary aspects of integers entirely intuitive. It serves as an instrument permitting natural movement between cardinal and ordinal numbering. If you need to show that a set contains three, four, seven or ten elements, you raise or bend simultaneouslythree, four, seven or ten fingers, using your hand as cardinal mapping. If you want to count out the same things, then you bend or raise three, four, seven or ten fingers in succession, using the hand as an ordinal counting tool.

Though not all of human civilization had utilized a base 10 number system. As noted, the Babylonians used a sixagesimal number system, the Mayans a vigesimal(foot AND toe?).

Quote

So, I decided to make up my own new number system - one with 12 digits. I called the 11th digit ‘zo’ [which would be equal to ‘10′ of the decimal system] and the 12th digit ‘ri’. At first, calculation was unbelievably difficult, but soon I got used to it, and it worked perfectly well. I didn’t see what the issue was with choosing ‘10′ as the base over every other digit. I was calculating on my fingers when it struck me - we have 10 fingers! Well, not exactly 10 ‘fingers’ but 8 fingers and two thumbs, which are collectively called ‘digits’. It was a moment of revelation for me. We have 10 digits, because nature and evolution gave us 10 ‘digits’!

Yes, that although base 10 was the prevailing base for most of the world, other bases were still in use. Various Nigerian tribes such as Janji, Piti,  and Gwandara, used a duodecimal base system. The Chepang language of Nepal, and the Mahl Language of India are known to use duodecimal base systems. 

Why 12? There are 12 lunar cycles, 12 finger bones(ok, ok, more abstruse but whatevers). Therefore, it IS possible to count to 12 on your hands with your finger bones.

Quote

When we run out of ‘digits’, we put add +1 to the ten’s place and return to 0 in the unit place and when we cross 9 ‘tens’, we put +1 in the hundred’s place and so on. This works perfectly well with any given number of digits [though I have only tried with even; I shall try a 7 digit [odd] number system soon].

An odd number system is generally more of the same but troublesome. 1/2 in base 5, for instance, is 0.2222, and other typically nice fractions turn messy.

Quote

This makes us think. When we come across ‘weird’ quantities like  299,792,458, it’s not that they are weird, unnatural or odd, it’s just that’s the way our number system represents those particular quantities.

Good observation. pi in base pi is far more typical, as you might expect, though a few innovations are still to be made..

Quote

I did the calculations on a piece of paper. All the arithmetic calculations - addition, subtraction, multiplication, division work perfectly well. Unlike the decimal system though, for the number to have a terminating decimal value in my 12 digit number system [e.g. 58.27], it would have to be divisible by “3^n × 2^2n”, unlike the decimal number system’s “2^n × 5^n” [because the factors of 12 are 2*2*3, whereas that of 10 are 5*2]. This would imply that numbers having [recurring?] non-terminating value in our 10-system would have a terminating value in some other system. At least for rational numbers. I am not sure about irrational numbers.

You probably mean non terminating as in 2/3 is rational, etc etc.. 

It is useful to cite the definition of a rational number.

A rational number is a number that can be expressed in the form of a/b, where a and are b are integers. The definition has absolutely nothing to do with the base, which is just like saying " will my gameboy play my Pokemon Blue in a totally different way if it is painted green?"(spoiler: no.)

It is also the case for a rational base n expansion; if, and only if it is a rational integer, then a irrational integer has a non terminating(i.e, as in repeating.) expansion in EVERY rational base.

Suppose a number x has a base n expansion that begins with a terminating(non repeating..) sequence of digits 

a1 a2 a3....\

It turns out it is rational, and we can find a fraction for it.

For example suppose we are working in base 8, and we want to find a fraction for the number 0.13456456456… where the digits are understood base 8. Then i=2i=2, and a1a2=a1a2= 13; and j=3j=3, and b1b2b3=b1b2b3= 456. Then we can calculate that

I will expand this lengthy post with more info about negative bases, imaginary, and irrational bases.

