Hobbit
Senior Member
Hehe, really got you pulled in here with the thread topic, huh? Anyway, I'm a World of Warcraft player and I hit the forums from time to time. Every time I go in there, there's a thread about the mathematical theorem .9~=1 (the '~' is a keyboarding substitute for the line above the number that indicates the number is repeated ad infinitum). In every one of these threads, links to major mathematical instruction sites pieced together by Ph.D.s and college math departments are thrown in all over the place by people who actually know that .9~=1. There are maybe one or two blogs linked that were made by brain-dead high school dropouts explaining in no specific terms how this can't possibly be true, and that .9~ only practically equals once, but that it isn't exact. The people who believe this theorem to be false outnumber us 10:1, and it makes me sad.
Now, while we're at it, I must address the inevitible squealer coming in here saying, "THER IS NO WAY .9~ EQUALZ 1!!!1!!!111!!" I must also show off. So here, without further delay, is a full explanation of why .9~ is exactly equal to 1.
This next section addresses the idea of infinity and why it doesn't follow the normal rules for a number. If you truly feel you have a grasp of infinity, you can skip it, but if you think that adding to or multiplying infinity alters its value, you need to read this.
Infinity isn't really a number. Its properties are perplexing simply because it's hard to grasp the idea of something going on forever. Now, adding to infinity or multiplying it by any number does not alter its total value in any way, and here's a model to prove why. It's called the "Hilbert Hilton," named after the German mathemetician David Hilbert, who first invented this model.
The Hilbert Hilton is a hotel of infinite size with an infinite number of rooms, but all of these rooms are completely full. Now, the first problem is this: If another person comes to the hotel seeking a room, can he be given one?
Give up?
The answer is yes, he can be given one. All that has to occur is for all the current occupants to move over one room, freeing up room number one. This can be proven by set pairing, which proves that two sets of numbers are the same size by matching each component of one set to exactly one component of the other set. For every number in the set of all whole numbers (1,2,3,etc., which would be the room numbers of our hotel), there is a number equal to that number's value plus 1, meaning that infinity plus one is still just infinity, and our hotel can easily clear out one room. The person in room 1 moves to room 2, room 2 to room 3, and so on.
Now comes the next problem. If another full hotel of the same size next door must close down, can all of the occupants find room in the Hilbert Hilton?
Correct, they can, but do you know why?
Same reason. For every number in the set of whole numbers, there is a number equal to twice its value, meaning all occupants can simply occupy the room with a number double the value of their old room number. The new residents can occupy the odd numbered rooms.
The third proof of infinity comes by proving that 1/infinity=0. This can't be done with the hotel, but think about long division. When you divide a number by a larger number, you keep multiplying the base number by ten until it divides, then divide the answer by the total number you multiplied by, represented by continually moving the decimal point with each multiplication. For 2/10, you add a zero to the 2, making it 20/10, which is 2. Move the decimal (divide by ten), and it's .2. So, basically, a number, n, is divisible by another number, k, if, for some value, x, n*10^x>k. There is no value for x that can make 1, or any number, for that matter, greater than infinity. Since it never divides, you simply add zeros over and over ad infinitum, which we can all agree is zero.
Ok, enough about infinity. Here are the three proofs that .9~=1.
The first is the theorem of ninths. It states that .x~ is equal to x/9. 1/9 is .1~. 3/9, or 1/3, is equal to .3~. Therefore, .9~ is equal to 9/9, and, as we all know, 9/9 is most definitely equal to one. This is the easiest proof to understand, but is typically the first people tend to ignore, as it just doesn't seem 'mathematical' enough. It looks too abstract, but, nonetheless, holds true. However, this doesn't help if the little brat you're explaining this to calls you and idiot. You'll need the other two to make him look like an idiot.
The next one is a simple algebraic proof.
.9~=x
9.9~=10x (multiplying by ten moves the decimal, and since the sequence is infinite, this doesn't change the fact that it's infinitely repeated)
9=9x (subtract .9~, which equals x, from both sides)
1=x
1=.9~
There, it's as simple as that, but you may require the third to make yourself look REALLY smart.
This one requires mathematical equivalence and is a quasi-algebraic proof. First, assert that .9~ is equal to 1-.0~1, or an infinite series of zeros with a one after it. The idiots typically accept this, thinking that the value .0~1 is the positive value subtracted from one that proves that .9~ is less than one. However, the point of this proof is to prove that .0~1 is equal to zero. First, it has already been established that 1/infinity is equal to zero. Well, any decimal with zeros follow by a 1 is equal to 1/10^(n+1), where n is equal to the number of zeros. Well, if the number of zeros is infinite, then the value is equal to 1/10^(infinity+1). As previosly established, working simple arithmetic on infinity doesn't change its value, so we can eliminate all of that garbage by realizing that infinity+1 is infinity and that anything taken to the infinite power is also infinity, leaving 1/infinity as the value subtracted from 1 to get .9~. 1/infinity is zero, making .9~ equal to 1.
