Rationalizing Rationalizing the Denominator or Numerator

One of the topics in algebra, precalculus, and Calculus is rationalizing the denominator, or in some cases, rationalizing the numerator, of a fraction. For example, if we have a fraction such a

we note that the denominator is an irrational number. We rationalize the denominator, in this case, by multiplying our given fraction by the number 1 (so that we do not change the value of our fraction), but in the form of the square root of 5 divided by itself. Thus we get

Now, our new form of the fraction has a rational number in the denominator. This was our goal when we set out to rationalize the denominator.

This skill seems a bit strange when we start using a calculator. After all, the calculator is just as happy to do the original problem as it is to evaluate the "rationalied" form. For example, the following screen image, from a TI-83, shows both fractions being evaluated to 1.341640786.

The calculator does not seem to mind evaluating either form. In fact, it was easier (fewer keys to push) to write the original problem than it was to enter the the rationalized version. Why, then, do we spend time learning how to "rationalize" fractions?

This skill is not just a new torture of mathemtics. There are actually some problems in Calculus that take advantage of the rationalizing the denominator or the numerator. However, the skill of rationalizing denominators used to be important long before we study and learn Calculus. We used to teach this skill because it greatly simplified evaluating expressions. This was at a time when we did not have calculators to do the arithmetic. At that time we had tables of square roots. We would look up the square root of 5 and find that 2.236 is a good approximation. If we were to try to evaluate the original problem we would have to divide 3 by 2.236, and that is an ugly, messy task. However, if we "rationalize" the denominator for the original problem to produce

then we can evaluate that fraction by multiplying 3 times 2.236 and then dividing that result by 5. Such a task is much easier to compute by hand. It should be clear from the example above that the advent of powerful calculators has removed the need to "rationalize" denominators as a means for simplifying arthetic computations. However, as noted in the text and above, there are a number of situations in Calculus where we do need the skill.