Yet more maths. All you didn't want to know about real numbers.
In comments to my previous post, we got onto 1 = 0.999... recurring. This is in fact true, and the answer is in my reply comment to the previous comments.
Further about infinities, we have that we can just fill in a hole if we have a well-behaved area of maths. As Babso has pointed out, there should be a little bit extra left over when you calculate 1 - 0.999.... = 0.0...01, where there are an infinite number of zeros. However, since the real numbers are well behaved, then actually there is no difference, and 1 - 0.99..... = 0 exactly.
The difference is in maths in terms of limits of sequences. If you can get close enough for any element along the sequence, then the limit exists, and you can just fill in the whole. It all comes down to infinity again... 0.0...01, with an infinite number of zeros is in fact a well defined concept, since it is a limit of a sequence. There's a concept in maths which is 'dense' ness. The best example of denseness is in terms of rational numbers and real numbers. The rational numbers are dense in the real numbers. This means that between each pair of rational numbers, we can find a real number between the two, and between each pair of real numbers we can find a rational number.
No matter how close these numbers are, we can always find a real number between a pair of rational numbers and a rational between each pair of real numbers. This is an amazing fact of the real numbers. In fact, we define real number as the limit of sequences of rational numbers. So we fill in the holes between the rational numbers to obtain the real numbers. This is a definition of the real numbers that get around the problems above.
Just as a brief revision, rational numbers are all numbers that can be written as p/q, where p and q are both integers, and real numbers are those numbers that cannot be written as p/q, such as pi or the square root of two.
Another amazing fact is that there are as many rational numbers as integers, and there are as many integers as natural numbers, and there are as many natural numbers as even numbers.... :)
Further about infinities, we have that we can just fill in a hole if we have a well-behaved area of maths. As Babso has pointed out, there should be a little bit extra left over when you calculate 1 - 0.999.... = 0.0...01, where there are an infinite number of zeros. However, since the real numbers are well behaved, then actually there is no difference, and 1 - 0.99..... = 0 exactly.
The difference is in maths in terms of limits of sequences. If you can get close enough for any element along the sequence, then the limit exists, and you can just fill in the whole. It all comes down to infinity again... 0.0...01, with an infinite number of zeros is in fact a well defined concept, since it is a limit of a sequence. There's a concept in maths which is 'dense' ness. The best example of denseness is in terms of rational numbers and real numbers. The rational numbers are dense in the real numbers. This means that between each pair of rational numbers, we can find a real number between the two, and between each pair of real numbers we can find a rational number.
No matter how close these numbers are, we can always find a real number between a pair of rational numbers and a rational between each pair of real numbers. This is an amazing fact of the real numbers. In fact, we define real number as the limit of sequences of rational numbers. So we fill in the holes between the rational numbers to obtain the real numbers. This is a definition of the real numbers that get around the problems above.
Just as a brief revision, rational numbers are all numbers that can be written as p/q, where p and q are both integers, and real numbers are those numbers that cannot be written as p/q, such as pi or the square root of two.
Another amazing fact is that there are as many rational numbers as integers, and there are as many integers as natural numbers, and there are as many natural numbers as even numbers.... :)