0.999 = 1
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- KwesiJ0
it means perfection is a reality
- MrDinky0
"With the rise of the Internet, debates about 0.999... have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics."
from same wiki
- mitee_0
_alpha=.999
- ribit0
From http://www.straightdope.com/colu…
Now then. Lint has already provided proof that .999~ = 1. From grade school math we know that .333~ = 1/3, .666~ = 2/3, and 1/3 + 2/3 = 1. Clearly .333~ + .666~ = .999~. Ergo, .999~ = 1.
The mind (yes, even mine) instinctively rebels at this conclusion. We readily concede that .999~ gets infinitely close to 1--to put it in mathematical terms, 1 is the sum of the converging infinite series .9 + .09 + .009 + . . . But, we protest, .999~ never quite reaches that limit. If at any step we halt the progression to infinity to take a sum, we find that we remain separated from 1 by some infinitesimal amount.
But that's just the point, the mathematicians say. When a decimal repeats ad infinitum, you never stop.
The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.
Nonsense. The fraction 1/3 is an ordinary number, and .333~ is the same ordinary number; an infinite series of 3s simply happens to be the only way to express said number given the limitations of decimals. Granted, decimals let us express the quantity 1 without difficulty, but the process of infinite repetition produces the same result; .999~ is merely another way of saying 1. Likewise, pi is an ordinary number; it's just a quirk of the real number system that we have to express it as 3.14159 etc (without ever repeating or stopping). Rational numbers, which by definition can be expressed as fractions, translate to repeating or terminating decimals; irrational numbers (like pi) never repeat or terminate in their decimal form.
If you're still having trouble, consider another example involving a converging infinite series: Zeno's paradox, proposed by the Greek philosopher Zeno in the fifth century BC. Suppose Achilles and a tortoise have a footrace. Achilles is ten times faster than the tortoise, but the tortoise has a ten-meter head start. In the time Achilles runs those ten meters, the tortoise crawls one meter. In the time Achilles runs that one meter, the tortoise plods another .1 meter. In the time Achilles runs that .1 meter, the tortoise lumbers ahead .01 meter. You get the picture. We seem to be reasoning ourselves to the conclusion that Achilles can never pass the tortoise.
But common sense says he does, and common sense is right. The expression 10 + 1 + .1 + .01 . . . is a converging infinite series whose sum is 11.111~ (or, to express it as a mixed number, 11 1/9). Common sense also tells us that Achilles does not merely approach this limit (as Zeno's paradox would have us believe), but reaches and then passes it--i.e., that Achilles overtakes the tortoise at 11 1/9 meters. We thus see (I hope) that there's nothing magical and unattainable about limits, and so no barrier to grasping that .999~ = 1.
Doesn't that enhance your quality of life? Of course it does. Not that body paint doesn't have its place, but there's just no substitute for the pleasures of an infinite series.
- a_iver0
YOUR WRONG!! :-P
1/3= .33333
.3333/3= .11111 (1/9)1/3 + 1/3 + 1/3 can either equal 1 or .99999
come on, i figured this stuff out in high school
- Witt0
Don't we get the furthest from 1 as 9's are added to .999~?
- Noismith
- Nope.5timuli
- .999999 is closer to 1 than, say, .999 which is = to .9990000...ismith
- Unless the 9's you talk of are 9.0's, then yes.5timuli
- hmmm ok then. but the number is not getting bigger, you're just describing it's limits, no?Witt
- It's getting bigger but it'll never reach 1.0.5timuli
- one thing is certain 4.48am is like 5am to me. Goodnight!Witt
- ok i get it 5timuli. thx. gn.Witt
- Actually, I misread... if you're specifying farther into .999~ then it's not getting bigger, but you're not specifying a limit either.ismith
- either. You're just working harder than you need to because the ~ here is symbolizing an infinite string of 9ismith
- 9. If you said just .999 then yes, the number is actually changing as you specify extra digits.ismith
- .999~ is the same as .999999~ and .999999999~... but .999 ≠ .999999 hehismith
- 5timuli0
.1111 ≠ 1/9
Also:
.1111∞ ≠ 1/9
- Jnr_Madison0
I just like to pipe in and say I have a maths degree.
- this may not be true.Jnr_Madison
- at least you said maths right, that entitles you to en engrish degree5timuli
- I just checked my cv, I can confirm it's not true.Jnr_Madison
- But you do have an HND in Fire.5timuli
- It was just a module.Jnr_Madison
- I think we may have crossed paths. I took Radiator Technology & Central Heating Theory.5timuli
- 5timuli0
Equating fractions to decimals is like English to Japanese. Some translations exist, some don't.
- jfletcher0
lets argue if 1/x will ever hit the axis!!
- 5timuli0
I have one pound and take one guinea away, add three farthings, divide by six shillings and a ha'penny, can I afford the bus home?
- thinblkglasses0
Numbers don't exist, everyone knows that.
- they must be real otherwise we're talking bout nothing and thats unpossibleKwesiJ
- Llyod0
0.999 = half past a monkey's butt a quarter to his balls
- robs0
<-- math major, just graduated. my vote is with 5timuli, 1/9 only equals .1111 with limits through calculus. also, i say "math" instead of "math," and none of my classmates seemed offended.
- neverblink0
I agree with 5timuli and robs:
(1/3)*3 = 3/3 = 1
(0.333 ->∞ )*3 = 0.99 -> ∞ = 1 - 1/∞ != 1so however the difference is infinitly small, there is a difference.