Number bases

>>8302801
binary is used in computing because it's the only possible way to do computing
>this gate is off
>this gate is on
That's as complicated as you can actually get with circuitry, although ternary (base 3) computers have been proposed

>why do other number bases exist
that's like asking why different colors exist. they just do, and each one has its own way of being defined.

>>8302801
In base 9 you count to ten then move to a new digit
0-9
10-9
20-9
Etc...
Is the same thing with other bases.
Binary is
0
1
Instead of a 2, you add a new digit
01
11
001
011
111 etc...

For other bases, it's the same.
Computers use logic. True/false statements are physically generated using circuits, so binary is natural.

Bases are also useful for modular arithmetic, which is what you have with a clock, and used in encryption.
You can add/subtract multiply/divide etc.. in different bases too.
There's many more examples, but that's a start.

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>>8302815
>0-9
>10-9
>20-9
>base 9

>>8302819
It was supposed to be base 10 obviously. I only post when I'm on my phone and I'm only on my phone when I'm smoking. Hard for me to type 1 handed

>>8302801

The trick is you should drop your assumption that you know about base 10. If you knew about base 10, you'd know about all the other bases. Most people don't realize what they're actually doing when they work with base 10 because they got indoctrinated into mindlessly manipulating symbols without thinking about what they mean. Take the random base 10 number 9347 as an example. What does it mean to put those single digit numbers next to each other and to call it a new number named nine thousand three hundred forty-seven? There's in fact a very specific mathematical function you're performing on those different single digit numbers. It is:

x = digit
i = digit index
b = 10
f(x,i,b) = x * 10^i

x = 7
i = 0
b = 10
f(7,0,10) = 7 * 10^0
7 * 10^0 = 7

x = 4
i = 1
b = 10
f(4,1,10) = 4 * 10^1
4 * 10^1 = 40

x = 3
i = 2
b = 10
f(3,2,10) = 3 * 10^2
3 * 10^2 = 300

x = 9
i = 3
b = 10
(f(9,3,10) = 9 * 10^3
9 * 10^3 = 9000

9000 + 300 + 40 + 7 = 9347.

Non-10 bases are exactly the same, just change out 10 for whatever the base is and apply the function to each digit before summing them all together. So 4C2 in hexadecimal is (keeping in mind we use letters to refer to the extra number possibilities in a base greater than 10, so that A is 9+1 (10), B is 9+2 (11). C is 9+3 (12), etc.):

x = 2
i = 0
b = 16
f(2,0,16) = 2 * 16^0
2 * 16^0 = 2

x = C
i = 1
b = 16
f(C,1,16) = C * 16^1
C * 16^1 = 12 * 16^1
12 * 16^1 = 192

x = 4
i = 2
b = 16
f(4,2,16) = 4 * 16^2
4 * 16^2 = 1024

1024 + 192 + 2 = 1218

>>8302805
>That's as complicated as you can actually get with circuitry
See that part I can understand if it's a hardware limitation. Although I am confused as to how binary forms letters or more complex programs if each addition of a digit represents a new number and there are infinite numbers.

>that's like asking why different colors exist. they just do, and each one has its own way of being defined.
Yes but I mean colors exist as differing wavelengths of light but as far as I can tell number base systems only exist as a theoretical concept for the most part right?
Most people count numbers in base 10 because almost all human societies have decided to give each number a corresponding word and over time associated the same symbols for the numbers that happen to be combination of 0-9 but why would 5 be 8? I just don't get it?
I mean really there is nothing to define our numbers as being specifcally base 10 other than the Arabic symbols we use to represent them while any society uses base 4 would just use their own words for those numbers so there really wouldn't be any obvious distinction to any listener.

Just what is the point?

>>8302801
Another thing I don't understand is that if a numbers base is the total amount of symbols that can be allowed to exist why is it that in base 4 multiples of 4 can exist like 8 or 12?
And if every number must be either a multiple of 4 or divisible by 4 why can 3 exist when it would leave a remainder?

>>8302855
Every number can be represented in every base.

In base 4 you have
4^0 + 2*4^1 = 8

[math]
\displaystyle 0.999... = 0.1_3+0.1_3+0.1_3

[/math]

>>8302832
>why would 5 be 8?
huh?

