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I try to incorporate a little bit of history into my mathematics courses. For instance, why do we rationalize denominators? That is, why is 1/sqrt(2) traditio

I try to incorporate a little bit of history into my mathematics courses. For instance, why do we rationalize denominators? That is, why is 1/sqrt(2) traditionally considered bad form? There is a historical reason for this that few students today know about. I think understanding this history puts things in context and makes the subject less about arbitrary rules. Here's something that puts basic algebra into perspective. Every equation that we teach you to solve by hand is reducible to either a quadratic or linear equation. The rules we teach are all about transforming expressions/equations into quadratic or linear form. There is a purpose. It's not random. Does it help? I don't know.

waterhouse 10y ago
> rationalize denominators Hah. Ah, yes. Rationalizing denominators is sometimes useful for calculations and sometimes counterproductive. I think numbers of the form 1/√n are almost always simplest and best understood as 1/√n rather than √n/n. (Then you have symmetry in statements like "The diagonal of a square is √2 times its side"/"The side of a square is 1/√2 times its diagonal".) Probably the same for a/√n or 1/b√n. If you're about to add it to another such number and you need a common denominator, like 1/√2 + 1/√3, then you can turn it into 3√2/6 + 2√3/6; on the other hand, if you were to multiply 1/√2 and 1/√3, it's most sensible to keep them that way and get 1/√6; if you then needed to square it, 1/6 is simpler than 6/36. I do believe that teachers' insistence on rationalizing denominators regardless of context was an instance of cargo-culting—following an arbitrary rule without understanding where it came from or when it was appropriate. (And, incidentally, any denominator of the form "a ± b√c" should almost certainly always be rationalized, and that is a more difficult and valuable trick.) Edit: By the way, I would be interested to know what "historical reason" you have in mind. I have a feeling that it's of the same form as "at one point, mathematicians were sort of embarrassed by the idea of negative numbers, which were obviously not real, so they would prefer forms like x + 3 = 0 over x = -3". That's the only reason I can think of for always preferring that form. Was your comment about converting things to linear and quadratic form meant to apply to this? I hope you wouldn't assert that it was always to be preferred. (If a^2 + b^2 = c^2, tell me whether "a = 1/√5, c = 1/√2" yields simpler calculations than the alternative.) But my teachers did not communicate anything so nuanced—points taken off for any final answer anywhere with square roots in the denominator—and I don't think it was communicated to them, either. (At least one of them was led to assert that √5/5 was, in itself, "simpler" than 1/√5.) On the original subject of this thread, I might say that, whether or not it's directly passed down to students, the background of mathematics should be incorporated into what is taught to teachers, because otherwise a majority of them will be ignorant, and will come off to intelligent students as blindly following and enforcing arbitrary rules that they don't understand.
- yequalsx 10y ago
  Before computers became ubiquitous and cheap if one wanted to know the the first 5 decimals of 5/sqrt(2) you had to look up the result in a table of values. But to save space and printing cost that table of values did not contain 1/sqrt(2). It did contain sqrt(2). My comment about quadratic and linear equations was to point out that pretty much most of basic algebra is realizing that these are the only two types of equations that can be solved by hand (ignore 3rd and 4th degree formula) and what we do in algebra is to study equations that can be reduced to quadratic or linear equations.
- bikenaga 10y ago
  I've always assumed the "historical reason" was that when people had to do computations by hand, it was easier to find the decimal value of 1/2^(1/2) [how did you make those square root symbols?] by dividing 2^(1/2) by 2 than by dividing 1 by 2^(1/2) (assuming that you knew the decimal expansion of 2^(1/2) ). With calculators and computers, that isn't important. Most of the math teachers I know would not penalize anyone for leaving an answer as "1/2^(1/2)". But the idea of rationalizing is important - for example, so that you can express 1/[a + b * 2^(1/2)] in the form p + q * 2^(1/2), where a, b, p, q are rational.
  - waterhouse 10y ago
    Square root symbols: option-v on Macs. Don't know about other operating sysetms. ± is option-+ (aka option-shift-=). I just did option-[every key], and option-shift-[every key], at various points in the past, and learned the symbols that I liked.

waterhouse 10y ago

> rationalize denominators Hah. Ah, yes. Rationalizing denominators is sometimes useful for calculations and sometimes counterproductive. I think numbers of the form 1/√n are almost always simplest and best understood as 1/√n rather than √n/n. (Then you have symmetry in statements like "The diagonal of a square is √2 times its side"/"The side of a square is 1/√2 times its diagonal".) Probably the same for a/√n or 1/b√n. If you're about to add it to another such number and you need a common denominator, like 1/√2 + 1/√3, then you can turn it into 3√2/6 + 2√3/6; on the other hand, if you were to multiply 1/√2 and 1/√3, it's most sensible to keep them that way and get 1/√6; if you then needed to square it, 1/6 is simpler than 6/36. I do believe that teachers' insistence on rationalizing denominators regardless of context was an instance of cargo-culting—following an arbitrary rule without understanding where it came from or when it was appropriate. (And, incidentally, any denominator of the form "a ± b√c" should almost certainly always be rationalized, and that is a more difficult and valuable trick.) Edit: By the way, I would be interested to know what "historical reason" you have in mind. I have a feeling that it's of the same form as "at one point, mathematicians were sort of embarrassed by the idea of negative numbers, which were obviously not real, so they would prefer forms like x + 3 = 0 over x = -3". That's the only reason I can think of for always preferring that form. Was your comment about converting things to linear and quadratic form meant to apply to this? I hope you wouldn't assert that it was always to be preferred. (If a^2 + b^2 = c^2, tell me whether "a = 1/√5, c = 1/√2" yields simpler calculations than the alternative.) But my teachers did not communicate anything so nuanced—points taken off for any final answer anywhere with square roots in the denominator—and I don't think it was communicated to them, either. (At least one of them was led to assert that √5/5 was, in itself, "simpler" than 1/√5.) On the original subject of this thread, I might say that, whether or not it's directly passed down to students, the background of mathematics should be incorporated into what is taught to teachers, because otherwise a majority of them will be ignorant, and will come off to intelligent students as blindly following and enforcing arbitrary rules that they don't understand.

yequalsx 10y ago
Before computers became ubiquitous and cheap if one wanted to know the the first 5 decimals of 5/sqrt(2) you had to look up the result in a table of values. But to save space and printing cost that table of values did not contain 1/sqrt(2). It did contain sqrt(2). My comment about quadratic and linear equations was to point out that pretty much most of basic algebra is realizing that these are the only two types of equations that can be solved by hand (ignore 3rd and 4th degree formula) and what we do in algebra is to study equations that can be reduced to quadratic or linear equations.
bikenaga 10y ago
I've always assumed the "historical reason" was that when people had to do computations by hand, it was easier to find the decimal value of 1/2^(1/2) [how did you make those square root symbols?] by dividing 2^(1/2) by 2 than by dividing 1 by 2^(1/2) (assuming that you knew the decimal expansion of 2^(1/2) ). With calculators and computers, that isn't important. Most of the math teachers I know would not penalize anyone for leaving an answer as "1/2^(1/2)". But the idea of rationalizing is important - for example, so that you can express 1/[a + b * 2^(1/2)] in the form p + q * 2^(1/2), where a, b, p, q are rational.
- waterhouse 10y ago
  Square root symbols: option-v on Macs. Don't know about other operating sysetms. ± is option-+ (aka option-shift-=). I just did option-[every key], and option-shift-[every key], at various points in the past, and learned the symbols that I liked.