I have advocated Mathematica and other technologies in previous posts, but I think it is high time to advocate for doing math with the absence of technology other than an pencil and paper.
I like this rather old text I acquired from a used dealer: “Calculus With Analytic Geometry: A First Course” by UC Berkeley professors Murray H. Protter and Charles B. Morrey, published way back in 1963, about 49 years ago.
There is something to be said about a math text that isn’t full of distracting bull regarding applications of “technology” in aiding the completion of math problems. A real math text should be one where, if you were out on a desert island with nothing more than a pencil and paper (read: no calculator, internet or PC), you could try the examples and exercises and learn the ins and outs of math for yourself. Using technology trivializes math, while the use of serious thought makes the learning stick.
Related rates, which is a dying question type in Calculus (covered more in university these days), when they appear in modern texts, seem to require the use of a calculator. A book made like Protter and Morrey — written before the days of calculators — will often compensate by having questions you can do with a little thought and some scribbling of side calculations, making the calculator un-necessary. In doing their Related Rates problems, I haven’t felt the need for even a slide rule, which would have been the “technology” current with the text.
Sometimes going “desert island style” means you need to do long division, and at other times, you will need to be handy with dealing with radicals. And a book like this would be written with the expectation that you rarely need decimal answers, so an answer like is just left that way. No need to calculate further.
Here is an example:
A point moves along the curve in such a way that . Find when .
In this day and age, I would almost be made to feel like an inventor who had discovered the art of solving related rates problems by hand all by myself. Of course, far from being a trailblazer and breaker of new ground, I am really treading on paths that are well-worn by every first-year math student in the past several decades. This is by no means magical and neither you nor I would not be the first ones to master it. But that doesn’t take away from the joy of discovery and “getting it right”, at least not for me.
My solution, scribbled in ink on a blank, unruled sheet of paper, was thus:
I found writing an answer this way to be liberating. Over the past 30 years or so, we have come to expect to see all numbers in decimal form. Decimals are almost never exact, since not many rational numbers are nicely convertable into base 10. Base-10 is also not intuitive to computers, which thinks only in base-2, and require an arithmetic logic unit to make the conversions. Today’s faster processors and “textbook-style” calculators make this problem less visible, but it is always there, and will likely be there for some time to come.
Over the past 30 years, we have come to expect that all numbers are integers followed by decimals. Why should we be locked into this expectation? is also a number. And it is better than a decimal representation such as 3.795, because it is exact, no rounding necessary.