Math Stuff That Used To Make Sense Part I: Rationalizing Denominators

In the 21st century math teachers are still making students "rationalize the denominator," because a square root in the denominator is a crime against nature. Well, there actually used to be a good reason for not wanting a never-ending, never-repeating decimal in the denominator of a fraction, but we'd have to think back before calculators.

Supposedly, the first mathematician to insist the diagonal in a perfect 1x1 square can't be expressed as a whole number or a fraction was put to death. Irrational numbers have been given a bad name, but they come up very often in geometry and trigonometry, thanks to the Pythagorean Theorem. The "special triangles" so beloved by the SATs contain Ö2 and Ö3.

Why can't you just leave square roots in radical form? Well, in the real world, you never give your friend directions like "go Ö17 miles down the road and turn left." There is an effective but confusing and annoying algorithm to compute square roots which reminds me of long division and not in a good way.

So there used to be tables of square roots in the back of textbooks, or students used to memorize square roots using mnemonic devices, in case you weren't allowed to use your notes or a table (or a slide rule) on the test.

"For the square root of 2

I wish I knew" = The number of letters in each word is 1, 4, 1, 4. The square root of two is 1.414...

"To know the square root of 3

O charmed was he" = 1.732

"To know the root of 5

So we now strive" = 2.236

That was the reason for making students reduce irrational numbers like Ö75 to 5Ö3. They're both 8.660... but before calculators we memorized the square root of 3 but not the square root of 75. We wanted a form we could get a couple of decimal places out of, and multiplying 1.732 by 5 was a cinch compared to having to do the square root algorithm on 75 every time. Of course, now with calculators, there's no longer any need to reduce roots like this.

Similarly, before calculators, expressing the cosine of 45 degrees as 1/Ö2 would make your heart sink. Imagine trying to do long division, dividing 1 by a never ending, never repeating decimal. Echhh. So you "rationalized the denominator" and came up with Ö2/2. Dividing a long number by two is pretty easy in comparison. Both expressions give you 0.7071...

But nowadays? Come on: Ö2/2 and 1/Ö2 require the exact same number of keystrokes on your calculator to get 0.7071.... I submit there is absolutely no reason to continue to teach this outdated arithmetic to students, and to penalize students who write 1/Ö2 instead of Ö2/2.