We can not use decimal number on fraction. 1/3 is equal to 1/3 but not equal to infinite series numbers as 0.3333- - - that is from same fraction 1/3.
What in the world does that even mean?
$\displaystyle \dfrac{1}{3} = 0. \bar 3 \equiv \left ( \sum_{i=1}^{\infty}3 * 10^{-i} \right ).$
You can prove it if you like:
$\dfrac{1}{3} = \dfrac{10}{30} = \dfrac{9}{30} + \dfrac{1}{30} = \dfrac{\cancel 3 * 3}{\cancel 3 * 10} + \dfrac{1}{30} \implies$
$\dfrac{1}{3} = \dfrac{3}{10} + \dfrac{1}{10} * \dfrac{1}{3} = 0.3 + 10^{-1} * \dfrac{1}{3} \implies$
$\dfrac{1}{3} = 0.3 + 10^{-1} \left (0.3 + 10^{-1} * \dfrac{1}{3} \right ) = 0.3 + 0.03 + 10^{-2} * \dfrac{1}{3}.$
And we can keep that up forever.
I wonder whether that is a little backwards - which is probably the source of a lot of confusion on the subject. Would it be better to say:
- The real numbers are defined as the set of limit points of the rational numbers.
- Thus the existence of real numbers pre-supposes the properties of infinite sequences.
- Therefore we also presuppose the properties of infinite sums because infinite sums are (usually^{*}) evaluated as sequences of finite partial sums.
The only hole in this would be the supposed equivalence between a sequence of finite partial sums and infinite sums. While the equivalence seems to be valid for convergent sequences of finite partial sums, it is not necessarily the whole story as demonstrated by the whole controversy. This probably isn't a real problem because the definition of the real numbers as the limit points of the rationals ensures that we are talking about convergent sums anyway.
Not that anyone is going to teach it that way, of course.