Radian is a unit but not a base unit, and mole is "just a number" but is one. Why?

8 answers & replies

• [+][–]Arjun Nair12thSample· 2mo ago

  Angle is defined as arc length divided by radius, so it is length/length. Nothing is left over, that is why $[\theta] = [M^0L^0T^0]$. The radian is just a name we attach so we know the number is an angle and not, say, a ratio of two masses. Same for steradian ($\Omega = A/r^2$, area/area).

  Mole is different because SI chose to treat "amount of substance" as its own physical dimension. So $n$ has dimension [mol] in SI dimensional analysis. That makes Avogadro constant $N_A = 6.022\times10^{23}\ \text{mol}^{-1}$ carry the dimension mol$^{-1}$, so it is not dimensionless. Your reasoning was fine, not luck.

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  • [+][–]Aarav Kulkarni11thSample· 2mo agoAuthor

    This cleared it. So the logic is "unit is a name, dimension is the thing left after cancelling". Making that line #1 on my checklist.

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• [+][–]Harsh Gupta12thSample· 2mo ago

  Mnemonic time. "My Kind Student Always Keeps Mole Candy": Metre, Kilogram, Second, Ampere, Kelvin, Mole, Candela. Seven words, seven base units. It has saved me in more than one test. Also if the options in "not a base unit" include newton, the answer is newton, you're welcome.

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• [+][–]Tushar PawarDropperSample· 2mo ago

  I think radian and steradian are the 8th and 9th base units in SI, that's why they are listed along with the others in the NCERT table?

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  • [+][–]Aman YadavDropperSample· 2mo ago

    No. Base units are seven. Full stop. Radian and steradian were called supplementary units earlier. Now they are treated as derived, dimensionless units. NCERT puts them in a separate list for that reason.

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    • [+][–]Tushar PawarDropperSample· 2mo ago

      Ahh ok, I mixed up "listed in the table" with "base". Thanks, noted.

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• [+][–]Siddharth RaoDropperSample· 2mo ago

  One thing from my first attempt that cost me marks: the SI base unit of mass is the kilogram, not the gram. So when a numerical gives you grams you must convert before putting it in a formula that expects SI, or your answer is off by 1000.

  Also on dimensionless things, my list for tests: angle, solid angle, strain, refractive index, relative density, trigonometric ratios, and ratios of the same quantity in general (like Poisson's ratio). If the quantity is "something divided by the same kind of something", it has no dimensions. Fastest filter in the exam.

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• [+][–]Kavya NaikDropperSample· 2mo ago

  Aarav, don't spend a fourth read on that paragraph, it's not worth it 😅 NCERT mainly wants you to remember: 7 base quantities, their units, and that angle/solid angle are dimensionless. The mole bit is "SI decided amount of substance is its own dimension, done". Seen it asked as a one-line statement question, not as theory.

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