What Are Real Numbers? Real numbers are often introduced as “all numbers on the number line”. That is a useful picture, but it hides a deeper question: What exactly is this number line made of? From counting to measurement Natural numbers appear when we count: $$ 1,2,3,\ldots $$ Integers appear when we allow subtraction: $$ \ldots,-2,-1,0,1,2,\ldots $$ Rational numbers appear when we allow division: $$ \frac{p}{q}, \quad q \neq 0. $$ ...