What if our numbering system was not based on 10?

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What if our numbering system was not based on 10? :eek: What if it was based in 12, or 99, who knows. Would math theoreticly still work just as our mathematical system of 10 seems to work flawlessly?

Thanks in advance

BTW, what is .9i/2? is it .49i5? i=infinite!
 
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Arithmetic can be done in any base you'd like, with no differences in functionality; the numbers would just look a little different.

People commonly use binary (base 2), octal (base 8) and hexadecimal (base 16) when working with computers. Computers themselves do everything in binary.

0.9 i / 2 = 0.45 i.

i does not mean "infinite." It means "the square root of negative one."

- Warren
 
chroot said:
Arithmetic can be done in any base you'd like, with no differences in functionality; the numbers would just look a little different.

People commonly use binary (base 2), octal (base 8) and hexadecimal (base 16) when working with computers. Computers themselves do everything in binary.

0.9 i / 2 = 0.45 i.

i does not mean "infinite." It means "the square root of negative one."

- Warren

Ya, I just use "i" because I don't know what else to use. Can you please tell me what to use to represnt infinite? And I though about it, there is NO square root if -1, hehe. Because a negative times a negative equals a positive, and 1 is the standard unit, so I guess i means "imaginary". I think I remember something about negative roots in school, I forget.

So if we have 0 1 2 3 4 5 6 7 8 9 a b, what's b/2? Or something like that...
 
Gamish said:
Ya, I just use "i" because I don't know what else to use. Can you please tell me what to use to represnt infinite?
How about the infinity symbol, \infty.
And I though about it, there is NO square root if -1, hehe.
Of course there is -- it's i. It happens that i is not one of the reals, but that doesn't make it any less valid. It's unfortunate that the words "real" and "imaginary" have led so many people to think that complex numbers are somehow less valid than purely real numbers.
So if we have 0 1 2 3 4 5 6 7 8 9 a b, what's b/2?
Well, I think you were originally asking what "0.9 times infinity, divided by two" is. The answer is infinity. 0.9 times infinity is still infinity. One-half of infinity is still infinity.

Now, your new example, with {0,1,2,3,4,5,6,7,8,9,a,b} as digits in a non-decimal base. If I am to assume that there are twelve digits, then b/2 is halfway between 6 and 7, which would be represented as 6.6 in this number system.

- Warren
 
Basically, every number can be expressed in any base numbering system you want. We usually choose base 10 to be our numbering system (as you well know). This means that every digit represents a certain number of power of 10s. For example, 172 = 1*100 + 7*10 + 2*1. Thus, in the base 12 system (a.k.a the duodecimal system), every number is expressed in terms of powers of 12. However, all arithmetic stays the same! b/2 in duodecimal means 11/2 in decimal = 6 and a half. Converting back to duodecimal, we know that .6 would equal one half, and 6 stays the same. Our result, therefore, is 6 + .6 = 6.6.
 
Just to give you guys a heads up I think Gamish means 0.9 recurring when he talks about 0.9i and his 0.49i5 is an attempt to display infinitesimals, just so you know where this thread is probably going :frown:
 
Manchot said:
... However, all arithmetic stays the same! ...

Just to add a bit to what Manchot posted, I think it can be said that using a different base would not change any of the deeper results of number theory. There are some fun problems in "recreational mathematics" that do depend on the base, and of course they would have to be re-stated if you switched to some other base. An example would be issues involving the sum of the digits in a number. For instance, in base 10, the sum of the digits of 305 is 8. The same quantity written in terms of some base other than 10 could have different digit sum.
 
i

So, what is the text-based or ASCII form of expressing \infty? Let me give this example.

.9/2 = .45
.99/2 = .495
.999/2 = .4995
.9999/2 = .49995
.9\infty = ? perhaps .49\infty with a 5 at the end. Is this wrong? Is it inccorect to assume that a number can be after infinit?
 
To answer the original question, absolutly nothing would change, just how it was written. look

Binary (Base-2) Base 10
0=1
10=2
11=3
100=4
101=6
110=6
 
  • #10
Yap,folks,he meant recurring decimal... :cry: :biggrin:

Well,the notation i use is the one I've been taught in school,the one woth round brackets...
0.499...=0.4(9)=\frac{49-4}{90}=\frac{1}{2}

There,are u satisfied?? :approve:

Daniel.

PS.There's a 5 at the end,but not in the sense u meant it...
 