I hope I haven't caused too many headaches. Have a good night all.
Now, while we're at it, I must address the inevitible squealer coming in here saying, "THER IS NO WAY .9~ EQUALZ 1!!!1!!!111!!" I must also show off. So here, without further delay, is a full explanation of why .9~ is exactly equal to 1.
This next section addresses the idea of infinity and why it doesn't follow the normal rules for a number. If you truly feel you have a grasp of infinity, you can skip it, but if you think that adding to or multiplying infinity alters its value, you need to read this.
Infinity isn't really a number. Its properties are perplexing simply because it's hard to grasp the idea of something going on forever. Now, adding to infinity or multiplying it by any number does not alter its total value in any way, and here's a model to prove why. It's called the "Hilbert Hilton," named after the German mathemetician David Hilbert, who first invented this model.
The Hilbert Hilton is a hotel of infinite size with an infinite number of rooms, but all of these rooms are completely full. Now, the first problem is this: If another person comes to the hotel seeking a room, can he be given one?
Give up?
The answer is yes, he can be given one. All that has to occur is for all the current occupants to move over one room, freeing up room number one. This can be proven by set pairing, which proves that two sets of numbers are the same size by matching each component of one set to exactly one component of the other set. For every number in the set of all whole numbers (1,2,3,etc., which would be the room numbers of our hotel), there is a number equal to that number's value plus 1, meaning that infinity plus one is still just infinity, and our hotel can easily clear out one room. The person in room 1 moves to room 2, room 2 to room 3, and so on.
Now comes the next problem. If another full hotel of the same size next door must close down, can all of the occupants find room in the Hilbert Hilton?
Correct, they can, but do you know why?
Same reason. For every number in the set of whole numbers, there is a number equal to twice its value, meaning all occupants can simply occupy the room with a number double the value of their old room number. The new residents can occupy the odd numbered rooms.
The third proof of infinity comes by proving that 1/infinity=0. This can't be done with the hotel, but think about long division. When you divide a number by a larger number, you keep multiplying the base number by ten until it divides, then divide the answer by the total number you multiplied by, represented by continually moving the decimal point with each multiplication. For 2/10, you add a zero to the 2, making it 20/10, which is 2. Move the decimal (divide by ten), and it's .2. So, basically, a number, n, is divisible by another number, k, if, for some value, x, n*10^x>k. There is no value for x that can make 1, or any number, for that matter, greater than infinity. Since it never divides, you simply add zeros over and over ad infinitum, which we can all agree is zero.
Ok, enough about infinity. Here are the three proofs that .9~=1.
The first is the theorem of ninths. It states that .x~ is equal to x/9. 1/9 is .1~. 3/9, or 1/3, is equal to .3~. Therefore, .9~ is equal to 9/9, and, as we all know, 9/9 is most definitely equal to one. This is the easiest proof to understand, but is typically the first people tend to ignore, as it just doesn't seem 'mathematical' enough. It looks too abstract, but, nonetheless, holds true. However, this doesn't help if the little brat you're explaining this to calls you and idiot. You'll need the other two to make him look like an idiot.
The next one is a simple algebraic proof.
.9~=x
9.9~=10x (multiplying by ten moves the decimal, and since the sequence is infinite, this doesn't change the fact that it's infinitely repeated)
9=9x (subtract .9~, which equals x, from both sides)
1=x
1=.9~
There, it's as simple as that, but you may require the third to make yourself look REALLY smart.
This one requires mathematical equivalence and is a quasi-algebraic proof. First, assert that .9~ is equal to 1-.0~1, or an infinite series of zeros with a one after it. The idiots typically accept this, thinking that the value .0~1 is the positive value subtracted from one that proves that .9~ is less than one. However, the point of this proof is to prove that .0~1 is equal to zero. First, it has already been established that 1/infinity is equal to zero. Well, any decimal with zeros follow by a 1 is equal to 1/10^(n+1), where n is equal to the number of zeros. Well, if the number of zeros is infinite, then the value is equal to 1/10^(infinity+1). As previosly established, working simple arithmetic on infinity doesn't change its value, so we can eliminate all of that garbage by realizing that infinity+1 is infinity and that anything taken to the infinite power is also infinity, leaving 1/infinity as the value subtracted from 1 to get .9~. 1/infinity is zero, making .9~ equal to 1.
I hope I haven't caused too many headaches. Have a good night all.
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