# List of dimensionless quantities

This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.

## Biology and medicine

| Name | Standard symbol | Definition | Field of application |
| Basic reproduction number | $R_{0}$ | number of infections caused on average by an infectious individual over entire infectious period | epidemiology |
| Body fat percentage | | total mass of fat divided by total body mass, multiplied by 100 | biology |
| Kt/V | Kt/V | | medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time) |
| Waist–hip ratio | | waist circumference divided by hip circumference | biology |
| Waist-to-chest ratio | | waist circumference divided by chest circumference | biology |
| Waist-to-height ratio | | waist circumference divided by height | biology |

## Chemistry

| Name | Standard symbol | Definition | Named after | Field of application |
| Activity coefficient | $\gamma$ | $\gamma ={\frac {a}{x}}$ | | chemistry (Proportion of "active" molecules or atoms) |
| Arrhenius number | $\alpha$ | $\alpha ={\frac {E_{a}}{RT}}$ | Svante Arrhenius | chemistry (ratio of activation energy to thermal energy) |
| Atomic weight | M | | | chemistry (mass of one atom divided by the atomic mass constant, 1 Da) |
| Bodenstein number | Bo or Bd | $\mathrm {Bo} =vL/{\mathcal {D}}=\mathrm {Re} \,\mathrm {Sc}$ | Max Bodenstein | chemistry (residence-time distribution; similar to the axial mass transfer Peclet number) |
| Damköhler numbers | Da | $\mathrm {Da} =k\tau$ | Gerhard Damköhler | chemistry (reaction time scales vs. residence time) |
| Hatta number | Ha | $\mathrm {Ha} ={\frac {N_{\mathrm {A} 0}}{N_{\mathrm {A} 0}^{\mathrm {phys} }}}$ | Shirôji Hatta (1895–1973) | chemical engineering (adsorption enhancement due to chemical reaction) |
| Jakob number | Ja | $\mathrm {Ja} ={\frac {c_{p}(T_{\mathrm {s} }-T_{\mathrm {sat} })}{\Delta H_{\mathrm {f} }}}$ | | chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change) |
| pH | $\mathrm {pH}$ | $\mathrm {pH} =-\log _{10}(a_{{\textrm {H}}^{+}})$ | | chemistry (the measure of the acidity or basicity of an aqueous solution) |
| van 't Hoff factor | i | $i=1+\alpha (n-1)$ | Jacobus Henricus van 't Hoff | quantitative analysis (K$_f$ and K$_b$) |
| Wagner number | Wa | $\mathrm {Wa} ={\frac {\kappa }{l}}{\frac {\mathrm {d} \eta }{\mathrm {d} i}}$ | | electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell) |
| Weaver flame speed number | Wea | $\mathrm {Wea} ={\frac {w}{w_{\mathrm {H} }}}100$ | | combustion (laminar burning velocity relative to hydrogen gas) |

## Physics
### Solids

| Name | Standard symbol | Definition | Named after | Field of application |
| Coefficient of kinetic friction | $\mu _{k}$ | | | mechanics (friction of solid bodies in translational motion) |
| Coefficient of static friction | $\mu _{s}$ | | | mechanics (friction of solid bodies at rest) |
| Föppl–von Kármán number | $\gamma$ | $\gamma ={\frac {Yr^{2}}{\kappa }}$ | August Föppl and Theodore von Kármán | virology, solid mechanics (thin-shell buckling) |
| Rockwell scale | – | | Hugh M. (1890–1957) and Stanley P. (1886–1940) Rockwell | mechanical hardness (indentation hardness of a material) |
| Rolling resistance coefficient | C$_{rr}$ | $C_{rr}={\frac {F}{N_{f}}}$ | | vehicle dynamics (ratio of force needed for motion of a wheel over the normal force) |

