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I was doing a very simple equation, $\frac{1}{2}kx^{2}$, when I realized that, if I represented the distance in centimeters, $x^2$ would grow in size (because x is 10 centimeters), but if I represented it in meters, $x^2$ would shrink in size (because x is 0.1 meters). Why does this occur, if the values represent the same reality?

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  • $\begingroup$ $1\ {\rm m} = 10^3\ {\rm mm} = 10^{-3}\ {\rm km}$. Yet, $1\ {\rm m}^2 = 10^6\ {\rm mm}^2 = 10^{-6}\ {\rm km}^2$. Are you surprised that the number of squared millimeters in a squared meter is larger than the number of millimeters in a meter, while the number of squared kilometers in a squared meter is smaller than the number of kilometers in a meter? If not, you also should not be surprised by the scenario in your question. $\endgroup$
    – Andrew
    Feb 22 at 5:11
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    $\begingroup$ Do you mean that if you express your height as 1600 mm rather than 1.6 meter you grow in size? The two expressions represents exactly the same size. $\endgroup$
    – nasu
    Feb 22 at 5:12
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    $\begingroup$ If you change the units in your displacement, $x$, you should also change the units and numeric value of your spring constant $k$ accordingly, depending on what unit you want your spring's potential energy value in. $\endgroup$
    – notovny
    Feb 22 at 18:27
  • $\begingroup$ what do you mean by shrink and grow, in this context? compared to what? $\endgroup$
    – njzk2
    Feb 22 at 22:30
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    $\begingroup$ It does not make sense to directly compare numerical values in this way; you have to always keep the units. There is no meaningful way to say whether an area (something with units m^2 or cm^2) is greater or less than a length (something with units m or cm). $\endgroup$ Feb 23 at 14:23

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When you square a length of $10 cm$ you get an area of $100cm^2$, which is equal to the area of a square of side length $10cm$.

When you square a length of $0.1m$ you get an area of $0.01m^2$, which is equivalent to the area of a square of side length $0.1m$.

Both areas are the same. You were making a mistake by not taking into consideration the units and only judging the increasing or decreasing of the numerical value. Keep in mind that $1m^2 = 10^4cm^2$. Hence, it all makes sense.

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You are comparing 2 completely different things based on their numerical value.

if I represented the distance in centimeters, $ x^2 $ would grow in size

Grow in size, with respect to what? I assume that you are comparing w.r.t. previous length. But this is absurd. You can't compare 2 quantities with different dimensions.

If you are just comparing the numerical values, then this should not be surprising. Recall that the parabola $y = x^2$ is below the curve: $y = x$ when x is between 0 and 1.

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To avoid too many "ifs", let's consider non-negative values only.

If $x$ is a number then it's true that for some values of $x$ we have $x^2>x$, for some other values of $x$ we have $x^2<x$ (and there are cases when $x^2=x$). We can colloquially say that when a number gets squared, in some cases it grows and in some cases it shrinks, depending on the number.

However if $x$ is not dimensionless then neither $x^2>x$ nor $x^2<x$ (nor $x^2=x$) can be true, because e.g. we cannot compare meters to meters squared.

We can get some more insight from interpreting squaring as multiplying by a factor. For comparison let's start with a simpler example. Consider doubling, i.e. multiplying by the fixed factor of $2$:

$$double(x) \equiv 2 \cdot x$$

We can say that when a value gets doubled, it grows by the factor of $2$. Similarly we can imagine squaring as multiplying by a (non-fixed!) factor:

$$x^2 \equiv x \cdot x$$

Do you see how the right hand sides are similar to each other? Now we can colloquially say that when a value $x$ gets squared, it grows by a factor of $x$. This reasoning does not require $x$ to be dimensionless.

For dimensionless values we can observe e.g. that:

  • $10$, when squared, grows by a factor of $10$; the factor is greater than $1$, so it's really a growth;
  • $0.1$, when squared, grows by a factor of $0.1$; the factor is less than $1$, so it's a shrinkage, not a growth.

Note the values $10$ and $0.1$ are different and the two respective factors are different. But in your example:

  • $10 cm$, when squared, grows by a factor of $10 cm$;
  • $0.1 m$, when squared, grows by a factor of $0.1 m$.

The factors cannot be compared to the (dimensionless) number $1$, thus we cannot tell if it's a "real growth" or a "real shrinkage". Still the same value ($10 cm = 0.1 m$, right?) grows by the same factor (again: $10 cm = 0.1 m$, right?), so the outcome must be the same.

No matter what unit of length we choose for your $x$, the case will always be that $x^2$ is $x$ times larger than $x$, i.e. it's $x$ multiplied by the factor of $x$. The same value multiplied by the same factor gives the same outcome. We never can tell if the factor is greater than dimensionless $1$, because the factor is not dimensionless.

If we forget the factor is not dimensionless and try to treat is as a pure number, and compare it to $1$ to tell if it's a "real growth" or a "real shrinkage", then we may come to different conclusions, depending on the unit we neglected. It seems this is exactly what you did. Such invalid conclusions are one of the reasons we shouldn't neglect units in our calculations.

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  • $\begingroup$ This answer is the best among all of the answers I've seen. Thank you very much. $\endgroup$ Feb 23 at 4:03
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If the big square is 1cm by 1cm, it's the same area as 100 square mm

enter image description here

$0.5x^2$ could represent the area of the red triangle, where $x$ is the length of the side.

If $x=1$ (in cm) the area is $0.5$ square cm, but it has to be $50$ square mm, so it's true that the numerical value for the area must change, if the units change, but both values represent the same area.

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My intuition is to think of $cm^2$ as two units multiplied together:

$$10\ cm^2 = 10\ cm \cdot cm$$

Given that $1\ m = 100 \ cm$, you can convert one of the $cm$ above by dividing by 100:

$$10\ cm\cdot cm = 0.1\ m\cdot cm$$

But that doesn't give us a value in square meters, because $0.1\ m\cdot cm \neq 0.1\ m^2$. To fully convert the value to square meters, you need to replace the second $cm$ too:

$$0.1\ m\cdot cm = 0.001\ m \cdot m= 0.001\ m^2$$

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