>>8302830
This really makes you think

>>8303023
that's 9,

you mean

[math]\displaystyle
20_4 = 2 \cdot 4^1 + 0 \cdot 4^0 = 8_{10}
[/math]

>>8303070
From what I understand in base 4, 5 is 8.
12348
Correct me if this is wrong

>>8303091
in base 4, 5 is 11
why the fuck would 5 be 8

>>8302801
>>8302832
Regarding binary, it's somewhat of a hardware limitation, and somewhat about signal clarity. COmputers use transitors to track states, which as you may be familiar are either on or off, 0 or 1. You can count 10 things with your hands because you have 10 fingers. Computers can count 2 things because they only have 2 fingers.

There have been analog systems which can count higher, but binary (digital) works better because it's easier to clearly recognize a signal as one thing or the other. If you just have 2 cards, one black and one white, it's very easy to see from far away whether it's one card or the other. If you have 10 cards, from black to white with varying degrees of gray inbetween, it's very hard to know which specify card you are looking at. Is it gray #7 or is it slightly lighter and actually gray #6? Digital signals are hard to corrupt than analog.
Regarding bases as a whole it's important to separate NUMERALS from NUMBERS. Most modern systems use the familiar Arabic numerals. I can represent the numer four as the numeral "4", the Romans represented the number four as the numeral "IV".So we can choose to represent a number with any set of symbols we want. The other important thing to keep in mind is that our system of system of numerals goes from 0-9, not 1-10.

Finally as >>8302830 pointed out you need to understand the way we represent numbers in base ten to understand how we can represent them in any base.

It's base ten because each digit is a power of ten, and you sum all the digits together to arrive at the number. the number one hundred twenty-three point 4 is represented in based ten as "123.4" because each digit is a power of ten. It is 1*10^2 + 2*10^1 + 3*10^0 + 4* 10^-1.

Base six would have each digit represent a power of six, and we could use numerals 0 through 5. In this system the number thirty-two would be represented as "52" because it is 5*6^1 + 2*6^0.

>>8303091
>>8303096
In a base, you only use digits up to n-1 where n is the base.
5 is > 4 so 5 doesn't show up at all in base 4

>>8303091
in base 10
you use symbols 0....9
note: 9=10-1

in base 4
you use symbols 0....3
note: 3=4-1

in base n
you use symbols 0....m
note: m=n-1

base 4 counting:
0,1,2,3,10,11,12,13,20,21,22,23,30,31,32,33,100,101,102,103,110 and so on

[math] \displaystyle
8_{10} = 20_4
[/math]

>>8303097
As for why we would want to use a base other than ten, it's because certain systems exist in those other bases.

Western vocal music was originally done with 7 notes to an octave, "do re me fa so la ti", which fits well with base 7. Modern western music uses mostly twelve notes to an octave and is now better suited to base twelve.

It's also helpful to understand what is arbitrary and artificial about our number system and what aspects are truly fundamental.

>>8303106
Okay I see.
So why did most civilizations decide upon base 10?
What made 10 symbols the best fit rather than making say a symbol for every number or maybe base 5?

>>8303112
https://en.wikipedia.org/wiki/Decimal#History

OP is looking for this: you use a specific Base of X when you have X symbols (or in the case of computers, states) to represent numbers with.

We have ten symbols, 0 thru 9, so we use Base ten. It could easily have been Base 3 or base 14... but we went with ten. Probably because we have ten fingers. Base is otherwise completely arbitrary and has no effect on mathematic functions.

That's why you know all spooky pattern stuff like numerology is bullshit: those patterns evaporate when you change base, which is arbitrary, while mathematical relationships like everything that underlies physics, remain.

>>8303117
Funny thing is,

no matter what base an alien life form uses,
they will think of it as "10-based"

>>8303116
This is actually a very interesting read.

>>8303127
That's retarded. 10 represents a number which they can define in their base and which may or may not be the Base they use.

You mean to say, that whatever their base is, they will find it just as natural and simple as we find base 10.

>>8303117
What about 1 and 0?

>>8303142
Base 1 is simple. 1 is 1. 2 is 11. 3 is 111. It gets tedious as you can see..

Base 0 is impossible. It means you have no symbols to represent numbers. You can't even talk about numbers, let alone do math.

>>8303138
think again
is an alien uses base 4, they will count
"0,1,2,3,10,11,12,13,20,...."
so they write the number 4 as "10"

>>8303148
So straight one, or else nothing?

