The proof resolves a nearly 80-year-old problem known as the Duffin-Schaeffer conjecture. In doing so, it provides a final answer to a question that has preoccupied mathematicians since ancient times: ...

Most people rarely deal with irrational numbers—it would be, well, irrational, as they run on forever, and representing them accurately requires an infinite amount of space. But irrational constants ...

When you estimate a number you decide approximately, or roughly, how big that number is. To estimate accurately, you need to have a good understanding of the value and order of numbers. A number line ...

The decimal expansion of pi goes on forever. But an infinite number of fractions can approximate it to ever-increasing accuracy.KuoCheng Liao/Quanta Magazine The deep recesses of the number line are ...