Mach number is a function of temperature and true airspeed.
Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature.
Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is found from Bernoulli's equation for  (above):
The formula to compute Mach number in a supersonic compressible flow can be found from the Rayleigh supersonic pitot equation (above) using parameters for air:
where:
:q is the dynamic pressure measured behind a normal shock.
As can be seen, M appears on both sides of the equation, and for practical purposes a root-finding algorithm must be used for a numerical solution (the equation is a septic equation in M and, though some of these may be solved explicitly, the Abel–Ruffini theorem guarantees that there exists no general form for the roots of these polynomials). It is first determined whether M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then the value of M from the subsonic equation is used as the initial condition for fixed point iteration of the supersonic equation, which usually converges very rapidly. Alternatively, Newton's method can also be used.