I find it funny that people have no problem understanding that 9/9 = 1 or .5 * 2 = 1 or 1/9 * 9 = 1 but suddenly when represented as 0.999... = 1 the matter becomes debatable.
I don't see how that's really surprising. Your examples are all easily explainable to a very young child and can be described in physical terms, "9 apples split between 9 people", "half a cake times two" - even people who don't need such simplistic thoughts, the fact that it is relatable to real life does make it easier to grasp.
Whereas 0.999... just isn't relatable at all, it's a concept that purely exists for the sake of mathematics.
Not to mention, your examples are all sums, 0.999... = 1 isn't. You can call them all equations, sure, but the difference is that between "if I make changes to X then it can equal Y" and "X already equals Y, even though they appear to be different numbers".
the .999 = 1 issue is not a math problem so much as a symbol issue. Those fractional representations of ninths are presented to 4th graders if i recall correctly, yet adults will argue that there must exist some number between .999... and 1 even though both symbols represent the same value.
Apologies, not all of us are able to see straight through to the truth in every case. This argument convinced me (not least because it occurred to me too a few years ago ;) - I hadn't seen it anywhere before at the time, though I'm sure this is because (a) I hadn't spent much time looking, and (b) to those skilled in the art, it is so obvious as to be barely worthy of comment.
If you're simply not convinced by his line of thought (it's hard to tell...), an alternative argument might be: noting that 3 multiplied by one third is 1, consider what that would look like if written out in decimal notation.
(I am sure there are plenty more arguments too, probably much better than mine.)
(assuming sarcasm) edit: <- directed to a post that was deleted
Some things just click better for others and some of them require more of an explanation? I don't see what's so baffling about this or why people are complaining about an explanation. Proofs are definitive but they don't really help you understand what's going on a lot of the time.
0.999... confused me when I first encountered it but when I thought about what an infinite number of 9s would mean it clicked better. Something else I didn't initially believe despite proofs was the Monty Hall problem, but imagining it with 1,000 or more doors instead of 3 made it clear why the odds weren't 50/50.
In line with how you were able to understand the Monty Hall problem, one key to understanding this issue is realizing that:
infinity + 1 = infinity
I think the hurdle here is: "0.0001 is between 0.999 and 1, so for every additional 9 you put after that decimal point, I can put another 01 thus making a number between 0.999... and 1." This turns into a race to a non-existent finish line.
Personally, picturing the digits never did it for me. I think I work better when thinking about quantities and values: the infinite number of 9s means the value will continue increasing until it gets to 1. Or just understanding that 0.000...001, an "infinitesimal" quantity, is the same as 0. Something like that.
This question was posed by my math teacher to the class during a lean time in class for fun during my engineering days. This was my answer.
x = 0.999999....... (1)
10x = 9.999999....... (2)
Perform (2) minus (1)
9x = 9
x = 1
When I wrote it on the board, I still remember the awe in my friend's faces. And all I was doing was using the standard method to convert a recurring decimal into a fraction. :)
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[ 2.9 ms ] story [ 53.0 ms ] threadahh .5 * 2 = 1
Whereas 0.999... just isn't relatable at all, it's a concept that purely exists for the sake of mathematics.
Not to mention, your examples are all sums, 0.999... = 1 isn't. You can call them all equations, sure, but the difference is that between "if I make changes to X then it can equal Y" and "X already equals Y, even though they appear to be different numbers".
Was Mt. Everest still not the highest mountain even before it was discovered? The need for "seeing is believing", applied to math is pretty childish.
If you're simply not convinced by his line of thought (it's hard to tell...), an alternative argument might be: noting that 3 multiplied by one third is 1, consider what that would look like if written out in decimal notation.
(I am sure there are plenty more arguments too, probably much better than mine.)
and since 1/inf is 0
0.(999) = 1
Check the same reasoning in mathematical notation: http://upload.wikimedia.org/math/6/f/a/6fa510b44742046a167b4...
Some things just click better for others and some of them require more of an explanation? I don't see what's so baffling about this or why people are complaining about an explanation. Proofs are definitive but they don't really help you understand what's going on a lot of the time.
0.999... confused me when I first encountered it but when I thought about what an infinite number of 9s would mean it clicked better. Something else I didn't initially believe despite proofs was the Monty Hall problem, but imagining it with 1,000 or more doors instead of 3 made it clear why the odds weren't 50/50.
infinity + 1 = infinity
I think the hurdle here is: "0.0001 is between 0.999 and 1, so for every additional 9 you put after that decimal point, I can put another 01 thus making a number between 0.999... and 1." This turns into a race to a non-existent finish line.
1 * 10^(-inf)
x = 0.999999....... (1)
10x = 9.999999....... (2)
Perform (2) minus (1)
9x = 9
x = 1
When I wrote it on the board, I still remember the awe in my friend's faces. And all I was doing was using the standard method to convert a recurring decimal into a fraction. :)