  • #11
\infty is not a real number and arithmetic is not defined on it.
It is meaningless to think about \infty*2or \infty/54000
 
  • #13
poolwin2001 said:
\infty is not a real number and arithmetic is not defined on it.
It is meaningless to think about \infty*2or \infty/54000

Well, I did not put \infty by itsself. What I meant by .9\infty
is .999...~

So, let me ask this. If \frac {.999\infty} {2} does not equal \frac {1} {2}, instead, 4(9)5, then .999... does not equal 1?
 
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  • #14
Yep,it's because recurring 9 is taken as unity that everything does make sense...

Daniel.
 
  • #15
Gamish said:
Well, I did not put \infty by itsself. What I meant by .9\infty
is .999...~

So, let me ask this. If \frac {.999\infty} {2} does not equal \frac {1} {2}, instead, 4(9)5, then .999... does not equal 1?


why do you insist on reinvineting the wheel in this manner? there is a perfectly good notation of recurring decimals without you inventing bizarre and conflicting new uses of symbols.


you're also doing the usual mistake of thinking that you can have a decimal point, a four, an infinite number of 9's and then a 5. you can't whilst talking about decimal expansions of real numbers, so don't.
 
  • #16
matt grime said:
why do you insist on reinvineting the wheel in this manner? there is a perfectly good notation of recurring decimals without you inventing bizarre and conflicting new uses of symbols.


you're also doing the usual mistake of thinking that you can have a decimal point, a four, an infinite number of 9's and then a 5. you can't whilst talking about decimal expansions of real numbers, so don't.

OK, can you please tell me then what .999.../2 is? I though that dextercioby confirmed my theory, unless you know the answer to this rather simply math problem?
 
  • #17
yes, it is, in the real numbers , (equivalent to) 1/2, as we established quite a while ago. When he said there's a five at the end but not inthe sense you mean, i suppose he means that it is also represented by 0.5
 
  • #18
Gamish said:
Well, I did not put \infty by itsself. What I meant by .9\infty
is .999...~

So, let me ask this. If \frac {.999\infty} {2} does not equal \frac {1} {2}, instead, 4(9)5, then .999... does not equal 1?
Fortunately .4999... =.5 so all of your concerns are for naught.

Think about it, how can you have an infinite number of digits followed by anything other then that same digit. As soon as you place a different digit you have ended the series of digits, so it is not infinite.
 
  • #19
Sorry for the double post, it is just that I posted it, and then it did not show up when I refreshed the page, because the post was on page 2 of the thread, lol.

OK, so we have established that .999.../2 = .4999...? And the "5" at the end is rather imaginary, because you cannot have a number at the end of infinite, which means that the (9) was not really infinite. So, with all this said, it would be innacurate to assume a number can be after infinite, so .999.../2 would have to equal .1/2 :-p
 
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  • #20
Gamish said:
And the "5" at the end is rather imaginary,

what 5 at the end? who on Earth apart from you thinks there is a 5 at the end in any way shape or form?

because you cannot have a number at the end of infinite, which means that the (9) was not really infinite.

steady on there, no one has said any such thing at all. that sentence doesn't even make an logical, mathematical sense. in fact i'd even go so far as to say that it was self contradictory.


So, with all this said, it would be innacurate to assume a number can be after infinite, so .999.../2 would have to equal .1/2 :-p

you appear to be doing maths as no one else does, who knows what is going on in your system.
 
  • #21
matt grime said:
what 5 at the end? who on Earth apart from you thinks there is a 5 at the end in any way shape or form?



steady on there, no one has said any such thing at all. that sentence doesn't even make an logical, mathematical sense. in fact i'd even go so far as to say that it was self contradictory.




you appear to be doing maths as no one else does, who knows what is going on in your system.

I have no idea where you get these absurd and inaccurate remarks from, so I will break it down for you, maybe you can understand it in lamen terms.

.9/2 = .45
.99/2 = .495
.999/2 = .4995
.9999/2 = .49995

So, my question was what is .999.../2? now, if you have not realized already, .999... means an infinite number of 9's to the right of the decimal place. So, everything else has a "5" at the end, but what about .999...? It seems that Integral and dextercioby explained it fairly clearly, but you seem to rather criticize my choice of words, as if you did not really understand the point in the question in the first place. I suggest that you think about the question a little more, maybe you will get it :smile:
 
  • #22
matt grime said:
you appear to be doing maths as no one else does

Is it because he uses the same digits as us that u'd go that far to call that "maths"? :-p

Daniel.
 
