-1

Say you have a cube with sides equal to $1$ desimeter, dm. The volume $V$ will be given by $$V = 1dm \cdot 1dm \cdot 1dm = (1dm)^3.$$

This equals to $1^3 \cdot d^3 \cdot m^3$.

How come all places write cubic desimeters as $1dm^3$ then, without raising $d$ in $3$ as well?

Davood
  • 4,223
Erik
  • 119
  • 1
    There's no such thing as a $d$: recall $1$ decimetre is $1/0$ metres. – Angina Seng Sep 24 '17 at 13:12
  • You should think of "dm" as one word describing the unit. Short for "decimeter". The unit of volume is then $\text{decimeter}^3$. – Ethan Bolker Sep 24 '17 at 13:14
  • @EthanBolker This is the conflict I am facing. I picture "1dm" divided between to parties: the number or quantity, 1, and the unit, m. But which party does the prefix belong to? Intuitively it seems that it belongs to the quantity. To picture 1dm better, we say 10^(-1)m instead. However, then we end up with (1dm)^3 = 10^-3 * m^3, not 1dm^3 – Erik Sep 24 '17 at 13:17
  • See my comment expanded into an answer. – Ethan Bolker Sep 24 '17 at 13:19
  • @LordSharktheUnknown I think the OP asks a good question. See my answer. – Ethan Bolker Sep 24 '17 at 13:34

1 Answers1

1

You should think of "dm" as one word describing the unit. Short for "decimeter". The unit of volume is then $\text{decimeter}^3$.

But you can in fact "cube the d" if you do it right. The metric prefix "d" really means "multiply by $1/10$" so $$ (1 \text{dm})^3 = 1^3 \times (1/10)^3 \times \text{m}^3 . $$

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
  • This is exactly the way I picture it. But if then replace (1/10)^3 with d^3, you get 1 x d^3 x m^3. This is very confusing when reading volumes in books or online, where one cubic desimeter is displayed as 1dm^3, not 1d^3m^3. In my mind, reading 1dm^3 equals to 0.1m^3. But if we follow the writing in your answer and the one I picture as correct, we get d^3m^3 = 0.001m^3. Two very different answers – Erik Sep 24 '17 at 13:22
  • 1
    $1 \text{dm}^3$ really is $0.001\text{m}^3$. The convention is to think of the unit as a single word, so you don't exponentiate the individual characters. It's just completely unconventional to write $\text{d}^3$ but it makes sense if you do it right. That's separating the prefix from the unit. You could write $\text{megaton}^3$ as $\text{mega}^3\text{ton}^3$. – Ethan Bolker Sep 24 '17 at 13:27