1 hour ago, Kanad said:

The decimal number system is the one that we all most commonly use. Its popularity is because of its simplicity and ‘easiness’, compared to the other number systems.

So, I was wondering - why does the decimal number system have 10 digits? Why exactly 10? My brother commented that that’s because its called the DECimal system. But still, why did we choose 10? Why not some other number? Some of you might say that it’s because ‘10′ is simple, if you are one of them, you fail to notice that the reason ‘10′ is ‘simple’ and ‘easier’ is because we took it as the ‘base’ and have been calculating on it since we were toddlers.

So, I decided to make up my own new number system - one with 12 digits. I called the 11th digit ‘zo’ [which would be equal to ‘10′ of the decimal system] and the 12th digit ‘ri’. At first, calculation was unbelievably difficult, but soon I got used to it, and it worked perfectly well. I didn’t see what the issue was with choosing ‘10′ as the base over every other digit. I was calculating on my fingers when it struck me - we have 10 fingers! Well, not exactly 10 ‘fingers’ but 8 fingers and two thumbs, which are collectively called ‘digits’. It was a moment of revelation for me. We have 10 digits, because nature and evolution gave us 10 ‘digits’!

When we run out of ‘digits’, we put add +1 to the ten’s place and return to 0 in the unit place and when we cross 9 ‘tens’, we put +1 in the hundred’s place and so on. This works perfectly well with any given number of digits [though I have only tried with even; I shall try a 7 digit [odd] number system soon].

This makes us think. When we come across ‘weird’ quantities like  299,792,458, it’s not that they are weird, unnatural or odd, it’s just that’s the way our number system represents those particular quantities.

I did the calculations on a piece of paper. All the arithmetic calculations - addition, subtraction, multiplication, division work perfectly well. Unlike the decimal system though, for the number to have a terminating decimal value in my 12 digit number system [e.g. 58.27], it would have to be divisible by “3^n × 2^2n”, unlike the decimal number system’s “2^n × 5^n” [because the factors of 12 are 2*2*3, whereas that of 10 are 5*2]. This would imply that numbers having [recurring?] non-terminating value in our 10-system would have a terminating value in some other system. At least for rational numbers. I am not sure about irrational numbers.

I think it would be better to make a video about this. Only if I had a camera…  

I am well aware that this is not a ‘discovery’. This is something very basic that some people and most mathematicians most probably would already know. Still, finding it for myself was quite a feat, one that I am proud of.

This brings me to a question: Is it possible for irrational numbers [numbers with non-recurring decimal value] to have a terminating value in some other n-digit decimal system? If yes, then what should be the value of n for the square root of 2 to have a terminating value? @Master Flap @Donald J Trump

1. No.

2/ sqrt(2).

[Shawnic 2,977]

Incredible Kanad! Not because you discovered this independently but because of your dedicated time and research to this in spite of it not being required of you. You made my day.  

As for the base thing, I don't think any particular base is better than the others, at least for us humans. As you said, we learned counting on our fingers and hence base 10. Did you arbitrarily choose base 12? Or was it because we have 10 fingers plus 2 legs? You should also look into base 2. 

 

I don't think that using another base will get rid of irrationality. Because while the base is different, it ultimately represents the same number. Pi is still pi. That's just what I think though. I hadn't looked into this part of math much. Since you've sparked my interest, I'll look into it and tell you what I find. 

As for the speed of light thing, I don't think it'll serve any purpose to find a system where it's more elegant. Universal constants are pretty arbitrary in appearance since it's physics not math. Unless you suspect a more profound involvement of math in our Universe (I do too, but I have a hunch it's not because of the weirdness in the numbers) 

[Conan Edogawa 837]

6 minutes ago, Master Flap said:

Incredible Kanad! Not because you discovered this independently but because of your dedicated time and research to this in spite of it not being required of you. You made my day.  

As for the base thing, I don't think any particular base is better than the others, at least for us humans. As you said, we learned counting on our fingers and hence base 10. Did you arbitrarily choose base 12? Or was it because we have 10 fingers plus 2 legs? You should also look into base 2. 

