value of 0.9999... (1 Viewer)

[sammeh]

just one of the interesting little bits of scrap math doodling i came across whilst trying to revise:

let x = 0.9 repeating
10x = 9.9rpt
10x - x = 9.9rpt - 0.9rpt
9x = 9
x = 1 = 0.9rpt ?

or

consider 1/3 = 0.3rpt
0.3rpt * 3 = 0.9rpt
1/3 * 3 = 1
but 1/3 = 0.3rpt
therefore 1 = 0.9rpt ?

or

x = 0.9rpt
10x = 9.9rpt
9x + x = 9.9rpt
x(9 + 1) = 0.9rpt(10)
10x = 9.9rpt
x = 0.9rpt

just a little demonstration of how you can outsmart yourself in some cases by overworking the question. i think its probably pertinent to note how easy it can be to forget some of the basic principles of mathematics when you start with the approaching-pure mathematics - altho this question is hardly difficult its still something to note.

besides, its interesting - its quite difficult to pick out the fallacious steps in the 1st and 2nd results of 0.9rpt. the 3rd is an illustration of a no more "correct" method but it gives a true answer.

of course, probably the best value to give for 0.9rpt is
0.9rpt = lim(x->inf) (10^x - 1)/10^x

questions/comments/flames/anything, im procrastinating here, help me out ;p

 

[Euler]

well, i would interpret 0.99999.... as a geometric series,

first term 9/10, ratio 1/10.

Then 0.999999.... is the sum of the geometric series, which is 1.

In other words, 0.999999..... is an infinite sum that converges to 1.

 

[MyLuv]

I dont get the 3rd one ...all u did was to prove 0.(9)=0.(9) which is obvious (and I didnt get the proof either).For me,0.(9)=1 as the 1st proof.
Btw,u on W3XBnet Forum? =p

 

[sammeh]

lol no, i 1st saw this on the gotfrag forums of all places. and the 3rd one is just an example of how ridiculous the other two are - they all use questionable reasoning.

a value like 0.9rpt is only expressible using other methods, eg limits or a geometric series like euler said (altho i hadnt thought of that b4 - so cheers )

 

[KeypadSDM]

3rd one doesn't do anything

x = 0.9rpt
10x = 9.9rpt
9x + x = 9.9rpt
x(9 + 1) = 0.9rpt(10)
10x = 9.9rpt
x = 0.9rpt

I.e.
x = 0.9rpt
blah de blah de blah
x = 0.9rpt

 

[George W. Bush]

By definition, two numbers are only different if there is another distinct number between them

i.e. by definition .999999999999999999999999etc. = 1

 

[KeypadSDM]

Originally posted by George W. Bush
By definition, two numbers are only different if there is another distinct number between them.

I.e. there are infinitely many numbers between two different numbers. Neat.

 

[Grey Council]

lol, nice Bush. Never thought of it that way.

But then that leads to every number being the same. Cause if 0.9rpt = 1, then 1 = 1.1rpt, adinfinitum.

hrm

 

[George W. Bush]

No, 1.1 is between 1 and 1.1rpt , as well as numerous other numbers.

Pretty sure the only time it comes up is with the 9 repeaters.

 

[Grey Council]

lol! my bad

 

[sammeh]

3rd one doesn't do anything

exactly my point

 

[Affinity]

Originally posted by George W. Bush
By definition, two numbers are only different if there is another distinct number between them

Only works for real and rationals

 

[George W. Bush]

There's a distinct number between pi and e

 

[KeypadSDM]

They're trancedental, they're different