Can fractions actually be converted to decimals?

Question

I was working on a spreadsheet in Excel (I'm a plebe, I know), and I came across a fraction that actually equated to 33.3% of a total number. While looking at it, and looking at the number that went with it, I realized that fractions of 1/3 continue on ad infinitum.

My question is this: Can fractions accurately be converted to decimals without rounding or assuming that .000 ad infinitum = 0 and still have the numbers reach the intended and proven result?

I don't wish for this to be specifically for Excel. I'm asking for more of a theoretical question about the continuation of fractions to an infinite number of places.

From what I know of fractions, 1/3 would equate to .333... ad infinitum. By my reasoning, the difference between 1 and "3/3" would be infinitesimally smaller the further you go. With that logic TECHNICALLY 9.99... != 10.0 or is there something I'm missing? I understand that in practice the numbers match, but given infinite places, when graphed the numbers would never intersect, but would get infinitesimally smaller.

Example: 3.33.... + 6.66.... = 9.99.... != 10.00....

Given an infinite number of decimal places, then

Note: I have no idea how to use the MathJax equations. If anyone wants to edit that in, I'd be appreciative.

Answer 1

Technically, $$9.\overbrace{9\ldots9}^n < 10$$ for any finite number $n$ of $9$s after the decimal point. Also, technically, $$9.\overbrace{9\ldots9}^n < 9.\overline9$$ for any finite number $n$ of $9$s after the decimal point; the number on the right has an infinite number of digits and the number on the left doesn't.

Certainly if you put any finite number of $9$s on the right of the decimal point, no matter how large a finite number of $9$s you put there you will always have a number less than $10$. But you also won't have $9.\overline9$.

So what is $9.\overline9$ after all? To quote from this answer to the question "Is $0.999999999\ldots = 1$?",

Symbols don't mean anything in particular until you've defined what you mean by them.

If we want to define $9.\overline9$ so that it has a meaning and actually represents a real number rather than a different mathematical object altogether, the definition that generally makes sense is that it is a limit of an infinite series, and when we look more carefully to see what that limit is, we find that it is $10$. Not just very, very close to $10$, but actually equal to $10$.

Answer 2

If the number of digits is finite (which is always the case in an Excel sheet), the answer is no.

For example, the fraction $1/3$ can not be expressend with a decimal number with finitely many digits.

In general, if the fraction is in its lowest terms and the denominator has a prime factor different from $2$ or $5$, the corresponding decimal number will have infinitely many digits.

Answer 3

The decimal representation of rational numbers come in two forms: either they terminate or a finite pattern emerges. In your case this pattern in $1/3$ is just $3$ repeating over and over again.

Whether a fraction terminates is interesting and depends on the number system you use. If you have a reduced fraction of the form $$\frac{1}{a} $$ This fraction will terminate whenever the prime divisors of $a$ are only $2,5$, the same one's as $10$'s, the base of our decimal system.

One of the advantages of working say base 60 (as the babylonians did), is you have a lot more terminating fractions, since $60=2^2*3*5$ has more prime divisors.