If you've taken any course that covers number systems, you've probably learned the "repeated multiplication" method for converting a decimal fraction to binary. Multiply by 2, record the digit before the decimal point, keep the remainder, repeat. It works. You can pass an exam with it. But it also feels a little like a magic trick — you follow the steps, you get the right answer, and you have no real intuition for why it works or why it sometimes never ends. Here's the thought process that gave