1 != 1

[Tim_Lou]

.3333~ definitely = 1/3

 

since as the numbers 3 is repeated infinite times, its approaching the limit 1/3... and .333~ means .3333~ where the numbers of 3 approach infinity...

.333~ does not really represent a number, but rather, a limit in a way.

 

mathmetical proof would be:

.3333~=x

3.333~=10x

3.333~ - .333~ = 10x - x

3=9x

x=1/3

its just like the prove shown by pgrmdave.

[Tim_Lou]

An alternative expression of .333~ would be:

.33333~ = sigma (3)*.1^(n), where n goes from 1 to infinity

 

applying the equation of sum of gemotric series: (t1)(1-r^n)/(1-r)

(.3) (1-.1^n) / (1-.1)

where as n approaches infinity, 1-.1^n becomes 1

therefore, it = .3/.9 = 1/3

[pgrmdave]

So then is my original idea correct? If numbers can equal something other than themselves, does this mean that math is not precise?

[Robust]

I think we need to bring root 2 into this discussion, which also runs to infinity. Root 2 rules to my way of thinking. If you give me a bit, I believe I may be able to show where it resolves the area of the circle by a whole number....which the irrational pi is unable to do by itself. Tomorrow perhaps. Interesting topic!

 

 

"All things number ande harmony." - Pythagoras

[sanctus]

So then is my original idea correct? If numbers can equal something other than themselves, does this mean that math is not precise?

You still don't get the idea numbers are only equal to themselves.

 

The may be equal to a limit of a series, but not to the term in the series.

[Bo]

As for the original idea; i completely agree with sanctus. 0.3333....3 is always an approximation to 1/3, because you specify the end of the decimal. This end lies in infinity; for this reason the rest of the arguments fail. (a definition of infinity is for example: if you think you'r at the last 3, there is always one more, so you can't do the itarative approach). In mathematics, comparing inifinities is one of the most dangerous things to do (in physics it still is dangerous, but quite common :))

 

Just like we work with imaginary numbers. the sqrt of -1 doesn't exist

no it is not the same, the imaginary number is well defined.

 

Bo

[Tim_Lou]

as for pi,

it can be written as something else:

lim n--> infinity: sin(pi/n)*n (in radians) (as a function of sin, it can be expended into infinite series)

try it yourself, put calculator in radians mode, take sin (pi/9999999) *9999999, it comes out nicely close to pi.....

 

and as for sqrt(2),

its = (2)^.5, or... (a+:)^.5, which can be expended according to the binomial theorm....

 

in my opinions, endless decimals numbers can always be expressed as some form of infinite sum....

to me, they are like limits rather than numbers...

(these are all my original ideas, im not 100% sure if they are truely accurate... if im wrong, correct me)

[sanctus]

The rational numbers are dense in the irrational numbers therefore you're right Tim-Lou.

[nWorld]

I think the idea behind 0.999.... = 1 is that 0.9999.... is intfinitely close to 1, so it is one. There are an infinite amount of values between any two values, so if there are no values between 0.999.... and 1 then it must be true that 0.999..... = 1. I hope that was the topic.

[sanctus]

yes that's the intuitive explanation.

[infamous]

Let's just write 1/3 or cir./dia. and let the computer do the rest. I have enough work to do.

[hashes]

It is like

100/3=33.333...3

33.333...3*3=99.999...9

Then, 100=33.333...3*3=99.999...9, 100=99.999...9

[maddog]

I know that the end of infinity doesn't exist, but we still need to work with it.

 

At the end of infinity it should be perfectly exact.

 

To Tormod, I actually have a tossup. I am not sure which statement I think is more hilarious!

Actually, I request a place/thread where we can post the silliest statements made by someone

or heard and have this post/thread pinned so it always at top of whatever category it is in. :)

 

1 != 1 ??? (Yeah I figured the ! doesn't mean factorial after reading the post)

 

I think we are having a problem with representation here and what numbers mean.

 

1/3 != .33333... is an approximation and is not exact. Since you are apparently a programmer,

I would expect you know that. There is no EXACT representation for 1/3 in floating point

or double precision. Any 32-bit or 64-bit value only comes close.

 

Mathematically you can do things exact yet the representation in decimal digits fall short in

accuracy. This is not the short sightedness of the number system, just the user. :)

 

Maddog

[beccareb]

The problem really all stems from the fact that .3333.... does not really equal 1/3 because no matter how many threes there are, there will always be a remainder. Sigh. I just wonder how this thread has gone on so long, it's the same post repeated over and over with some Dave-bashing thrown in for good measure.

[pgrmdave]

but the reason for the inexactness is because of the fallibility of the computer, no ending decimal can be exact, but why not a decimal that didn't end? Each extra decimal place that one goes, one gets closer to being exact. With infinite decimal places, the number should be exact.

[AsaTaiyo]

I actually almost agree with dave. .33333.......... does not equal (1/3). But theoretically as the limit approaches infinite it will equal (1/3). Its our human minds that can not comprehend the idea of a repeating decimal, or a number that is always approaching another number as a limit. Infinite is used to solve that problem.

[maddog]

With infinite decimal places, the number should be exact.

 

I have a new silly one... :Alien:

 

Please tell me how much memory it would take to store a number with an infinite number

of digits, hmmm ?

 

Now I am aware of some algorithms using the gamma function to calculate the next

prime number where they are out about 10^100 digits at the moment. This does not

fit in any integer 36 or 64 bit or whatever. What is used is the notion unlimited extended

precision by writing the value out in ASCII of each digit and computing the value. Not as

efficient though does allow variable precision. :Alien:

 

Maddog