Why is the number of shares held by stockholders discrete?

It is discrete because if it did have decimals, lots of money would be lost. For example, if you have a stock at $20.36, and you buy 100.2 shares, the value of those shares is $2040.6828. If over a period of time, the stock goes up $2, it it will be at $22.36. This makes the value of shares $2240.472. Basically, you have earned $199.7892. But now what do you do with the .7892? If you decide to round up to .79, but a thousand people are selling their 100.2 shares, then they are owed a total of $.8. If this happens every day over a year, they are owed $29.2. If every stock did that then the extra amount owed due to the rounding would be massive.

Looking the other way, if you round down every time to .78, then each person holding 100.2 shares will lose $.0092. Again if there are 1000 people, collectively they will lose $9.2. And over a year, that is $3358 that just disappears. Because you can neither round up or down (because it will result in disaster) the amount of shares held by the shareholders must be discrete.