4

$$x \propto y^2$$

How is it different from saying:

$$x \propto y$$

That is; when we say that Two variables are proportional then it means that two variables are related such that when one is zero other is too. And change in one variable is accompanied by change in other. This is a general definition for proportionality. Then if we write $x \propto y^2$, by definition, we implicitly mean that $x \propto y$. So, why write $x \propto y^2$ instead of a simple one $x \propto y$?

Is it due to the calculation of constant of variation? Viz., the constant of proportionality onliy lies between $x$ and $y^2$ relation and not between $x$ and $y$ relation? Is it for that purpose that we specify them?

quid
  • 42,135
Sufyan Naeem
  • 2,298
  • Proportionality means that when I double one side, the other side also doubles. So $x \propto y$ is not the same as $x\propto y^2$: if I double $y$, in the first $x$ doubles, while in the second it becomes $4$ times as large. – Milind Hegde Dec 01 '15 at 09:27
  • I am not asking that whether $x \propto y$ is same as $x \propto y^2$. I know that it is not and it is explicit. In a very general way, we say that if $x$ increases, $y$ increases too then it means $x \propto y$ mathematically. In $x \propto y^2$ we mean same that is; as $x$ increases $y$ increase too. So Why to specify $x \propto y^2$ and $x \propto y$? – Sufyan Naeem Dec 01 '15 at 09:49
  • 1
    Actually, the proportional symbol means more than if one grows, so does the other. It also says that the growth is linear, i.e. by a constant, which is stronger. – Milind Hegde Dec 01 '15 at 09:59
  • For example: If $A$ is the area of a circle and $r$ is the radius, then $A\propto r^2$ as the size of the circle varies. However, $A\not\propto r$. (That symbol means "is not proportional to") – Akiva Weinberger Dec 01 '15 at 11:45
  • @AkivaWeinberger can you please explain rigorously why is it wrong to say $A \propto r$? Area increases as radius. When radius vanishes, circle vanishes and so Area of the circle. – Sufyan Naeem Dec 01 '15 at 15:40
  • 2
    But that's not what $\propto$ means; it's only part of it. We know $A=\pi r^2$. That is, $A$ is a constant multiple of $r^2$. That's what $A\propto r^2$ means. $A$ is not a constant multiple of $r$, so we can't say $A\propto r$. – Akiva Weinberger Dec 01 '15 at 15:42
  • 2
    $A\propto r^2$ means that, if $A$ doubles, then $r^2$ doubles. This is true. $A\propto r$ would mean that if $A$ doubles, $r$ doubles. This is not true. – Akiva Weinberger Dec 01 '15 at 15:43

3 Answers3

5

If $x$ is proportional to $y^2$, then when $y$ doubles, $x$ is four times as much.

If $x$ is just proportional to $y$, then $x$ doubles when $y$ doubles.

For example, the position of a ball falling towards the ground is proportional to $t^2$ ($t$ is time). So if the length of time doubles, the ball has fallen four times as far. This is very different from the ball's velocity, which is proportional to $t$. If the amount of time doubles, the final velocity of the ball also doubles.

EDIT: It may be tempting to think this way: $x\propto y$ means that there is some constant such that $x=cy$. But if $x=y^2$, then $x=y\cdot y$, so we could just write $x$ is proportional to $y$ since $x$ is something times $y$. This is not correct. We only call things proportional when one relates to the other via a constant.

pancini
  • 19,216
  • @Sir. How the definition of proportionality given at Wikipedia covers all the aspects, you are talking about? Definition says, "Two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier." – Sufyan Naeem Dec 04 '15 at 10:37
  • What is your question? – pancini Dec 04 '15 at 20:00
  • Suppose putting radius $x$, we get area of the circle $y$ then increasing radius 2 times viz, $2x$; must give us area of the circle 2 times as previous i.e $2y$. Isn't so? – Sufyan Naeem Dec 04 '15 at 21:22
  • No, every time the radius doubles, the area gets four times larger. Thus, $area\propto x^2$ – pancini Dec 04 '15 at 21:23
  • To put matters straight. think of an arbitrary relation between two variables. Assume it is reported that by increasing one of them to $x$, other reaches $y$ then I have no need to resume and see the relation because I can imagine that if something is $y$ due to $x$ of other thing then must increasing $x$ of other thing 2 times increase something 2 times i.e. it must vary to $2y$. – Sufyan Naeem Dec 04 '15 at 21:30
  • This is not true. Consider $x$ is a side of a square with area $y$. If $x=1$, then increasing $x$ by $1$ increases $y$ by $3$. Increasing $x$ by $2$ increases $y$ by $8$. You can't think of "increasing by." You have to think of multiplying by. – pancini Dec 04 '15 at 21:37
  • Now, I can ask you that how the definition I stated above is covering all you arguments? Especially, this one just above. – Sufyan Naeem Dec 04 '15 at 21:40
  • That's what I'm trying to say: you do not have the correct definition of proportionality. "x is proportional to y" means that $x=cy$ for some non-zero constant $c$. – pancini Dec 04 '15 at 21:42
  • 1
    $x\propto y$ if and only if $x=cy$ for some $c\neq0\in\Bbb{R}$ – pancini Dec 04 '15 at 21:49
  • And, increasing two times and multiplying by 2 is same. You said, "you can't think of 'increasing by'. You have to think of 'multiplying by'." Explain this as well. – Sufyan Naeem Dec 04 '15 at 21:54
  • I'm not sure. I think we may be on the same page using different words. – pancini Dec 04 '15 at 22:22
  • As you said $x \propto y$ is only valid if $x=cy$. Then taking $A \propto r^2$ instead of $A \propto r$ must have just one and only one reason that is: there must be no constant between $A$ and $r$ i.e. $\frac {A}{r} \neq k$. Isn't so? Now I think I am just one step away. If you just answer me to this question, I 'll be there. – Sufyan Naeem Dec 05 '15 at 15:51
  • 1
    @SufyanNaeem yes, and $A\propto r^2\implies A=kr^2$ – pancini Dec 05 '15 at 15:52
  • OMG! The same thing I wrote in my original post earlier and you said "it may be tempting...". Well, Better late than Never. :) – Sufyan Naeem Dec 05 '15 at 19:09
  • You could have read the full paragraph and avoided this conversation :) – pancini Dec 05 '15 at 19:10
4

The proportionality symbol carries the implication that if $x \propto y \implies x=ky$ for some nonzero real k. That being said, it's clear that $ x \propto y \ne x \propto y^2 $ since $ ky \ne ky^2 $

matias
  • 75
1

While dealing with Functions we can do that.

For example. Whether $f(x)=x$, $f(x)=x^2$ or $f(x)=x^3$ we can say that $f(x)=y$ , in general.

While dealing with "$\propto$", don't forget the implication.

For example. $x \propto y \implies x=ky, k \neq 0$

I am pretty sure that you are thinking like we think while dealing with functions. $k$ is not a function there but a simple multiple. We can easily relate (with "equals to" sign) any two quantities in mathematics with prefixing just a $f$ (or any symbol denoting function). But we can not express two quantities in equation arbitrarily with placing a constant. So , we have to describe enough when we place a constant in contrast to the situation, when we place $f$ symbol. So, it is obligatory there to write $y^2$ in stead of $y$.

Man_Of_Wisdom
  • 649