>>8303149
You are wrong and the reason is subtle and has less to do with numbers than it does with language. I'll give you a little time to think about it.

>>8303151
Like counting years on your prison wall. As soon as you attribute an empty space with zero, however, you have moved up to base 2.

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>>8303154

>>8303169
>still not understanding that joke depends entirely on language
>not grasping the universality of numbers which causes such misconceptions to evaporate

You still don't understand bases.

>implying base 10 isn't base A

>>8303180
>no matter what base an alien life form uses,
>they will think of it as "10-based"

disprove the claim instead
of ad hominum hand waving

>>8302805
It isn't the only possible way to do computing, why would it be? Binary is just used because of hardware constraints, familiarity and it's just easier to work with in general.
https://en.wikipedia.org/wiki/Ternary_computer
However, it's a good idea for us to experiment more with different systems and ideas regarding computing, especially as advancement in performance slows.

You could have n-base computer systems with existing hardware. The problem lies with interference/noise in electronics. Every extra bit you add increases the error rate which makes it worse overall, dispite the ability to carry more data per bus etc.

With optics it is possible to create n-base systems which maintain sensible error rates because of theit higher resistance to niose sources such as em radiation and magneric fields etc.

Basically anything other than a 1 or a 0 is too error prone to reliably send with electronics.

>>8303169
slightly more to this than I thought originally

>>8303202
so there is no number so large that we cannot have an n-base computer system?

>>8303213
I don't see how that follows from what I said.

Think of it this way, the 1 and 0 we refer to are just simplifications of voyage levels. Eg. 3v = 1 and 0v = 0

So if you check a voltage and it's a 0.5v, you can usually assume that this is a 0. 0.5v is not close enough to 3 for it to be confused for a 1.

In a system with more voltage levels eg 0v = 0 1v=1 2v=2 3v=3 , now a difference of 0.5v due to noise is going to start fucking your shit up. You have to retransmit the data etc. and it ends up worse for performance.

>>8303184
Man, he points out that (twice) it is language based, as in dependent on the symbology used, and that if you correctly use the numerical truths that are being communicated, which are universal and do not depend on language, the joke doesnt work.

And after that went over your head, you erroneously claim the proof was an ad hominem attack..... maybe you are retarded?

>>8303287
it's not a joke,
it's a statement on how language works

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>>8303287
>more ad hominem attacks
>no detailed explanation
'it's language based' explains absolutely nothing

What would the alien call [math]4_{10}[/math] is the central question here. Also, >>8303169

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>>8302815
>In base 9 you count to ten then move to a new digit
Uh... No
In Base 9, you count 0-8

>>8302825
No excuse
>I only post when I'm on my phone and I'm only on my phone when I'm smoking.
That doesn't even make sense given the nature of your error.

>>8302832
> Most people count numbers in base 10 because
... because human hands have ten digits : fingers and thumbs; that's what "digit" actually means (toes are also digits, but conventionally we don't use them for counting).

There's absolutely nothing special about base ten. If we had six digits on each hand, we'd all be using base twelve.

>>8303313
It depends what is the most natural base for the aliens life and how their civilisations and languages evolved around that.
This of it this way, we mostly use base then and have 10 unique symbols to represent that, 0,1...9
Now 0,1,2 and so on are arabic numerals, and I think it is logical to deduce that aliens wouldn't exactly have the same arabic culture and people on their home planet. Thus there is no reason for them to use digits drawn as 0,1,2..

If they mostly use let's say base 4 (maybe because they have 2 fingers on each hand) they would have 4 unique symbols for their numbers, for example |,/,\,-... (starting with 0). so The number 4 in base 10 would be "4" in our language but "/|" in their language.

similarily, if they used base 16, they would actualy have 16 unique symbols to represent numbers. numbers don't end at 9 because they have to, but because that's the system we use. we say "15" they might say "@" or whatever symbol they created for the singular number 15. since in their language 15 is not a number made of a 10 and a 5, but it simly is "@" and every other number is made of symbols up to "@"... simply said 15 is a single digit for such a hypothetical alien.

if the joke is based on misrepresenting numbers because of base then that is impossible as such a mistake is language specific.

