Do dimensionless quantities have units or not? I am confused a bit about this concept.

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    $\begingroup$ By definition, dimensionless quantities are quantities without units. $\endgroup$ – Noiralef May 7 '17 at 19:40
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    $\begingroup$ Dimensionless means unit-less. $\endgroup$ – Wrichik Basu May 7 '17 at 19:44

The tricky examples are angles and percentages. The SI-brochure says these quantities are dimensionless, also termed quantities of dimension 1. These dimensionless quantities do have units: 1 rad is equal to 1; 1° is equal to 2π/360. The unit of percentage is %, which is equal to 0.01. Hence 1 rad = 1 = 100%.

  • $\begingroup$ It might also be worth pointing out that, even if two terms have the same dimensionality, that doesn't mean they're the same "stuff." We typically think of radians as being "different" than percentages, even though they are both dimensionless. Likewise you can measure time in seconds, or you can measure the Specific Impulse of a rocket fuel, but they are decidedly different things. The one guarantee units do give you is that, if the units don't match, then they must be different types of "stuff" and cannot be added or subtracted. The guarantee is more hazy when the units do match. $\endgroup$ – Cort Ammon May 8 '17 at 17:17

The mere definition of a dimensionless quantity states that a dimensionless quantity is such a quantity to which no physical dimension can be assigned.They can't obviously have any units then.There is a difference between dimensions and units. A dimension is a measure of a physical variable (without numerical values), while a unit is a way to assign a number or measurement to that dimension. For example, length is a dimension, but it is measured in units of feet (ft),meters (m),etc.So when we define a quantity it is necessary to assign both a unit and a dimension to it.Leaving any one of them will render the quantity meaningless


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