Talk:Rationalisation (mathematics)

This article is a translation. If something is wrong, I would like that this article is corrected. --Creole99 (talk) 16:32, 21 March 2008 (UTC)

I've been playing around with this and I don't know how to rationalize things like this, because the difference of two squares won't work:
 * $$\frac{ 1 }{ 1 + \sqrt{3}-\sqrt{5}}$$

76.25.36.49 (talk) 22:46, 9 August 2009 (UTC)


 * How about this:

\frac{1}{1 + \sqrt{3} - \sqrt{5}} \cdot \frac{1 + \sqrt{3} + \sqrt{5}}{1 + \sqrt{3} + \sqrt{5}} = \frac{1 + \sqrt{3} + \sqrt{5}}{2\sqrt{3} -1} $$
 * and then one more step will do it. Michael Hardy (talk) 23:57, 9 August 2009 (UTC)

What is the history of this subject?
Most people with training / education in mathematics know how to do this, but the reasons given for the necessity of it are not generally understood. We need an actual history of why there is a ban on radicals in the denominator. I have a guess, but it is not a historical fact (yet).

Here is my guess. Think of Euclid. The numerator could be any length or quantity that one wanted to divide into parts. It could easily be an irrational number. But the total number of parts, and therefore the denominator, must be an integer. In other words, I can divide a cookie evenly for three kids, but how often do you have square-root-of-two number of kids? Seen in this light, the ban makes sense to me.

Any ideas where to search? Tomgear (talk) 01:42, 15 September 2009 (UTC)


 * Your math is wrong. To say that the total number of parts is an integer is inconsistent with the denominator being an integer. Michael Hardy (talk) 03:41, 15 September 2009 (UTC)


 * I wouldn't really call it a "ban". It's a canonical form.  One can judge whether certain kinds of expressions are equal to each other by putting them in canonical form and seeing whether they're the same when they're in such a form.
 * Euclid did not have a concept of real number. He had a concept of what it means to say line segment A is to segment B as segment C is to segment D.  He also had a concept of commensurability: If A and B can each be divided into an integer number of segements of length E, then A and B are commensurable, and E is a "common measure" of A and B.  If A and B have no common measure, then they are incommensurable. Michael Hardy (talk) 03:46, 15 September 2009 (UTC)
 * Euclid did not have a concept of real number. He had a concept of what it means to say line segment A is to segment B as segment C is to segment D.  He also had a concept of commensurability: If A and B can each be divided into an integer number of segements of length E, then A and B are commensurable, and E is a "common measure" of A and B.  If A and B have no common measure, then they are incommensurable. Michael Hardy (talk) 03:46, 15 September 2009 (UTC)


 * Spent a bit of time looking in a history of Math text book. I really think the issue has something to do with the fact that for a long time, magnitudes were conceived of as totally seperate from quantities. For example, the unit, "one" was not considered a number. Root two is a magnitude but not a quantity. I suspect but have not yet found that the "canonical form" of a fraction had to be a magnitude OR quantity in the numerator and a quantity in the denominator. Tomgear (talk) 19:05, 18 September 2009 (UTC)

Surds
An uncommon term is not helpful in the first sentence of an article on developmental mathematics. Especially when there is no link to describe it. We should remove this archaic term and replace it with something descriptive. Cliff (talk) 06:05, 22 March 2011 (UTC)

negative square roots
I think it's very un-pedagogical to give things like $$ \sqrt{-5}$$ as example. Usually √ is defined as a function from nonnegative reals to nonnegative reals. Otheerwise you have at least to specify the "branch cut" or rather range of the argument of the result you choose, given there are multiple solutions to x² = -5. Why not write rather $$ i\,\sqrt{5}$$ instead, to keep things unambiguously well-defined? &mdash; MFH:Talk 14:08, 3 December 2020 (UTC)
 * Even if the elementary courses, square roots are considered only for positive numbers, they are defined for every complex numbers. In calculus, it is useful to choose a uniquely defined square root, the "principal square root", and therefore to rewrite $$\sqrt {-5}$$ as $$i\sqrt 5.$$ In algebra, this rewriting is not convenient for several reasons. For example, for studying $$\mathbb Q[\sqrt {-3}]$$ (Eisenstein numbers), using $$i\sqrt 3$$ amounts introducing two numbers that are not Eisenstein numbers; this would make the study much less clear, and hides the fundamental automorphism of $$\mathbb Q[\sqrt {-3}]$$ (algebraic conjugation). Also, in many case, the quantity under the radical sign may contain variables, and its sign is difficult to determine; see for example Cardano formula. So, in an algebraic context, it is common to define $$\sqrt x$$ as any of the two square roots of $x$ (this is also common in complex analysis). So it depends on the context whether $$\sqrt x$$ is the principal square root or any of the two square roots.
 * In this article, using $$i\sqrt 5$$ instead of $$\sqrt {-5}$$ would make the rationalisation much more difficult, as implying to "rationalize" separately the two square roots ($i$ is a square root). Also this would require to know whether the argument of the square root is positive real, real, or nonreal complex, although the rationalization works exactly in the same way in all these cases (think of the rationalization of $$\textstyle\frac 1 {1+i}$$). So your suggestion may be more pedagogical only for people who will definitely remain at a very elementary level of algebra. For the others this will make their further studies more difficult; this is not the aim of a good pedagogy. D.Lazard (talk) 15:31, 3 December 2020 (UTC)

Mathematics
Reason of rationalizing 41.115.28.110 (talk) 17:42, 25 January 2022 (UTC)