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Form Champernowne constant by using concatenation to join the decimal representations of the positive integers $1,2,3,\dots$ after the decimal point. Champernowne proved in 1933 that this particular real number is a normal number in base ten. The construction is striking because its rule is completely explicit, while all finite words have the prescribed limiting digit block frequencies. ^[ref] Copeland and Erdős explicitly recall this result at the start of their own paper. ^[ref]

The first $100000$ digits of Champernowne constant have the following exact counts, in digit order $0,1,\dots ,9$:For the $99999$ adjacent pairs fully inside that prefix, the words $00,01,12,99$ occur $642,1753,2088,753$ times respectively. Thus, for example, $F_{100000}(12)=2088/99999$, which is about $0.02088$, rather than $0.01$. The prefix ends partway through the decimal representation of $22222$; the many leading ones in earlier positive integers help explain the visible imbalance.

These are finite counts, not a proof of the normal number theorem: a finite prefix can be followed by an all-zero tail. The theorem for Champernowne constant concerns the infinite limit, and establishes base-ten normality, rather than absolute normality. Nor do computations prove normality for familiar constants: the normality of Pi and Euler's number remains unproved, as discussed by Bailey and Borwein. ^[ref]

Use the curvature-minus-one Poincare disc model, whose Riemannian metric isIts geodesics are Euclidean diameters and arcs of circles meeting the boundary circle orthogonally. The hyperbolic area of a measurable region $D$ isAn isometry can move the center of a circle to the origin. Its hyperbolic radius $r$ and Euclidean radius $\rho$ then satisfy $r={\int }_0^{\rho }2du/(1-u^2)=2\mathrm{artanh}\rho$, so $\rho =\tanh (r/2)$. Integrating the area element of a surface in polar coordinates yieldsIntegrating the line element around the circle givesA geometric definition of Pi valid in this geometry isIndeed $\sinh r=r+O(r^3)$ and $\cosh r=1+r^2/2+O(r^4)$. The circumference-to-diameter ratio at a fixed positive radius is not constant, so the limiting qualification is necessary.