>>8304561
it's not a joke

base-4 aliens don't have the words four,five etc
the just count ...,two,three,ten,eleven,twelve,twenty,...
the number after running out of symbols will always be called "ten" no matter what the base

>>8304575
what does that even mean?
"ten" means nothing. they may as well call it "blurgh" I don't see how that is relevant.
They may or may not have new names for numbers beyond their base. Their brain structure may as well make it easier for them to pronounce the digits one by one, so instead of ten they would say "one,zero"... why the fuck now? no "tens" or "teen-s" or "ty-s" need to be introduced to communicate a number. They may as well not have a name for anything and sing in different frequencies based on the number they want to say.

so no, the first number in the higher order doesn't need to be called "ten" no matter what.

>>8304601
Well, if you don't get it, too bad.
I can't make it any simpler.

But I repeat, this is not a joke.

>>8304575
the only reason we call the numbers 10,11,12 20 etc..."ten" "eleven" "twelve" "twenty" etc... is because we recognize that bases can be changed, and have names for them in base 10.

If aliens really did use base 4 as their main number system, it's entirely more likely that a number like 111 would not be "one hundred eleven" than just "one one one" in their language because one hundred eleven implies they know and use base 10.

I don't know why you're stretching this so far, your argument holds no water.

>>8304626
One more last clue
saying "ten" "hundred" etc means different things in different bases.
You associate them with 10 100
a base 4 creature would associate them with 4 and 16
a base 17 creature would associate them with 17 and 289

and now, I'm done.

>>8304653
All you are saying is that numbers have names.
well doh...

>>8304653
This is wrong though.
10^2 =/= 4^2
just because each digit of a particular base has the same exponent, doesn't mean you'd call it the same number.

in binary, I don't say 10 is "ten"
I say it's "two"

"11" is 3
and so on and so forth.
Eventually, I'd get to 110 and have a name for it. I might call it "ten" or I might not.

My point is, bases map the reals to the reals as a function. Obviously there's a 1:1 correspondence that follows, and each output could be given a name.

It's also equally likely that instead of naming the numbers, they are left as an abstraction.

For a longer binary string like 10101011110101, you're not going to say the actual number, you're just going to repeat the string.

>>8304717
supposed to be "six" not ten, but w/e

>>8304653

>One more last clue

I fucking hate anons like this. If he actually had a coherent idea he wouldn't be presenting it like a riddle.

>>8304575
>the number after running out of symbols will always be called "ten" no matter what the base
That's not even true on earth
Diez
Dix
Dieci
Deich
Zehn
Ju

ternary is pretty under-represented here, I only see even bases being discussed here, which saddens me

base 3 is literally the most economic number system that exists, dont believe me?
wiki: radix economy

there is much potential in this number system, even more than decimal

>>8305582
> your IQ

>>8303297
its a stupid fucking joke, because you are incorrectly translating it into 10, (a symbol that means the number 10), when in fact whatever symbols they used would look nothing like a 10, and when translated correctly (in this case assuming base 4), would mean the number 4.

so its a stupid joke that hinges on a fallacy and it evaporates when you talk about the math.

all bases are not base 10 unless you are fucking sophomoric in your comprehension of both math and language.

>>8302830
This helped me a bit on a project euler problem thanks anon

>>8305674

>>8303169

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Here is the best authority on the subject:

https://www.youtube.com/watch?v=_uJsoZheTR4

>>8302801
Imagine no number bases
1=|
2=||
3=|||
5=|||||
8=||||||||
10=||||||||||
25=|||||||||||||||||||||||||
52=||||||||||||||||||||||||||||||||||||||||||||||||||||


Shit's hard to read and gets annoying to write. So lets do what kids do and group them into 5's

1=|
2=||
3=|||
5=̵|̵|̵|̵|̵
8=̵|̵|̵|̵|̵ ||||
10=̵|̵|̵|̵| |̵|̵|̵|̵
25=̵|̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵
52=|̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ |̵|̵|̵|̵ ||

Better but still annoying as numbers get bigger. So let's repeat the grouping of 5 5's. To avoid making new symbols let nest it and remember the number of groupings by spacing them over:

1=|
2=||
3=|||
5=| 0
8=| |||
10=|| 0
25=| 0 0
52=|| 0 ||

Now add in 4 usual symbols for | through |||| and we have base 5:

1=1
2=2
3=3
5=10
8=13
10=20
25=100
52=202

Notice the number of grouping is arbitrary so we can replace it by any base. Next time we'll learn addition which is just going group by group and trying to build a larger nested group with them combined.