# Can fractions actually be converted to decimals?

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.

• You can represent fractions by a finite numbre of decimals accurately if you allow special notation for denoting periodic digits (typically, an overlline). Otherwise, no: The example with $\frac13+\frac13+\frac13=1$ works for any number of decimal digits. – Hagen von Eitzen Jun 24 '16 at 19:22
• As an aside, $\frac{10}{3}=3.33333\dots$ is commonly denoted as $3.\overline{3}$, and $3.\overline{3} + 6.\overline{6}=9.\overline{9}=10$ is consistent. See this question for explanation of why. – JMoravitz Jun 24 '16 at 19:23
• But by definition, the overline postulate wouldn't be consistent were the number of decimal places infinite (as they are commonly assumed to be from what I'm aware), correct? – Anoplexian Jun 24 '16 at 19:31
• Some fractions cannot be represented exactly as decimals (eg 1/3). But Excel cannot represent floating point numbers exactly anyway. Only more specialised packages like Mathematica will give you answers accurate to hundreds of digits, and even they have limits. – almagest Jun 24 '16 at 19:45
• @almagest Does this then mean that comparing fractions to decimals is comparing apples to oranges, and that they can never truly be compared because of innate differences? Maybe peaches to persimmons would be closer? This is the entire question I'm asking. Can they really be compared? – Anoplexian Jun 24 '16 at 19:52

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 $10$.

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.

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.