What if our numbering system was not based on 10?

[Gamish]

What if our numbering system was not based on 10? 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!

 

[chroot]

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

 

[Gamish]

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...

 

[chroot]

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

 

[Manchot]

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.

 

[Zurtex]

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

 

[Janitor]

... 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.

 

[Gamish]

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?

 

[Cosmo16]

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

 

[dextercioby]

Yap,folks,he meant recurring decimal...

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??

Daniel.

PS.There's a 5 at the end,but not in the sense u meant it...

 

[poolwin2001]

$\infty$ is not a real number and arithmetic is not defined on it.
It is meaningless to think about $\infty$*2or $\infty$/54000

 

[dextercioby]

BTW,
$$0.999...=0.(9)\equiv 1$$

Daniel.

 

[Gamish]

$\infty$ 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?

 

[dextercioby]

Yep,it's because recurring 9 is taken as unity that everything does make sense...

Daniel.

 

[matt grime]

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.

 

[Gamish]

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?

 

[matt grime]

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

 

[Integral]

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.

 

[Gamish]

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 :tongue2:

 

[matt grime]

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 :tongue2:

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

 

[Gamish]

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 :rofl:

 

[dextercioby]

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"? :tongue2:

Daniel.

 

[matt grime]

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.

 

[Gamish]

I got your "inaccurate" remarks by quoting what you'd written.

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

And yes, lamen = layman just like you'd = you had

 

[matt grime]

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.

 

[dextercioby]

i think we don't need Mr.Integral to come back and tell u guys to "cut it out"... :tongue2:

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


Daniel.

P.S.Anyway,thisthread has diverted too much from the original idea...

 

[Gamish]

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.

 

[matt grime]

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.

 

[Gamish]

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

 

[dextercioby]

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"?? :tongue2:

Daniel.

 

[matt grime]

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...?

 

[Gamish]

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)?

 

[matt grime]

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.

 

[Alkatran]

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.

 

[chroot]

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