Proofs

• From the physical viewpoint, real numbers x describe values of different quantities. • We get values of real numbers by measurements. • Measurements are never 100 % accurate, so after a mea-surement, we get an approximate value rk of x. • In principle, we can measure x with higher and higher accuracy. • So, from the computational viewpoint, a real number is a sequence of rational numbers rk for which, e.g., |x − rk | ≤ 2−k. • By an algorithm processing real numbers, we mean an algorithm using rk as an “oracle ” (subroutine). • This is how computations with real numbers are de-fined in computable analysis