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# conjugation (mnemonic)

In pre-college mathematics, students typically learn how to rationalize the denominator (or, in some cases, numerator) of expressions such as $\displaystyle\frac{3}{\sqrt{11}+2}$ and $\displaystyle\frac{\sqrt{x+h}-\sqrt{x-h}}{2h}$. In order to do this, they multiply the numerator and denominator of the fraction by an algebraic conjugate (or, in some cases, its negative) to eliminate the square root(s) in the appropriate part of the fraction. Typically, the only algebraic conjugates that pre-college students encounter are those in some quadratic extension.

Most students who have advanced far enough in mathematics to encounter rationalizing denominators or numerators have also encountered some (usually Indo-European) foreign language. Such students are familiar with the concept of conjugation of verbs, in which the ending of the verb changes to make agreement with the person and number of the subject. A helpful mnemonic for students to calculate the algebraic conjugates that they need to use is pointing out to them that the procedure in mathematics is similar (and actually easier) than in foreign languages. The algebraic conjugates (or their negatives) that they need are nothing more than changing the ending of the number. For example, the way that a pre-college student is taught to rationalize the denominator of an expression such as $\displaystyle\frac{3}{\sqrt{11}+2}$ is:

$\begin{array}[]{rl}\displaystyle\frac{3}{\sqrt{11}+2}&\displaystyle=\frac{3}{% \sqrt{11}+2}\cdot\frac{\sqrt{11}-2}{\sqrt{11}-2}\\ &\\ &\displaystyle=\frac{3\sqrt{11}-6}{11-4}\\ &\\ &\displaystyle=\frac{3\sqrt{11}-6}{7}\end{array}$

## Mathematics Subject Classification

97D40*no label found*11R04

*no label found*

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## Comments

## hopefully not offensive

In the object "congugation (mnemonic)", I make reference to students learning foreign languages in which they must conjugate verbs. I know that this procedure is not referred to as conjugation in all foreign languages. To play it safe, I qualified that statement using "Indo-European". I don't want to offend somebody by not including a language that they know from being considered simply from the fact that I don't know very much about the language. For example, I know that we have someone active from Finland, and that Finnish is not an Indo-European language. Also, if someone speaks a language in which the changing of verbs is not referred to as conjugation, then this mnemonic is obviously not helpful; however, I feel that it is a helpful mnemonic for those who encounter Indo-European languages.

## Re: hopefully not offensive

Not to worry, we have in the Finnish language a quite rich conjugation of verbs -- BTW I have made the conjugation and syntax program "Tuuli" which gives 1064 different forms of an arbitrary verb (there exist still more forms). But unfortunately, 'conjugation' is "taivutus" in Finnish :(

Also the Basque verb conjugation is versatile. On the contrary, the Chinese (hanyu) has no conjugation, and I think there are in PM some persons with this mother tongue.

Regards,

Jussi

## Re: hopefully not offensive

> contrary, the Chinese (hanyu) has no conjugation, and I

> think there are in PM some persons with this mother tongue.

Indeed. And just to add another example, Japanese has verb (and adjective) conjugation. I hear Korean has too, but I have no experience with this language.

// Steve