10 / 3 = 3.333333333333R.............
However..
3 * 3.3333333333333333R...... = 9.99999999999R........
Where has the 0.11111111111R.....
disappeared to ?
Printable View
10 / 3 = 3.333333333333R.............
However..
3 * 3.3333333333333333R...... = 9.99999999999R........
Where has the 0.11111111111R.....
disappeared to ?
It is not 0.111111111111111... that disappeared but ratherQuote:
Originally Posted by leohopkins
0.000000000000.............................1
And it did not go anywhere.
0.999999999999999999999999999999.................. . = 1
How on earth can 0.9999999999.......... = 1 ???Quote:
Originally Posted by DrRocket
Okay so your calculator will not go on to infinity but do this......
0.99999999999 + 0.11111111111 = ? guess ?
My "guess" is that it equals 1.11111...Quote:
Originally Posted by leohopkins
In any event, it is quite true that .999999... = 1. All this means is that 1 has more than one decimal expansion. This is true for every rational number.
A decimal expansion is simply a way of writing a number as an infinite sum of powers of ten, where the coefficients of each power of ten take values between 0 and 9. For rational numbers (like 1, 3.24, etc) it turns out there's more than one way to write it as such a sum.
In this case, we can write 1 = 1+0*10^-1+0*10^-2+...
or 1 = 0+9*10^-1+9*10^-2+...
Both infinite series can be shown to converge to the same number (and the first series very clear is the number 1).
To make all this rigorous however requires some knowledge of real analysis.
Quote:
Originally Posted by leohopkins
x=0.999999999999999999999..................
10x=9.999999999999999999999999999
10x-x = 9.9999999999999......... - 0.999999999999..........
=9
9x=9
x=1
1/9 = 0.1111111111111...
9 * 1/9 = 0.9999999999999...
9 * 1/9 = 1
Therefore,
1 = 0.9999999999999...
Yes I see your point....
So I would be correct in also saying that. 4.9999999 ... = 5 ?
YesQuote:
Originally Posted by leohopkins
4.999999999.......... = 4 + 0.999999........ = 4 +1= 5
Quote:
Originally Posted by leohopkins
And since any constant expression is a function, than 0.999... = 1 (that is, there can only be one output for x, therefore making 0.999... and 1 non-unique numbers, relatively).
Would that mean then that if your would write 0.9' as an answer to a question, It would actually mean that the answer was 1. Hence, the proof that I'm familiar with, that DrRocket wrote in a post, or is that open for interpretation regarding the question?
So if i would write 9.9' as an answer, when the answer actually was 10, would I be correct?
If by 9.9', you mean 9.999..., then yes, you'd be correct.Quote:
Originally Posted by IAlexN
But if I were grading it, I'd contemplate -1 point for writing the answer in a less simple form (graders eyes get tired after a while, so try to help them out).
.99 +.11 =1.1
Clearly .99999999999999999999...... +.111111111111111111111111111.....
Is more than one.
YepQuote:
Originally Posted by Duelix
.999999........ + .1111111111....... = 1 + 1/9 = 10/9 > 1
humans made up numbers... In nature there is no such symmetry as a 100 trillion digits of a centimeter probably not even in quarks but it is something i have always been bothered by!
maybe continuum numbers don't really exist meaning perhaps they have to be written out as fractions to be correct?
There is a faint possibility that the universe really is discrete on the smallest scales, but that still wouldn't make the real numbers any less useful.
lol of course not!
People made up numbers to comfort themselves who knows maybe a long time ago some cavemen kids were sitting around with 2 rock cups and the mom went and poured a little more water in one of the caveman kids cups and the caveman kid got pissed and made up a way of measuring it lol just a funny example :lol: its like atheists say about religion