No, 1/3 IS 0.333... recurring. There is no rounding error, people just can't wrap it around their head that 0.999... recurring and 1 are the same number. It is just ingrained in their heads that for each number there is only one way to write it (without using fractions), which is simply not true.
As AndroidXP remarked there is more than one way to write a certain number: 1, 2/2, exp(0), 0.999...
Even if I write 9s all my life, I could NEVER write enough decimals, so I will never write down the number that is 0.999... recurring. That's inifinity for you.
You can look at it this way: if you take the smallest positive number you can imagine, then it is still larger than the difference between 1.0 and 0.999... And if you take half of that, it's still too big. Or half of that ...
How many numbers must equal 1 under a decimal system before students realise that if you're not going to use an appropriate representation for fractions then you're not going to get the correct final result. There's more to maths than calculators.
The fraction 1/3 is valid number but 0.3~ is only an accepted notation within the limitations of our base 10 number system to repesent a value that can't otherwise be represented in base 10. It is a rounding fudge for base 10.
In the context of 1/3 it means that at some point it is not posssible to represent the last digit and that some inifinite portion is left unstated.
The NUMBER 0.9~ though does not equal 1. Forget the context based "shorthand" and look at the base 10 number system.
Every DP is 10 times less than the DP before it. Given this you MUST accept that
0.9 < 1
0.99 < 1
0.999 < 1
0.9~ < 1
It DOES NOT MATTER that it has a never ending number of significant digits as you must also accept that every DP is 10 times less than the one before it therefore 0.9~ < 1.
Someone said work out 1 - 0.9~. Fine
1 - 0.9 = 0.1
0.1 - 0.09 = 0.01
.......
It will never reach 0, just closer to at each dp to 0. It only reaches zero when limits come into play!
If you accept that the NUMBER 0.9~ = 1 it means you believe every DP is NOT 10 times smaller than the one before.
This is what is meant by the number tends towards 1, getting ever closer BUT never reaching 1.
Don't confuse the NUMBER 0.9~ and the SHORTHAND REPRESENTATION for 3 * 1/3
This is all just semantics but needs to be pointed out
Woz - Yes but 0.9 recurring is only infinitely smaller than 1.0, so there is no calculation in which using 1.0 would not give, to all intents and purposes, the same answer.
it seems you are looking at it as a very very very very very very very very very very very very very very very very very very very very very very very very very long finite number.
If the turtles destination is at 1, and 1 has a size, and that size is anything other than 0 (which would meen the destination does not exist), then the turle has reached it's destination.
In other words, give numbers a size, and suddenly 0.9(9) is no longer a problem, so
1 = 1 + ( -0.0(0)1 ) to ( +0.0(0)1 )
EDIT: I just made that up, but blow [and suck] me it works
But if you are achieving a result of 0.9 where 9 continues for infinity, why not cancel the infinity by giving 1.0 (infinite 0's) then a 1..? Thus removing infinity from the riddle, and proving once and for all that 0.9(9) not only equal 1, but as a concept simply doesn't exist, because the answer is 1.0(0)1 not 0.9(9).
Of course, in the same way 0.9(9) also has a size of 0.9(9)(0)1, but it's twin infinities make it only of comparable value to other results with a twin infinity.
As a person who is allergic to mathematics I really have a hard time getting my head around the whole thing. I understand the idea behind 0.999(inf) = 1, but it still doesn't seem to me as if the former is the same type of number as the latter. It's not possible to write an infinite string of numbers, so to me the idea of 0.99999 = 1 remains a completely dry and intangible.
Would it mean that 1+0.999999 (infinite) = 2? or 1.99999999 (infinite) or both? If it's both, then why do we even use 0.999? Is it just to make up for the inconvenient 'paradox' of 1 / 0.3 = 0.33333 * 3 = 0.999?
Doesn't that just mean our whole foundation for mathematics is flawed in some way? I'm sure its my tendency to think visually instead of mathematically that is causing me some conflicting thoughts when thinking about this, but it all seems odd and wrong to me :P
Lerts, suppose you want to post a reply in this thread. There is some distance between your mouse pointer and the "Reply" button.
You move the mouse halfway. Distance left: 0.5 part of original distance
You move the mouse halfway again. Distance left: 0.25 part.
Again. Distance left: 0.125 part.
And so on.
Do you ever reach the "Reply" button? No.
I have now proved that you did not reply to this thread.
Both. Writing it as 2 is more convenient, but 1.9(9) is still a valid notation.
You can also write the number 16 as 17, 20, 22, 24, 31, 40, 121 or 1000 (in base 9, 8, 7, ..., 2). It's still the same number.