### Optics

| Name | Standard symbol | Definition | Named after | Field of application |
| Abbe number | V | $V={\frac {n_{d}-1}{n_{F}-n_{C}}}$ | Ernst Abbe | optics (dispersion in optical materials) |
| f-number | N | $N={\frac {f}{D}}$ | | optics, photography (ratio of focal length to diameter of aperture) |
| Fresnel number | F | ${\mathit {F}}={\frac {a^{2}}{L\lambda }}$ | Augustin-Jean Fresnel | optics (slit diffraction) |
| Refractive index | n | $n={\frac {c}{v}}$ | | electromagnetism, optics (speed of light in vacuum over speed of light in a material) |
| Transmittance | T | $T={\frac {I}{I_{0}}}$ | | optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample) |

### Other

| Name | Standard symbol | Definition | Named after | Field of application |
| Fine-structure constant | $\alpha$ | $\alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}$ | | quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction) |
| Havnes parameter | $P_{H}$ | $P_{H}={\frac {Z_{d}n_{d}}{n_{i}}}$ | O. Havnes | In dusty plasma physics, ratio of the total charge $Z_{d}$ carried by the dust particles $d$ to the charge carried by the ions $i$, with $n$ the number density of particles |
| Helmholtz number | $He$ | $He={\frac {\omega a}{c_{0}}}=k_{0}a$ | Hermann von Helmholtz | The most important parameter in duct acoustics. If $\omega$ is the dimensional frequency, then $k_{0}$ is the corresponding free field wavenumber and $He$ is the corresponding dimensionless frequency |
| Lundquist number | S | $S={\frac {\mu _{0}LV_{A}}{\eta }}$ | Stig Lundqvist | plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma) |
| Perveance | K | ${K}={\frac {I}{I_{0}}}\,{\frac {2}{{\beta }^{3}{\gamma }^{3}}}(1-\gamma ^{2}f_{e})$ | | charged particle transport (measure of the strength of space charge in a charged particle beam) |
| Pierce parameter | $C$ | $C^{3}={\frac {Z_{c}I_{K}}{4V_{K}}}$ | | Traveling wave tube |
| Beta | $\beta$ | $\beta ={\frac {nk_{B}T}{B^{2}/2\mu _{0}}}$ | | Plasma and fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma. |
| Poisson's ratio | $\nu$ | $\nu =-{\frac {\mathrm {d} \varepsilon _{\mathrm {trans} }}{\mathrm {d} \varepsilon _{\mathrm {axial} }}}$ | | elasticity (strain in transverse and longitudinal direction) |
| Q factor | Q | $Q=2\pi f_{r}{\frac {\text{Energy Stored}}{\text{Power Loss}}}$ | | physics, engineering (Damping ratio of oscillator or resonator; energy stored versus energy lost) |
| Relative density | RD | $RD={\frac {\rho _{\mathrm {substance} }}{\rho _{\mathrm {reference} }}}$ | | hydrometers, material comparisons (ratio of density of a material to a reference material—usually water) |
| Relative permeability | $\mu _{r}$ | $\mu _{r}={\frac {\mu }{\mu _{0}}}$ | | magnetostatics (ratio of the permeability of a specific medium to free space) |
| Relative permittivity | $\varepsilon _{r}$ | $\varepsilon _{r}={\frac {C_{x}}{C_{0}}}$ | | electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum) |
| Specific gravity | SG | | | (same as Relative density) |
| Stefan number | Ste | $\mathrm {Ste} ={\frac {c_{p}\Delta T}{L}}$ | Josef Stefan | phase change, thermodynamics (ratio of sensible heat to latent heat) |
| Strain | $\epsilon$ | $\epsilon ={\cfrac {\partial {F}}{\partial {X}}}-1$ | | materials science, elasticity (displacement between particles in the body relative to a reference length) |

