2

In this answer I wrote that perhaps the principal utility of canonical forms is to tell whether two things are equal.

Is $\dfrac 1 {1-\sqrt 2}$ equal to $1+\sqrt2$, or $\dfrac 1 {2\sqrt 3}$ to $\dfrac{\sqrt3} 6$? Just put the expression in "simplest radical form" and you can see whether they're equal.

Is there another purpose?

Is the quadratic equation $\dfrac 5 x = 3+x$ the same as $x^2+3x-5=0$? Put them in the form of quadratic term first, then linear term, then constant term.

Is $\dfrac{51}{68}$ equal to $\dfrac{21}{28}$? Put them in lowest terms.

Perhaps another purpose can be seen in this: $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\cos x - \sqrt 2}\right)^2 = 1 $$ is better than the form $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\sqrt 2 - \cos x}\right)^2 = 1 $$ despite their saying the same thing.

Likewise $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(c+a-b)} $$ is better than $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(a+c-b)} $$ because of the rotation of $a,b,c.$ Does this count as a different purpose? Or might it be a purpose of a different thing?

Community
  • 1
Michael Hardy
  • 1
  • From what I understand, rationalizing the denominator was beneficial in the pre-calculator era. –  Oct 27 '16 at 17:32
  • @Bye_World : "Beneficial" for what purpose? Some purpose that they don't serve when we have calculators? – Michael Hardy Oct 27 '16 at 17:33
  • Consider $\frac{\sqrt{a}}{a}$ vs $\frac{1}{\sqrt{a}}$. Before calculators, to evaluate this, you'd have to do a series expansion (or Newton's method or whatever approximation scheme you prefer) to approximate $\sqrt{a}$. Then it's much easier to take that approximate result (which may be 10+ digits long depending on your needs) and divide by $a$ than to divide that number by $1$. I assume, anyway. Luckily I grew up in the calculator/ computer age. –  Oct 27 '16 at 17:36
  • Are you suggesting that it is only for numerical calculation that these things are useful? What if you're doing algebra and you don't have a specific numerical value of $a$? – Michael Hardy Oct 27 '16 at 17:40
  • Here's MJD's version of this argument. Note that most of the answer to that linked question, though, see to postulate like you that having things in standard form makes them easier to recognize -- which of course is certainly a benefit. –  Oct 27 '16 at 17:42
  • I'm giving one other possible explanation. I'm not saying that it's the only reason or even that it's more important than being able to more easily recognize when two expressions are equal. –  Oct 27 '16 at 17:44
  • @DavidQuinn : Fortunately I phrased that as a question. – Michael Hardy Oct 27 '16 at 18:05

1 Answers1

1

$$\frac1{1-\sqrt2}=1+\sqrt2\iff \color{red}1=(1-\sqrt2)(1+\sqrt2)=\color{red}{-1}$$

Ok, so that was just goofing around a trivial typing mistake. To try to actually and seriously address your nice question:

No, the equation $\;\frac5x=3+x\;$ is not the same equation as $\;x^2+3x-5=0\;$...not even close: the first equation has a rational non-polynomial fucntion on the left side whereas the second expression above is a polynomial . What happens is that we usually are interested in the equations' solutions , and both equations above have the very same solutions, though one of them is defined on onje point more than the other one.

About your question about fractions: simple definition, and

$$\frac{51}{68}=\frac{21}{28}\iff51\cdot28=21\cdot68(=1,428)\;\;\;\color{green}\checkmark$$

Of course, you know all this and all the rest of things you "asked" (better wondered), and I'm not sure I can see what the actual intention of all this could be, yet when we have some uses we go with one thing, and for other ones we may go with another one

FOr example, if there are $\;68\;$ people and we bought $\;51\;$ pizzas to share among them all, it may be easier and much clearer to actually write $\;\frac{51}{68}\;$ instead of an apparentely easier $\;\frac34\;$. The last fraction tells me very little in the first exposed situation.

I insist with my students, either from univeristy or from hig school (when I have them), that if possible and reazonably easy and quick they must write an expression (and I'm thinking of functions now" in several equivalent ways depending on what the task is, for example

$$f(x)=x-\frac1x=\frac{x^2-1}x=\frac{(x-1)(x+1)}x$$

The first form is nice to realize what the domain is and, more important, to differentiate it in case of necessity. The second and third forms are better to find out where the function vanishese and also for asymptotes .

Finally, whether $\;c+a-b\;$ is better or worse than $\;a+c-b\;$ or $\;a-b+c\;$ is mostly a matter of tste, though I think that alphabetical order usually makes things easier to grasp, so I'd go with the third formk unless some conditions are given that may make other form easier to work with.

DonAntonio
  • 211,718
  • 17
  • 136
  • 287