I had a meeting with Bill Barnier, a Math Professor at Sonoma State University. He has a PhD (Algebraic Topology), 1967, UCLA. He teaches many of the upper division courses and is considered one of the smartest teachers here. He has won various awards, including two within the last few months. He also wrote a textbook for Discrete Mathematics which is used in colleges all over the country. He also came up with a hyper-geometric equation used in probability with a colleague. Needless to say, he was a good choice to talk about the subject matter.
Although he was most helpful, I was unable to write all that I needed to explain the topic the best way possible. It?s not that crucial though, as the truth remains the same.
Most of this has been said in the other thread but I will write what I was told by a reliable source as to eliminate doubt.
To start, we?ll keep it simple:
Lets say m=.999?
Multiply both sides by 10 to get 10m = 9.999?
Subtract m from both sides to get 9m = 9
Divide and you find that m=1.
Although that is not a ?proof? per se, it is completely valid and is an easy way of seeing .999??s value.
Here?s a more in depth look:
.999? is an infinite series show here: 9/10 + 9/100 + 9/1000 + ?.
Factor out the 9/10 to get 9/10(1 + 1/10 + 1/100 + ?)
The rate of change inside the parenthesis is 1/10 (shown as r below). Since it?s less than one we know the formula converges and we can use an infinite sum equation. (Calc 2)
The equation then looks like this:
9/10(1/(1-r))
= 9/10(10/(10-1))
subtract and cancel and you?re left with 1.
Another way to look at the same formula is to assume you have a line from 0 to 1 and you?re going 9/10?s of the remaining distance to 1 each interval. Thus you get to 9/10 the first interval, then 99/100 on the second interval, then 999/1000?and so on.
If you take n intervals, you can represent this process with the same equation:
9/10[(1-(1/10)^n) / (1-(1/10))]
Since n goes to infinity, (1/10)^n equals 0.
Thus we?re left with the same as before;
9/10(1/(1-(1/10)) = 1
Finally, if you can agree that 1/3 = .333? (which it absolutely does),
Then we can multiply that by 3: 3(1/3) = 3(.333?)
Thus 1 = .999?
So if you believe that 1/3 = .333? then it is impossible for 1 != .999?
I know it?s hard to read, by try and write these equations on paper if you need to.
To sum it up, 1 is EXACTLY equal to .999? If you do not agree, please please please talk to a math professor who will sit down and work through it with you. It is not a topic up for debate in the math field because it?s a fact and has been proven. Please listen to the experts!
Although he was most helpful, I was unable to write all that I needed to explain the topic the best way possible. It?s not that crucial though, as the truth remains the same.
Most of this has been said in the other thread but I will write what I was told by a reliable source as to eliminate doubt.
To start, we?ll keep it simple:
Lets say m=.999?
Multiply both sides by 10 to get 10m = 9.999?
Subtract m from both sides to get 9m = 9
Divide and you find that m=1.
Although that is not a ?proof? per se, it is completely valid and is an easy way of seeing .999??s value.
Here?s a more in depth look:
.999? is an infinite series show here: 9/10 + 9/100 + 9/1000 + ?.
Factor out the 9/10 to get 9/10(1 + 1/10 + 1/100 + ?)
The rate of change inside the parenthesis is 1/10 (shown as r below). Since it?s less than one we know the formula converges and we can use an infinite sum equation. (Calc 2)
The equation then looks like this:
9/10(1/(1-r))
= 9/10(10/(10-1))
subtract and cancel and you?re left with 1.
Another way to look at the same formula is to assume you have a line from 0 to 1 and you?re going 9/10?s of the remaining distance to 1 each interval. Thus you get to 9/10 the first interval, then 99/100 on the second interval, then 999/1000?and so on.
If you take n intervals, you can represent this process with the same equation:
9/10[(1-(1/10)^n) / (1-(1/10))]
Since n goes to infinity, (1/10)^n equals 0.
Thus we?re left with the same as before;
9/10(1/(1-(1/10)) = 1
Finally, if you can agree that 1/3 = .333? (which it absolutely does),
Then we can multiply that by 3: 3(1/3) = 3(.333?)
Thus 1 = .999?
So if you believe that 1/3 = .333? then it is impossible for 1 != .999?
I know it?s hard to read, by try and write these equations on paper if you need to.
To sum it up, 1 is EXACTLY equal to .999? If you do not agree, please please please talk to a math professor who will sit down and work through it with you. It is not a topic up for debate in the math field because it?s a fact and has been proven. Please listen to the experts!
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