  • #23
I got your "inaccurate" remarks by quoting what you'd written.

You want to explain it to me in "layman's terms"?

I understand perfectly what you're arguing. It is one of the standard errors: look at the finite, terminating decimals, something happens, why doesn't it happen for an infinitely long string? You missed out the truly clever bit of noting that n nines divides to give n-1 nines, so you missed a chance to talk about infinity minus 1.
 
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  • #24
matt grime said:
I got your "inaccurate" remarks by quoting what you'd written.

What do you mean there? You must back up what you post.

And yes, lamen = layman just like you'd = you had
 
  • #25
Post 20 contains only direct quotes from your previous post plus my comments on them. You are seeming to claim that the quotations are inaccurate.
 
  • #26
i think we don't need Mr.Integral to come back and tell u guys to "cut it out"... :-p

I think the OP got the point...At least i hope... :rolleyes:


Daniel.

P.S.Anyway,thisthread has diverted too much from the original idea...
 
  • #27
You are seeming to claim that the quotations are inaccurate.

Notice that I said "remarks", NOT "quotations"! look.
I have no idea where you get these absurd and inaccurate remarks from

So, tell me exactly where I was innacurate and you where NOT innacurate.
 
  • #28
Take this one, then

" because you cannot have a number at the end of infinite, which means that the (9) was not really infinite."

steady on there, no one has said any such thing at all. that sentence doesn't even make an logical, mathematical sense. in fact i'd even go so far as to say that it was self contradictory.

explanation:

(9) symbol by definition means repeats a infinite number of times. So you are saying that as it's not really infinite, presumably you mean it is finite, thus it terminates, contradiciting the definition of dextercioby's symbol 0.(9)

Of course, "really" infinite is not a term with a clear definition.
 
  • #29
matt grime said:
Take this one, then

" because you cannot have a number at the end of infinite, which means that the (9) was not really infinite."

steady on there, no one has said any such thing at all. that sentence doesn't even make an logical, mathematical sense. in fact i'd even go so far as to say that it was self contradictory.

explanation:

(9) symbol by definition means repeats a infinite number of times. So you are saying that as it's not really infinite, presumably you mean it is finite, thus it terminates, contradiciting the definition of dextercioby's symbol 0.(9)

Of course, "really" infinite is not a term with a clear definition.

what I meant was this. If we have .999.../2=.4999...with a 5 at the end, then the (9) is not really infinite. I lamen terms, if there is a number after the 9, the 9 is finite
 
  • #30
Gamish said:
what I meant was this. If we have .999.../2=.4999...with a 5 at the end, then the (9) is not really infinite. I lamen terms, if there is a number after the 9, the 9 is finite


This is pure maths...Can u prove that "the 9 is finite"?? :confused: :-p :eek: :rolleyes: :wink:

Daniel.
 
  • #31
What you've said there is

"if <state something that is false and contradicts the notation> then <the notational assumption is contradicted>"

The last bit of that post refers to "the 9". Which of the (possibly infinitely many) nines are you referring to with the definite article? ("the 9" apparently is finite too...?)

I mean in 0.99... there is a number after the first 9, the second 9,... and so on.


Or perhaps you're just pointing out that if we take a terminating decimal (ie one with a "last" non-zero digit) then multiply by two, the answer must be a terminating decimal too. So...?
 
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  • #32
Why must such a simple thing be so complicated? When I meant is that it is innacurate to say that a number can occur after an infinite number.

BTW, you express an infinite number by ()? like 1/3=.(3)?
 
  • #33
that is the convention that dextercioby explained to you. One may indicate repeated digits in many ways.

It is not inaccurate to say that a number can occur after an infinite number, it is just plain wrong when talking about decimals. These things are not complicated, and if you took the time to learn the things other people would explain to you rather than presuming to tell them they are wrong then you wouldn't make such mistakes nor would you try and explain the real numbers in layman terms to a professional mathematician.
 
  • #34
The most common way to denote it on a computer is:
.999~ / 2 = .4999~

So, here's my challenge to you:

You say that there is a 5 at the end of the string of 9s for (.999~/2). I want you to write out the number. When you reach the 5, tell me.
 
