070216-4
Math trivia: .999... = 1
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11 comments:
Only if your rounding to the nearest integer?
I'd like to see that proof.
No rounding.
1/3 = .333...
2/3 = .666...
1/3 + 2/3 = .999...
BUT 1/3 + 2/3 = 3/3 = 1.
So, .999... = 1.
Here's another way of looking at it.
0/9 = 0
1/9 = .111...
2/9 = .222...
3/9 = .333...
...
9/9 = .999...
and 9/9 =1
So 1/3 is .333 without rounding?
Sorry about that last comment, it had a snarly ring to it.
This problem does show the range of infinity with the repeating result given by dividing and how it never quite reaches an end.
Snarly? Ha, never! Question everything, accept nothing at first glance. :)
1/3 is not just .333, it is .333... (where the ellipses represent repetition into infinity.) Just like 2/3 = .666... And of course 3/3 = .999.... and we know that 3/3 also equals 1.
If you'd like a more advanced proof that .999... = 1, check out this site. The proof illustrated involves limits.
http://mathforum.org/dr.math/faq/faq.0.9999.html
I am conversant with limits having struggled through three calculus courses. Your unique use of the ellipses was never a part of my mathematical training but seems more a symbol of wordly expression. But I accept the possibility. Thanks for the link.
You're welcome. It was fun talking about this. I am glad someone was interested.
The more accepted symbol for repeating numbers is of course the little line over the last number, but I have no idea how to type one out on the computer unless I search for it somewhere in the character mapping.
Try again:
_
.9999 Trouble is, you have to use the line above for the line above.
Here is one I like " ° " Its ASKII 248 gotten by hitting Alt 248 using the keypad.
Finally warming up. ♪ ♫ ♫
That's a sneaky little way of getting around it. Let me try:
-
.999
I remember when I found out that alt + 0222 makes a nice little
:Þ
I think that this blog word-processor is deleting preceding spaces. Testing, testing:
qqqqqqqqq_
.999999999
I guess the tools are only worth what we pay for them and I am constantly amazed that we can do these blogs for free (I have seen a few that you have to pay for, but they don't seem to offer any extra amenities.)
Not to beat a dead horse, but I think that using a limit to prove that .9 . . . equals 1 is cheating. Sure .9999. . . approaches 1 as a limit but it never quite gets there. Kind of like an asymtote?
Here is a handy math symbol: Σ which is 228. I found this list of PC charactors in a Peter Norton book on the IBM PC. He was the guru back when we needed help with the Microsoft DOS nonsense!
The best to you and yours.
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