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Say you have a cube with its volume being 27 centimeters cubed. All its dimensions are equal to 3cm, since 3x3x3 = 27 (I know it could have different values, but that isn't the focus here), and we know it is 3 dimensional because the measuring unit is raised to the power of three. So, what would happen if I were to raise the cm to the power of 0? Does this question even make mathematical sense? Is it an impossibility, does it equal one or what other possibilities are there?

Please keep in mind I am still in high school and am not a native english speaker,so if possible please explain in a more "easy to understand" language. Thank you guys!

Edit: Someone commented it would be just a point, this was my first thought too, but later I thought it didn't make much sense. A point has coordinates, it can't have a value (right?), otherwise it becomes a segment of a line, not only that but wouldn't the number before the cm multiply it and turn it into another value then? Maybe I don't understand enough mathematics, but if $cm^0$ was indeed just a point, how would you represent it numerically?

Justquestionasker
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  • If cm is to the power of $0$ then it would be a point, I guess. – bobeyt6 Oct 22 '22 at 23:03
  • Relevant questions: https://math.stackexchange.com/questions/1980010/, https://math.stackexchange.com/questions/1592102/, https://math.stackexchange.com/questions/3483152/ – Jam Oct 22 '22 at 23:09
  • "$cm^0$" makes no sense. A bit more trick is something like "$kg^2$". A physical constant can contain this, but "$kg^2$" itself does not make sense since we cannot multiply weights. – Peter Oct 23 '22 at 08:22

2 Answers2

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Yes: $(3 \text{ cm})^0 = (5 \text{ s})^0 = 1$.

Raising a unit to zeroth power is the same as not having that unit.

Tomek Czajka
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Physical quantities commonly have an associated dimension, for instance a dimension of time $\text{T}$, mass $\text{M}$, or length squared $\text{L}^2$. Others have no dimension at all, such as angles, sound pressures, and constants. In general, math tends to only be interested in these dimensionless quantities.

Quantities with dimension obey a few fairly intuitive algebraic rules. Two quantities may be added or subtracted only if they are of identical dimensions (e.g. $2.5\,\text{cm}+0.7\,\text{cm}$ but not $2.5\,\text{cm}+0.7\,\text{s}$) but the quotient or product can be taken for quantities of any dimension (e.g. $2.5\,\text{cm} / 0.7\,\text{s}$, as in velocity). If we want to describe the mathematical structure of these rules, we could do so using the principles of vector spaces, as is done in MSE Q1980010 and MSE Q3483152. Along the same lines, physical units are essentially unit vectors in these vector spaces.

But for all intents and purposes, a unit acts like an algebraic variable. Any unit raised to the power of zero represents a dimensionless quantity and is what would measure a constant or the quotient of two quantities of equal dimension. For instance,

$$\frac{2.5\,\mathrm{cm}}{0.7\,\mathrm{cm}}=\frac{2.5}{0.7}\,(\mathrm{cm})^{1-1}=3.6\,(\mathrm{cm}^0)$$

which has dimension $\text{L}/\text{L}=1$.

Jam
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