4.7.4 Symbols

We close this section with an observation about the power and convenience of abstract symbols. Every explicit calculation you will ever do with numbers — whether with pencil and paper, calculator, or computer — will be done with rational numbers. From the point of view of practical calculation, any other numbers are fictional. In particular, we can never use an exact value of `pi`, or `2^(pi)`, or even `2^(3.141)` in an arithmetic calculation. But think of the extra work we would create for ourselves if we had to write out a rational approximation for every number that came along. Worse than that, all our familiar algebraic, trigonometric, and other rules — including those of calculus — would be, at best, approximately true. The real number system that is the basis of this course is not real at all, but merely a figment of our collective imagination. On the other hand, irrational numbers enable us to do exact calculations with circumferences and diameters of circles, with angles and sides of triangles, and with many other important relationships. We have just seen that they enable us to talk about power functions without worrying about whether base, exponent, or value is rational — and the key derivative formula remains the same for all exponents, rational or not.