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Dimensionless numbers
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Ratios such as Reynolds or Mach numbers that govern similarity.
Dimensionless numbers are ratios where all units cancel. They determine how a system behaves independent of size or units. Examples: Reynolds number (inertia vs viscosity), Mach number (speed vs sound speed), the fine-structure constant (≈1/137).
Flow in a pipe turns turbulent when the Reynolds number Re = ρvd/η exceeds a few thousand, regardless of the fluid.
Two systems with the same dimensionless numbers behave alike, which is why scale models of ships and aircraft work.
In more detail
A dimensionless number has no units because its dimensions cancel out, such as a ratio of two lengths. Such numbers describe the shape or regime of a system independently of the units used.
Real-world uses
- The Reynolds number tells engineers whether a flow in a pipe is smooth or turbulent.
- The refractive index of glass is a dimensionless ratio of the speed of light in vacuum to its speed in glass.
- Aircraft wind-tunnel tests use the Reynolds number to scale results from a small model to a full-size wing, provided the numbers match.
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Frequently asked questions
Is a radian dimensionless?
Yes. It is the ratio of arc length to radius, so the metres cancel out.
Why are dimensionless numbers useful?
They let a result from one experiment apply to a scaled-up or scaled-down system with the same behaviour.
Is a dimensionless number always a pure ratio?
Yes. Its units cancel exactly, so the value is the same in any consistent unit system.
Related topics in Measurement & Math Tools
Checking equations by dimensions · Deriving formulas by dimensions · Limitations of dimensional analysis · Order-of-magnitude estimates · Unit conversion · Scalars and vectors