 

I don't think that using another base will get rid of irrationality. Because while the base is different, it ultimately represents the same number. Pi is still pi. That's just what I think though. I hadn't looked into this part of math much. Since you've sparked my interest, I'll look into it and tell you what I find. 

As for the speed of light thing, I don't think it'll serve any purpose to find a system where it's more elegant. Universal constants are pretty arbitrary in appearance since it's physics not math. Unless you suspect a more profound involvement of math in our Universe (I do too, but I have a hunch it's not because of the weirdness in the numbers) 

 Maybe there are 12 finger bones, I suppose..? 

Irrational numbers are irrational irregardless of their base, except for when it is base (irrational number..)

 

[Agent P 2,480]

This was really interesting. Like the others said, as far as I know we follow a ten base system because we have 10 fingers. Some Inuit people use a 20 base decimal system because they count with their fingers and their toes together.

[kiba 374]

I'm sorry i m really tired to read this but it sounds smart I just think we chose 10 because 10 fingers idk (oh and thats why i should have read it cause you mentioned im sorry ill be leaving now)

Edited October 3, 2016 by Kibe
cause me dumb

[Shawnic 2,977]

2 hours ago, Donald J Trump said:

 Maybe there are 12 finger bones, I suppose..? 

Irrational numbers are irrational irregardless of their base, except for when it is base (irrational number..)

 

An irrational base is impossible since obviously, base number has to be a natural number. 

[Shawnic 2,977]

A friend and I were working on a project (top secret mission, currently suspended) and we did a lot of research in base 7. But what made you decide to try base 7 next? 

Also, irrationality is not defined by what goes in the denominator. As Trump I mean, Field said, any number that can be expressed as a fraction is rational. Pi is not a real fraction but a quotient of an irrational number with another number to give another irrational number. If the denominator is not of the form (2^m)*(5ⁿ) then the number will be non terminating repeating. Irrational numbers are non terminating non repeating. The digits of pi don't recur. If they did, one could express it as a fraction. 

[Conan Edogawa 837]

On 10/3/2016 at 6:41 AM, Master Flap said:

An irrational base is impossible since obviously, base number has to be a natural number. 

A non integer representation base can indeed be possible. 

 

For a non integer radix(base), the non integer radix   β  > 1,  the value of the following decimal expansion

 .

is 

These d's are non negative integers less than   β.

This is not a impractical idea of a base, for their are myriads of applications for  β- expansions, such as in coding theory(Kautz) and quasicrystals.

General information about  β expansions

 β expansions are generalities of decimal expansions. While infinite expansions are NOT unique(.9999.. = 1.), all finite decimal expansions are unique.

Note that finite  β expansions are NOT unique, for example   φ + 1 = φ² for β = φ, the golden ratio!

I will help clarify this terminology by exemplifying this.

Phi-base, or Golden base.

Any non negative real number can be expressed as a base phi number only using the digits 0 and 1. 200 is not allowed, for it is not in  standard form.

Despite using an irrational base, all non negative numbers have a unique representation that terminates.

Decimal	Powers of φ	Base φ
1	φ0	1
2	φ1 + φ−2	10.01
3	φ2 + φ−2	100.01
4	φ2 + φ0 + φ−2	101.01
5	φ3 + φ−1 + φ−4	1000.1001
6	φ3 + φ1 + φ−4	1010.0001
7	φ4 + φ−4	10000.0001
8	φ4 + φ0 + φ−4	10001.0001
9	φ4 + φ1 + φ−2 + φ−4	10010.0101
10	φ4 + φ2 + φ−2 + φ−4	10100.0101\

 

1/2 ≈ 0.010 010 010 01

 

WILL EXPAND ON THIS

Edited October 24, 2016 by Donald J Trump
educate the masses

[Kanad 351]

20 hours ago, Donald J Trump said:

For example suppose we are working in base 8, and we want to find a fraction for the number 0.13456456456… where the digits are understood base 8. Then i=2i=2, and a1a2=a1a2= 13; and j=3j=3, and b1b2b3=b1b2b3= 456. Then we can calculate that

I don't understand how/why you took i as 2 and j as 3. Please explain.

