 Just as the real numbers emerge naturally as the tools to measure finite quantities, as the limits of rational approximations, the extended real numbers emerge naturally as the tools to measure potentially infinite quantities that can be approximated by finite quantities. \(+\infty\) can be taken to be the area of an infinite region, the length of an infinite line, the limit of \(1/x^2\) as \(x\) goes to 0, and so on. Although we are used to thinking of lengths and areas as positive numbers, it is sometimes useful to consider them as negative as well, when we care about the direction that they are pointing, and in this sort of situation \(-\infty\) is useful as well. Just as the real number operations of addition, multiplication, subtraction, and division correspond to certain operations on the quantities they measure, these operations can often be extended to these extended real numbers, as long as we are careful about a few cases.