Do You Think .9~=1?
Thread Starter
Senior Member
Joined: Jan 2003
Posts: 983
From: redondo beach
[quote name='psyclo' date='Sep 17 2005, 01:43 PM']Damned mathematicians thinking there is a diffrence, us engineers know it doesn't matter.
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mathmaticians know theyre the same.
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
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mathmaticians know theyre the same.
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
Thread Starter
Senior Member
Joined: Jan 2003
Posts: 983
From: redondo beach
Thread Starter
Senior Member
Joined: Jan 2003
Posts: 983
From: redondo beach
[quote name='7lufnis' date='Sep 17 2005, 03:43 PM']thats only as it approaches infinity. It never truly gets to inifinity its just an approximation.
[/quote]
its not constantly approaching it. its a set value.
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its not constantly approaching it. its a set value.
Thread Starter
Senior Member
Joined: Jan 2003
Posts: 983
From: redondo beach
Allright one last time.
in math we usually use base ten
hence 121 is read as 1*10^2 + 2*10^1 + 1*10^0 = 100+20+1 = 121
and 0.1 = 1*10^(-1)
However 10 is a somewhat arbitrary decision. Some ancient people did math in base 6. For my example we will use base 3.
121 in base 3 is 1*3^2 + 2*3^1 + 1*3^0 = 9+6+1 (base 10) = 16 base 10
Now consider 1/2 in base 10 this is written as 0.5. It has a definite value. In base 3 however it is written as 0.111~ because it cannot be represented with finite digits. The value is still exactly 1/2 not approaching 1/2 but exactly 1/2 but is symbolized by 0.111~.
Similarily in base 10 1/3 is written 0.333~. This is exacltly 1 part of 3 however one cannot write it in finite decimal expansion in base 10. In base 3 however the same number is 0.1
Finally 0.9~ is just one way of representing 1 in base 10. It's value is well defined like 1/2 is 0.11~ in base 3 is well defined. We do not do a function on 0.9~ nor is it rounded or an approximation. It is simply a valid representation of 1 in base 10
By the same argument 0.2222~ = 1 in base 3.
in math we usually use base ten
hence 121 is read as 1*10^2 + 2*10^1 + 1*10^0 = 100+20+1 = 121
and 0.1 = 1*10^(-1)
However 10 is a somewhat arbitrary decision. Some ancient people did math in base 6. For my example we will use base 3.
121 in base 3 is 1*3^2 + 2*3^1 + 1*3^0 = 9+6+1 (base 10) = 16 base 10
Now consider 1/2 in base 10 this is written as 0.5. It has a definite value. In base 3 however it is written as 0.111~ because it cannot be represented with finite digits. The value is still exactly 1/2 not approaching 1/2 but exactly 1/2 but is symbolized by 0.111~.
Similarily in base 10 1/3 is written 0.333~. This is exacltly 1 part of 3 however one cannot write it in finite decimal expansion in base 10. In base 3 however the same number is 0.1
Finally 0.9~ is just one way of representing 1 in base 10. It's value is well defined like 1/2 is 0.11~ in base 3 is well defined. We do not do a function on 0.9~ nor is it rounded or an approximation. It is simply a valid representation of 1 in base 10
By the same argument 0.2222~ = 1 in base 3.