  • #35
The fundamental disconnect here is that Gamish is stuck to the notion that a system's behavior in approaching infinity should be the same as that system's behavior at infinity. A simple confusion of the limit with the value at the limit. A calculus course should clear that right up; in the meantime perhaps a few examples of functions with
"discontinuous limits" would be educational.

- Warren
 
  • #36
Yes, the ~ is a common way of denoting a repeating decimal on a computer. And yes, .9~/2 = .5. You can prove it algebraically this way:

x = .9~ (we are defining x to be .9~)
10x = 9.9~ (by moving over the decimal point)

Then, we can subtract Equation 1 from Equation 2.
10x - x = 9.9~ - .9~
9x = 9
x=1

Obviously, then, .9~/2 = 1/2 = 0.5.

The trap that you've fallen into is a very common one, which I think is behind the reasoning that some people have. You see, when most think of division, they think of it as an algorithm that they learned as a child rather than a function. What is that algorithm, you ask? Long divison. When dividing two numbers, you're not performing long division, you're just performing the division function. It's hard to explain, but do you see what I mean?
 
  • #37
OK, I understand it a lot more now. I am just confused with one remark from matt grime.
It is not inaccurate to say that a number can occur after an infinite number, it is just plain wrong when talking about decimals
You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?

You say that there is a 5 at the end of the string of 9s for (.999~/2). I want you to write out the number. When you reach the 5, tell me.

My answer is kinda a joke, but I would say .49~5
 
  • #38
Gamish said:
OK, I understand it a lot more now. I am just confused with one remark from matt grime. You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?

I don't know exactly what he was referencing, but my guess would be the theory of transfinite numbers (cardinals and ordinals). Read up on them.
 
  • #39
Gamish said:
OK, I understand it a lot more now. I am just confused with one remark from matt grime. You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?



My answer is kinda a joke, but I would say .49~5

Alright, let's break the rules a bit and say that 5 is there.

The value it represents is 5*10^(-infinity) = 5/10^infinity = 5/infinity = 0

So it's entirely superfluous and might as well be another 9. (A bit of Occham's Razor I sees)
 
  • #40
Even an irrational number can be expressed as an infinite series of rational numbers.
 
  • #41
Gamish said:
You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?


No, I didn't say that. I said that saying it was "inaccurate" was not the best way of saying it, and that it is plain wrong to say that there is a number after infinity, though I can see how there may have been some confusion.

Crappy analogy: "it's inaccurate to say that a camel is a dodge charger" . Inaccurate is not a good word here as it indicates somehow there is some degree of accuracy involved when it is actually that you've just got the wrong idea entirely because you think they can both be modes of transport. Better to say "wrong" or even "idiotic". Litotes aside, of course.


So far no one else but you has talked about having something "at infinity" or "after infinity": mathematicians won't tend to say that since there is not, in this situation, any such place labelled "infinity".

There are for instance infinite ordinals where one can count "beyond infinity" of course the layman will be confused by this idea because he thinks that we actually mean you have to count as in count beans when you do maths. Witness the disbelief in Cantor's diagonal argument etc.
 
  • #42
So, I guess it was a miscommunication of words on your behalf. I was not trying to prove that a number can occur beyond infinity, I was trying to ask why not? With all this said, it would be wrong to say .9~5 meaning an infinite nuber of 9's to the right of the decimal, with a 5 at the end. :rolleyes:




void main {if (pf==1)functions.goto.href("http:\\www.physicsforums.com");}
/*hehe*/
 
  • #43
As an outside observer, I implore the initial poster to recognize his/her responsibility in the confusion of this thread due to "miscommunication of words".

It would be very helpful to have a basic understanding of the vocabulary and notation of a subject, but it is not essential - if one is open to learning.

But to invent one's own numerical notation (and even words (lamen??)), and to then feign condescension when one thinks others have not explained it well enough - is quite puerile.

Does the poster want to learn, or does the poster want to argue?
 
  • #44
Gamish said:
So, I guess it was a miscommunication of words on your behalf. I was not trying to prove that a number can occur beyond infinity, I was trying to ask why not? With all this said, it would be wrong to say .9~5 meaning an infinite nuber of 9's to the right of the decimal, with a 5 at the end. :rolleyes:/

If you were going to claim it was a standard decimal representation of a real number, of course it would be wrong. It appears you've not bothered learning the meanings of the symbols involved: Real numbers ARE NOT decimals. Decimals are a useful model of the real numbers, that is all. The objects you are asking about are not in that model, that is all, that is what's wrong.