## Geography, geology and geophysics

| Name | Standard symbol | Definition | Named after | Field of application |
| Albedo | $\alpha$ | $\alpha =(1-D){\bar {\alpha }}(\theta _{i})+D{\bar {\bar {\alpha }}}$ | | climatology, astronomy (reflectivity of surfaces or bodies) |
| Dieterich–Ruina–Rice number | $\mathrm {R_{u}}$ | $\mathrm {R_{u}} ={\frac {W}{L}}{\frac {(b-a){\bar {\sigma }}}{G}}$ | James H. Dieterich, Andy Ruina, and James R. Rice | mechanics, friction, rheology, geophysics (stiffness ratio for frictional contacts) |
| Love numbers | h, k, l | | Augustus Edward Hough Love | geophysics (solidity of earth and other planets) |
| Porosity | $\phi$ | $\phi ={\frac {V_{\mathrm {V} }}{V_{\mathrm {T} }}}$ | | geology, porous media (void fraction of the medium) |
| Rossby number | Ro | $\mathrm {Ro} ={\frac {U}{Lf}}$ | Carl-Gustav Arvid Rossby | geophysics (ratio of inertial to Coriolis force) |

## Sport

| Name | Standard symbol | Definition | Field of application |
| Blondeau number | $B_{\kappa }$ | $\mathrm {B_{\kappa }} ={\frac {t_{g}v_{f}}{l_{mf}}}$ | sport science, team sports |
| Gain ratio | – | | bicycling (system of representing gearing; length traveled over length pedaled) |
| Goal difference | GD | ${\text{Goal difference}}={\text{goals scored}}-{\text{goals conceded}}$ | Association football |
| Runs Per Wicket Ratio | RpW ratio | ${\text{RpW ratio }}={\frac {\text{runs scored}}{\text{wickets lost}}}\div {\frac {\text{runs conceded}}{\text{wickets taken}}}$ | cricket |
| Winning percentage | – | Various, e.g. ${\frac {\text{Games won}}{\text{Games played}}}$ or ${\frac {\text{Points won}}{\text{Points contested}}}$ | Various sports |

## Other fields

| Name | Standard symbol | Definition | Field of application |
| Capacity factor | | ${\frac {\text{actual electrical energy output}}{\text{maximum possible electrical energy output}}}$ | energy |
| Cohesion number | Coh | $Coh={\frac {1}{\rho g}}\left({\frac {\Gamma ^{5}}{{E^{*}}^{2}{R^{*}}^{8}}}\right)^{\frac {1}{3}}$ | Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles) |
| Cost of transport | COT | $\mathrm {COT} ={\frac {E}{mgd}}$ | energy efficiency, economics (ratio of energy input to kinetic motion) |
| Damping ratio | $\zeta$ | $\zeta ={\frac {c}{2{\sqrt {km}}}}$ | mechanics, electrical engineering (the level of damping in a system) |
| Decibel | dB | | acoustics, electronics, control theory (ratio of two intensities or powers of a wave) |
| Elasticity (economics) | E | $E_{x,y}={\frac {\partial \ln(x)}{\partial \ln(y)}}={\frac {\partial x}{\partial y}}{\frac {y}{x}}$ | economics (response of demand or supply to price changes) |
| Gain | – | | electronics (signal output to signal input) |
| Load factor | | ${\frac {\text{average load}}{\text{peak load}}}$ | energy |
| Peel number | N$_P$ | $N_{\mathrm {P} }={\frac {\text{Restoring force}}{\text{Adhesive force}}}$ | coating (adhesion of microstructures with substrate) |
| Pixel | px | | digital imaging (smallest addressable unit) |
| Power factor | pf | $pf={\frac {P}{S}}$ | electrical (real power to apparent power) |
| Power number | N$_p$ | $N_{p}={P \over \rho n^{3}d^{5}}$ | fluid mechanics, power consumption by rotary agitators; resistance force versus inertia force) |
| Prater number | β | $\beta ={\frac {-\Delta H_{r}D_{TA}^{e}C_{AS}}{\lambda ^{e}T_{s}}}$ | reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet) |
| Relative density | RD | $RD={\frac {\rho _{\mathrm {substance} }}{\rho _{\mathrm {reference} }}}$ | hydrometers, material comparisons (ratio of density of a material to a reference material—usually water) |