 

20 hours ago, Master Flap said:

Did you arbitrarily choose base 12?

I thought of adding an extra digit to the base 10 system, but I didn't want to deal with an odd number [11], so adding 2 extra digits seemed the to be the best option.

17 hours ago, Master Flap said:

what made you decide to try base 7 next?

I picked that randomly.

[Kanad 351]

The man was over 2 years younger than I am right now. Humbling.

[Shawnic 2,977]

39 minutes ago, Kanad said:

The man was over 2 years younger than I am right now. Humbling.

Technically not a man.

[Jazlyn 522]

What did i just read?

[Emfoofoo 1,389]

21 hours ago, Kibe said:

I'm sorry i m really tired to read this but it sounds smart I just think we chose 10 because 10 fingers idk (oh and thats why i should have read it cause you mentioned im sorry ill be leaving now)

well, at least you tried. i probably would've gotten to the same conclusion too. 

[Agent P 2,480]

On 10/3/2016 at 6:58 PM, Kibe said:

I'm sorry i m really tired to read this but it sounds smart I just think we chose 10 because 10 fingers idk (oh and thats why i should have read it cause you mentioned im sorry ill be leaving now)

 I skimmed through it too lol

I didn't get time to finish it

[Conan Edogawa 837]

11 hours ago, Kanad said:

I don't understand how/why you took i as 2 and j as 3. Please explain.

 

I thought of adding an extra digit to the base 10 system, but I didn't want to deal with an odd number [11], so adding 2 extra digits seemed the to be the best option.

I picked that randomly.

0.(13)(i = 2) 456456456(three digits, thus j = 3.)

[Conan Edogawa 837]

On 10/3/2016 at 6:41 AM, Master Flap said:

An irrational base is impossible since obviously, base number  have to be a natural number. 

Now I don't want budding mathematicians to get invalid information, so lo and behold, a complex base(2i).

The 2i base system had been submitted due to a mathematical talent investigation by Donald Knuth.

We all know that decimal expansions are known as

Note that

 

Thus...

To convert 1100_2i into a decimal, use the formula..../

 .

Converting into a 2i base system is obviously more difficult.

Most numbers have a unique decimal expansion, but just as 1 = .999999..., 1/5_2i =  1.(0300)…_2i = 0.(0003)…_2i.

To convert any digit into a 2i base system, one must split it into its imaginary and real components, convert them SEPARATELY, and interweave.

Example: -1 + 4i

 Since –1+4i is equal to –1 plus 4i, the quater-imaginary representation of –1+4i is the quater-imaginary representation of –1 (namely, 103) plus the quater-imaginary representation of 4i (namely, 20), which gives a final result of –1+4i = 123_2i.

Imaginary Part

Multiply by 2i, which obviously gives a real, and find the 2i base expansion, and finally shift 1 place to the left.

Finding  the real expansion unfortunately takes a convoluted systems of equations to solve.

Finding the real expansion has a tough prerequisite: guess how many digits the base 2i digit will have. Since 2i numbers generally have a lengthy expansion, it is wise to consider numerous digits.

EXAMPLE IN PROGRESS.

 

 

[Kanad 351]

On 10/5/2016 at 0:47 AM, Donald J Trump said:

0.(13)(i = 2) 456456456(three digits, thus j = 3.)

Assuming the number in decimal is 0.3333..., what would be the values of i and j? I'll be trying this formula out.

On 10/4/2016 at 2:29 PM, Master Flap said:

Technically not a man.

[Conan Edogawa 837]

3 hours ago, Kanad said:

Assuming the number in decimal is 0.3333..., what would be the values of i and j? I'll be trying this formula out.

I believe i would be 1 and j 1. I had only developed this formula for two decimal expansions(0.909223223..)