We know that pure numbers are dimensionless then how come universal constants like the gravitational constant have a dimension cause they are also equal to some numerical value and if the numerical values are dimensionless then shouldn't the constants also be dimensionless?
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2$\begingroup$ Think about speed, or velocity. Would you agree that this has dimensions? If you do, then you certainly must agree that the speed of light, $c$, has dimensions as well. Of course, there are certain systems of units (called natural units) that define things such that, for example, speed doesn't have a dimension $\endgroup$– DanDan面Commented Apr 27 at 7:55
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5$\begingroup$ There is no requirement of constants to be dimensionless. I do not fully understand why you'd think so $\endgroup$– SteevenCommented Apr 27 at 8:03
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$\begingroup$ John Baez's article on fundamental constants (and which constants are really "fundamental") addresses some of your questions, I think. $\endgroup$– Michael SeifertCommented Apr 27 at 11:49
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$\begingroup$ Voting to reopen. A perfectly clear question with good answers below. $\endgroup$– gandalf61Commented Apr 28 at 9:08
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$\begingroup$ A physical quantity is written as a number and a unit. When you change unit, both of these change and it remains invariant as a whole. Please clarify what you mean by "have a dimension cause they are also equal to some numerical value". $\endgroup$– Vincent ThackerCommented Apr 28 at 9:59
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3 Answers
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Take for example the gravitational constant. Expressed in SI units it is: $$G=6.674\cdot 10^{-11}\frac{\text{m}^3}{\text{kg}\cdot\text{s}^2}$$
You can also express this constant in imperial units. You do this by inserting $1$ m=$\frac{1}{0.3048}$ ft and $1$ kg=$\frac{1}{0.4536}$ pound in the equation above. We get $$G=1.0691\cdot 10^{-9}\frac{\text{foot}^3}{\text{pound}\cdot\text{s}^2}$$
The physical constant $G$ (i.e. the product of numerical value and measurement unit) in both cases is the same. But of course, when we change the measurement units, then also the numerical value changes, so that their product remains unchanged.
On the other hand, there are a few physical constants which are pure numbers (i.e. without a measurement unit). An example is the fine-structure constant. $$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}=\frac{1}{137.036}$$ This constant is a dimension-less constant, unlike the gravitational constant $G$ from above.
answered Apr 27 at 10:25
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$\begingroup$ It is helpful to think of these dimensional constants as "conversion factors" between physical quantities. The speed of light can be used to convert time and space, Boltzmann's constant for energy and temperature, Planck's constant for energy and frequency, and so on. $\endgroup$– paulinaCommented Apr 27 at 11:04
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Pure numbers are dimensionless, but numbers that measure things usually are not. A length can be $12$ inches or $1$ foot. Units matter here. If you say the length is $12$ or $1$, you don't know how long it is.
Sometimes the thing you measure is a ratio of like quantities. For example, you might want to measure the "oblongness" of a rectangle as the ratio of length to width.
You might find that that a particular rectangle is $24$ inches long and $12$ inches wide. Call its oblongness O. Then
$$O = l/w = \frac{24 inches}{12 inches} = 2 \frac{inches}{inches}$$
You could also calculate it this way.
$$O = l/w = \frac{2 feet}{1 feet} = 2 \frac{feet}{feet}$$
For ratios like this, the units don't make a difference, so we "cancel" them.
$$2 \frac{inches}{inches} = 2 \frac{feet}{feet} = 2$$
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dimension of constant such as
- speed of light (C)=M0L1T-1.
- dimension of planks constant=[ML2T-1].
- dimension of number such as 1,2,3,4......etc have no dimesion
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$\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$– Community BotCommented Apr 27 at 11:05