1. Introduction: A Fraction Already Knows How It Will End
You have been writing fractions as decimals since primary school. 1/2 = 0.5. 1/3 = 0.333... You did the long division and watched what came out.
Here is a question you probably never asked. Before you divide, can you tell whether the decimal will stop or run forever? Just by looking at the fraction?
Stop scrolling. Try it in your head before reading on. Will 7/8 stop or run forever? What about 7/6?
It turns out the fraction tells you in advance — if you know where to look. And once a decimal runs forever, there is a second question: does it repeat a fixed block, or not? That second question is what separates a rational number from an irrational one.
This section pulls all of that together. By the end you will be able to look at any number and place its decimal into one of exactly three boxes — and you will be able to travel both ways: fraction to decimal, and decimal back to fraction.