Doubt about dimensionless numbers

Dimensionless numbers are very important.

First, all "purely mathematical" constants are dimensionless. This includes

• Integers: $..., -1, 0, 1, 2, ...$
• Rational numbers: $frac{1}{2}$, $frac{1}{10}$, ...
• Real numbers: $sqrt{2}$, $e$, $pi$, $ln(2)$...
• Complex numbers: $i$, $1+i$, ...

Obviously all these numbers are crucial in both physics and mathematics.

Second, many physical quantities are naturally dimensionless. Examples include angles and the number of particles in a system.

Finally, dimensionless combinations of dimensionful constants are often crucial for determining the behavior of a system. The behavior of a system can often be divided into different regimes depending on the size of the dimensionless combinations that define it.

As an example, we can consider two bodies with masses $m_1,m_2$ orbiting their center of mass in Newtonian gravity. An important dimensionless parameter describing the system is the mass ratio, $q=m_1/m_2$. If $qapprox1$, then the two bodies are approximately equal in size, and both undergo a "large" motion to orbit their common center of mass. If $qgg1$ or $q ll 1$, then one object is much heavier than the other, and it becomes a very good approximation to say that the lighter object is orbiting the heavier object, which is approximately still.

Often, a useful approach analysis of a physics problem is to identify a small dimensionless quantity, and to perform a Taylor expansion in this small quantity.

Here is an answer for a slightly different take on your question, considering combining different physical variables to produce dimensionless ones. (Andrew mentioned this is his earlier answer.) Dimensionless numbers (more properly called dimensionless groups) arise from combining dimensional analysis with experimentation, to provide empirical formulas for evaluating complicated physical phenomena.

I am most familiar with those used in thermodynamics, such as: Nusselt number, Reynolds number, Prandtl number, Grashof number, and Stanton number. For example, the Nusselt number allows complicated convective heat transfer problems to be evaluated using a heat transfer coefficient, instead of having to evaluate the microscopic details of the heat transfer process. Or the Reynolds number which indicates whether flow is laminar or turbulent.

Some engineering texts on thermodynamics/heat transfer address dimensional analysis and dimensionless numbers in some detail; for example Elements of Thermodynamics and Heat Transfer by Obert and Young, and the old standard reference Heat Transmission by McAdams.

Another type of dimensionless number is a safety factor for a mechanical structure. The maximum load allowed for a bridge is less than the estimated failure load by the safety factor to account for stress raisers, fatigue, uncertainty in material properties, etc. The recommended safe speed for rounding a curve is a factor less than the maximum speed estimated for slipping to account for worn tires, slippery road, etc.