Wu Wei
ursus senum severiorum and ex-Bisy Backson
Nope, I still pick on computers...the newer the better
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The last post is the WINNER!
- Thread starter ninerbuff
- Start date
Nope, I still pick on computers...the newer the better
Sure you can. Figure out first what the infinite decimal *means*. Once that is done, the equality is clear.
As i said, invoking infinity means something has gone wrong.
And even in the mind that plays infinity 0.9. cannot equal 1 unless you include a margin of error. Or claim it as a close approximation
You could go on for infinity with that idea
But it is invoked in exactly the same way when you write 1/3=.3333....
Nope. No error. They are *exactly* the same
.99999.... means the infinite sum .9+.09 + .009 + .0009 +....
An infinite sum is *defined* to be the limit of the finite partial sums. And, in this case, that limit exists and is exactly 1.
In a similar way, the sum
1/2 + 1/4 + 1/8 + 1/16 + ....
is exactly 1.
In general a 'limit' is the thing being approximated by some process (a sequence of partial sums, for example).
Infinite sums and limits are used all the time is physics. there is nothing inherently wrong about them and they are often quite useful.
The movie was actually filmed in a little town not far from where I live. Each year, they have a groundhog day festival and show the movie for free.
Seems slight of hand to me one number equals another number.
Limits yes, infinite sums not so much.
It's just two ways of representing the same number. And that is nothing new.
For example, 1/2 = 2/4 = 3/6 = ...
and
2/3 = 4/6 = 6/9 = .....
Or that 14 base 10 is the same as 16 base 8 or 11100 base 2. These are ways to *represent* the same number.
Infinite sums are defined to be limits of partial sums. That is flat-out the definition.
So representing one number with another is how maths works???
I'll leave it here, I won't compute in my brain
I agree with you 99.999...%
It's a non mathematical win
No. Representing a number by more than one *expression* is one aspect of math. Often, there is more than one way to express a particular number in a certain way.
So, the fraction 1/2 is the same number as the fraction 2/4 or the fraction 12/24. These are just different ways of writing the same number. In particular, 1/2 = 5/10 =.5 is an expression in decimal form. But then, so is .50 or .500. But, as an infinite decimal, there is yet another way to write it: .49999....
Don't confuse the number and the expression representing the number. Different expressions have different rules and meanings. Infinite decimals mean the answer to a particular infinite sum.
So, 1/3 = .3 + .03 + .003 + .0003 + ...
and this last is what the infinite decimal .3333... *means*.
In the same way,
1= .9 + .09 + .009 + .0009 + ...
and this last is what the infinite decimal .9999... *means*.
Finite decimals are unique: there is no other way to present .034 as a finite decimal (which means base 10). But some infinite decimals have more than one way to write them base 10.
But, but, but... 0.333...2 is not 0.333...3
Which to me is the same difference as 8.999... and 9
Basically i don't do infinity which has never been shown to exist except in the imagination.
It might be easier from our froggy friend to understand
that same infinite series if expressed as....
1 = 9/10 + 9/100 + 9/1000 + 9/10000...
That's more obviously analogous to the traveling scenario.
Bob travels 1 mile.
1 mile = 1/2 mile + 1/4 mile + 1/8 mile + 1/16 mile....
We know that Bob completed his 1 mile journey
by observation. So we know that the infinite series
of Bob always having halfway to his destination
remaining to cover is no mere an approximation.
This infinite series is exactly equal to 1 mile.
(It also works for kilometers, furlongs, & even fathoms.)
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As i said at the beginning, approximations are never precise but are, by definition approximations
Subduction Zone
Veteran Member
You stopped. 0.333333 . . . does not stop. Ever.
1 = 1/2 + 1/4 + 1/8... isn't an approximation.
Tis an exact equality (because infinity exists
in mathematics.)
If 1 > 1/2 + 1/4 + 1/8.... by even 1 angstrom,
then Bob would never arrive at his destination.
He could only get really close.