My favourite equations

Ludwig Boltzmann’s equation links entropy (S) and the number of ways (W) the particles of a system can be arranged for a given macrostate. S has units of J/K. k is Boltzmann’s constant, 1.381 x 10*-23 J/K. Log W (actually the natural log, ln) is a dimensionless number. The equation implies that increasing the amount of order in a system requires the input of energy (rather like tidying up your desk).


This is from Boltzmann’s gravestone. He committed suicide after becoming despondent at the attacks on his theory by opponents of the atomic theory of matter. He was right and they were wrong.

another one by Euler
another one by Euler


A Machin formula for calculating π, quoted by 18th C Welsh mathematician William Jones.

infinity in an hour

This looks quite similar to Euler’s formula for π but goes to infinity!

How did he think of that?
How did Euler think of that?

2 + 2 = 5 (for very large values of 2)

This is a sort of joke based on rounding errors: 2.3 rounds down to 2 but 2.3 + 2.3 = 4.6, which rounds up to 5! This type of error resulted in my being informed by Professor Sir Michael Marmot (or rather his computer) that my cholesterol level fell outside the desirable range. The letter said it was 5 mmol/l but that figure was arrived at by adding HDL 1.9 to LDL 2.7, making 4.6 (within desirable range) but then rounding it up to 5, making it a cause for concern!

Einstein's Equation of General Relativity, relating energy (RHS) to curvature of space-time (LHS)
Einstein’s Equation of General Relativity, relating energy (RHS) to curvature of space-time (LHS)

The Michaelis-Menten equation for enzyme-catalysed reactions:


The thermodynamics of any process (e.g. a chemical reaction) is described by this equation (at constant pressure):


where ΔG is the change in the Gibbs free energy, ΔH is the enthalpy change (here, the heat gained or lost by the system), T is the absolute temperature in Kelvin, K, and ΔS is the change in entropy. If ΔG is negative, the process will be spontaneous (but may be fast or slow, depending on the energy barrier between the starting and finishing states).

The Hawking-Bekenstein equation:

S is entropy; k Boltzmann’s constant; c speed of light; h Planck’s constant; G universal gravitational constant; A area of event horizon of a black hole.



The Hawking equation for the temperature of a black hole:

where TsubH is the temperature and kappa is surface gravity (equivalent of g on Earth)

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