Finding Rationals Between Reals

Real Analysis · Axiom Academy

EXAMPLE Finding Rationals Between Reals Construct a rational number that lies strictly between and 1.415 , using the Archimedean property. Find a rational number with . Since is irrational and 1.415 is just barely larger, the two numbers are extremely close — yet density of promises a rational hides between them. Let's build one. The rational lands just to the right of and safely left of 1.415 . You built a rational number wedged into a gap thinner than one thousandth. The same recipe works between any two distinct reals: Measure the gap: the distance b-a tells you how fine a denominator you need. Archimedean property supplies the denominator: for the positive gap there is an n with , so fits inside the interval. Scaling boxes in the numerator: since , the open interval has width greater than 1 , so it must contain an integer m — then . Many answers exist: any of over 10000 works here, which is why is dense in — there are infinitely many rationals in every interval. Density of is a cornerstone of real analysis: the rationals are full of holes (no ), yet they crowd up against every real number, leaving no gap unfilled.

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