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I have found the following use of the word "evaluate" in several math books:

"To evaluate the continued fraction, start at the bottom and work your way up:"

$\huge \underbrace{2 + \frac{1}{1+\frac{1}{3}}}=2 + \frac{1}{\frac{4}{3}}=2+\frac{3}{4}= \underbrace{\frac{11}{4}}$ Why is this called an "evaluation" and not a simplification?

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This may be of interest. – goblin Nov 28 '13 at 11:11
Because clearly fractions and decimals are the right way of representing numbers, not continued fractions. This is entirely a product of how we've been trained to think about numbers. – Dustan Levenstein Nov 28 '13 at 11:52
@DustanLevenstein Have we not also been trained to simplify an expression involving fractions, you write it as a single fraction in simplest form? – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ Nov 28 '13 at 11:57
That doesn't contradict the use of the word "evaluate"; it simply means that "evaluate" and "simplify" are equally good words to use to describe this particular procedure. – Dustan Levenstein Nov 28 '13 at 12:03
I just checked the index of my Algebra textbook (Dummit & Foote) for the word "evaluate", and found the evaluation homomorphism from a polynomial ring. Indeed, you are right; I'm pretty sure that is not what we are discussing in this context. (NOTE: this is a joke.) – Dustan Levenstein Nov 28 '13 at 12:52
up vote 5 down vote accepted

Because people commonly use the word evaluate in that context; this is simply a fact of usage. You can see other examples here on the website of the National Institute of Standards and Technology, in the title of this paper in SIAM Review, and here, to pick three of the first few examples that turned up on a search. In this context evaluate simply isn’t the technical term whose definition you give in the comments; it’s another sense of the same word.

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Thank you for the examples, but wouldn't simplify or convert make more sense? – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ Nov 28 '13 at 19:51
@badass: No: this is a normal usage of the word evaluate, so by definition it makes sense. Simplify also makes sense; which one uses here is largely a matter of taste and context. If I were talking to a bunch of fifth-graders and dealing with relatively ‘short’ continued fractions, I might well use simplify; in most contexts, however, I’d probably use evaluate. – Brian M. Scott Nov 28 '13 at 19:54
In my opinion, those fifth-graders have the right to scream "CONTRADICTION!!!" after being preached the sentence: A variable expression is evaluated by replacing each variable with a given value and simplifying the resulting numerical expression. (NOTE: this is a joke.) – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ Nov 28 '13 at 20:30
I found this article if you're interested. – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ Nov 29 '13 at 0:48
@badass: I agree with Peterson that $\frac{m}n$ is proper if $|m|<|n|$. It’s also true that use of the term proper fraction is largely limited to the early grades, before the question of negative fractions arises, and there’s no telling how any given fourth grade teacher, say, would answer the question ‘Is $-2/3$ a proper fraction?’ – Brian M. Scott Dec 12 '13 at 20:56

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