We all love natural numbers. Let’s take a sequence of real numbers 0 = a_{0 }< a_{1 }< a_{2 }< …and corresponding intervals A_{n} = [a_{n}, a_{n+1}]. We call an interval A_{n} *lucky* if it contains a natural number:Here the intervals A_{1}, A_{2}, A_{4}, A_{5} are lucky and marked in green. We don’t want intervals to be too lucky and therefore insist that intervals are shorter than unit length. We also want each natural number to know where it belongs, so we don’t want our sequences to contain any natural numbers.

To increase happiness, we consider *two* such sequences a_{n} and b_{n} shown on top of each other at height +1 and -1, respectively. Corresponding intervals A_{n} and B_{n} are upper and lower edges of quadrilaterals Q_{n}.

In this particular example the sequences are linear, namely a_{n} = n(𝛗-1) and b_{n}= n(2-𝜑), where 𝜑 is the Golden Ratio once more. We note that either A_{n} or B_{n} is lucky, but never both. So each quadrilateral has precisely one lucky edge.

In order to understand how this can possibly be true, we consider more generally two increasing sequences a_{n} and b_{n} such that a_{n} + b_{n} = n, and such that intervals are shorter than 1. Then the average sequence is c_{n} = (a_{n} + b_{n})/2 = n/2, which we include in the next figure at height 0 between the sequences a_{n} and b_{n}.

As already noted, no interval A_{n} or B_{n} can contain two natural numbers. If both contain a natural number, we can join them by a red segment within Q_{n }which intersects the height 0 line in a point whose x-coordinate is of the form n/2, where n is a natural number. But all these points with half-integral coordinates are already taken by the (blue) sequence c_{n}. So at most one of the intervals A_{n} or B_{n} can be lucky.

Now suppose that neither A_{n} or B_{n} are lucky. Then both intervals A_{n} and B_{n} are contained in integer intervals of length 1, marked as green dots to the left and right of the orange quadrilateral. The endpoints average to points with integral coordinates (green crosses) that are 1 unit apart and outside the orange quadrilateral. But both blue crosses on the orange quadrilateral have half integral coordinates, which can’t be. Thus either A_{n} or B_{n} is lucky.

As promised, this proves **Raleigh’s Theorem:** If irrational 𝛂, 𝛃 > 0 satisfy 1/𝛂 + 1/𝛃 = 1, then 𝛂_{n} = [n 𝛂] and 𝛃_{n} = [n 𝛃] are complementary, i.e. each natural number occurs in precisely one of the two sequences.

To see this, let a=1/𝛂 and b=1/𝛃, a_{n} = n a and b_{n}= n b. Now observe that A_{n} contains a natural number k if and only if

a_{n }< k < a_{n+1} if and only if n< k 𝛂 < n+1 if and only if n = [k 𝛂] (and likewise for B_{n}).

The argument in terms of intervals is a mere geometric formulation of the standard proof of Raleigh’s theorem but gives a more general result that is not so apparent in the purely algebraic version.