Now addition, subtraction and multiplication work all the time, but division doesn't. A number system is closed under an operation if the operation can be applied to any two numbers and the result is still a number. The integers are not closed under division. Devising a number system closed under division is hard, but for most practical purposes close under division by non-zero numbers is close enough. To make the closure we just consider ratios as numbers and call them the rational numbers.
Real numbers complete the line.
From the days of ancient Greece it has been known that rational numbers do not suffice to give an account of magnitudes in geometry. A number system which remedied this defect was not devised until additional pressures arose from the development of the calculus. In this context it was clear that some calculations could get arbitrarily precise in their results and yet there be no rational number which could possibly be the result. The result of fixing this problem, the real number system, may be thought of, once again, as agreeing to add into the system solutions to this new class of problems. A real number may be thought of as the limit of a convergent series of rationals.