I can give different presentations (of the real numbers) where it is prefectly reasonable to talk about things positioned after an infinite number of terms, but we aren't dealing with them, are we?

I don't think you're in any position to claim anything about a mildly ambiguous sentence in English being a miscommunication.
 
  • #45
matt grime said:
If you were going to claim it was a standard decimal representation of a real number, of course it would be wrong. It appears you've not bothered learning the meanings of the symbols involved: Real numbers ARE NOT decimals. Decimals are a useful model of the real numbers, that is all. The objects you are asking about are not in that model, that is all, that is what's wrong.

I can give different presentations (of the real numbers) where it is prefectly reasonable to talk about things positioned after an infinite number of terms, but we aren't dealing with them, are we?

I don't think you're in any position to claim anything about a mildly ambiguous sentence in English being a miscommunication.

Well, just because we all use the english language doesn't mean it we cannot have some sort of miscommunication. If you read the above post, there was a miscommunication of the work "inaccurate". I know what real numbers are, but that is rather irrelevant to the argument I was establishing which was the theory that a number can occur after an infinite decinmal. I guess it would be expressed as .9~5

But, this thread seems to be an attempt to disprove the validy of my question, which is rather simple.
 
  • #46
You can't have a number at the end of an infinite series of number: there is no end. Period. By your logic I could:

.9~5~45454545~~~~~~~~~~~

Now what in the world would that mean?!
 
  • #47
Gamish said:
What if our numbering system was not based on 10? :eek: What if it was based in 12, or 99, who knows. Would math theoreticly still work just as our mathematical system of 10 seems to work flawlessly?

Thanks in advance

BTW, what is .9i/2? is it .49i5? i=infinite!

Yes, all the mathematical laws (commutative property, distributive, etc) would still apply, regardless of the base.

The Babylonians were pretty good mathematicians and they used a dual base system with base 60 as their primary base and 10 as a secondary base (in fact, this is where we got our 60 minute hours and 60 second minutes and our 60 minute degrees and 60 second minutes).

Base 5 numbering systems were most likely the first counting systems developed, followed by dual base 10-5 systems, and then other variations (Hindu and Chinese base 10 systems, Mayan base 20 systems, etc). Historically, the base of the numbering system has had little to do with the sophistication of the mathematics.

Base 5 was always a popular base because humans theoretically can perceive a maximum of 4 objects before they have to resort to counting. Only pre-school kids can be tested since knowing how to count biases the test results, so I'm not positive that's really the maximum number of objects an adult could perceive, but that idea is supported historically by the popularity of base 5 systems and the virtual non-existence of base 6 systems.
 
  • #48
Let me see if I can explain this both rigorously and simply. As you go down this list:

0.45
0.495
0.4995
0.49995
...

the numbers get closer and closer to 0.5 in a specific way. For any other number 'x', you can find a number 'e' such that all numbers on the list above a certain point are more than e away from x. For example, if x=0.51, you can pick e=0.009 and all numbers on the list are more than e away from x. If x=0.4999, you can pick e=0.00004, and all numbers on the list after the third are more than e away from x. But for x=0.5, there is no number e that will work. Go ahead, try to find it. Mathematicians say this means 0.5 is the "limit" of the series. And under decimal notation, the number 0.49...5 (although not usually defined, as others have mentioned) can be thought of as defined to be equal to the limit of the above series. So 0.499...5 = 0.5 in the same way 1/2 = 0.5. They are just different ways of writing the same number.
 
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  • #49
Prove that if you allow this "new representation" into the decimals that the algebraic operations still work.

Can we just forget this 0.4999...5 garbage? At least forget it in the context of the real numbers.

Incidentally how can someone be familiar with the defintion of the reals (complete totally ordered field, unique up to canonical isomorphism) and think that 0.9i is a reasonanle notation for a recurring decimal, never mind even start such a thread as this.
 
  • #50
are you talking about my "new representation" matt? If so, I only mean it to show the real reason the 5 at the end is meaningless: it doesn't affect the limit. To you, someone familiar with infinity, it is obvious that there can be nothing at the end. But I thought if I took it back to the basics and showed that infinity is only defined with limits, it would be more clear. You could have 0.(34)(56)(78) if you wanted, and define it as the limit of:

0.345678
0.343456567878
0.343434565656787878

obviously nothing would matter after the first ellipses, but it shows why it doesn't matter.
 
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