Suppose I own 1 share of company A. Company A then decides to undergo a 1 for 3 reverse stock split. After the reverse split, I will own 1/3 shares (i.e. 0.333333... shares). The number of shares I own will be a recurring decimal. How will this work for record-keeping purposes? Will the number of shares be rounded to some fixed number of decimal places (e.g. 0.33 shares), or do the regulatory authorities forbid reverse stock splits that create these sticky situations for lots of shareholders?
No you won't have a repeating decimal number of shares in the company.
You will end-up with an exact integer number of shares. When the reverse split is done, all fractions are turned into cash.
There are two reasons to do a reverse split. The first is to get the share price over a hard limit that is set by the stock exchange. The other reason is to get rid of small investors.
If it is the second reason then anybody that only has a handful of shares will see their entire investment turned into cash. This reduction in the number of investors can reduce the amount of oversight the company has via organizations like the SEC as the company edges towards being a non-public company.
The other way fractional shares in company could happen is during a merger or split. In those cases again fractional shares in the new company are issued as cash.
Now inside a stock or bond fund there can be a fractional shares. But they don't have repeating decimals. They cut off the fraction after a few decimal points.
Here is how Vanguard handles it:
For example, say you purchased $ 3000 of the Vanguard 500 Index Fund at $ 107.65. When we received your $ 3000, we would divide that sum by the price per share to calculate the number of shares you received. In this particular case, the fund comes to 27.86809103576405.... shares. On our website and in your paper statements, we would then round this number of shares to three decimal places and state that you now owned 27.868 shares.
When you downloaded your recent transactions to Quicken, the OFX file creation process brings your share balance out to a much higher level of precision so that your transaction balances match. For example, we would pass a share value of 27.86809 to Quicken, so that when Quicken calculated the total of the transaction, it would equal $ 3000.00.