This article presents background and proofs of the fact that the recurring decimal 0.9999… equals 1, not approximately but exactly, as well as some of the most common arguments that claim that 0.999… is less than 1, or that it is not exactly 1.
Should we include a counter-example for (each of) these claims? or an example?
Possibly the simplest argument offered as to why works along the lines of "because it starts with zero, is obviously less than one". However, as many mathematics and science teachers are keen to point out, that which is "obvious" is not necessarily true, and such a statement is not particularly useful unless backed with some evidence that it can be proven.
The argument may be strengthened by a suggestion along the following lines:
0.9 < 1
0.99 < 1
0.999 < 1
and following this pattern, < 1. The problem with this argument, however, is that it suggests that a statement S(n) that "a zero followed by a decimal point followed by n 9s represents a number less than 1", which is true for all integer n is also true when n is an infinite value. The problem here is that such a deduction is not generally valid (compare with a statement such as "n is finite", which by definition is true for any integer n but false if n is infinite). In fact, the behaviour of infinite properties when compared to their finite counterparts means it is actually possible to prove statements such as , and with greater rigor, in some cases, than it is to prove .
See also:
References:
This article presents background and proofs of the fact that the recurring decimal 0.9999… equals 1, not approximately but exactly, as well as some of the most common arguments that claim that 0.999… is less than 1, or that it is not exactly 1.
Should we include a counter-example for (each of) these claims? or an example?
Possibly the simplest argument offered as to why works along the lines of "because it starts with zero, is obviously less than one". However, as many mathematics and science teachers are keen to point out, that which is "obvious" is not necessarily true, and such a statement is not particularly useful unless backed with some evidence that it can be proven.
The argument may be strengthened by a suggestion along the following lines:
0.9 < 1
0.99 < 1
0.999 < 1
and following this pattern, < 1. The problem with this argument, however, is that it suggests that a statement S(n) that "a zero followed by a decimal point followed by n 9s represents a number less than 1", which is true for all integer n is also true when n is an infinite value. The problem here is that such a deduction is not generally valid (compare with a statement such as "n is finite", which by definition is true for any integer n but false if n is infinite). In fact, the behaviour of infinite properties when compared to their finite counterparts means it is actually possible to prove statements such as , and with greater rigor, in some cases, than it is to prove .
See also:
References: