Some benefits of irrationality: solution

You were asked to investigate the dimensions of the B paper size series. The area of a sheet of paper of size BN is the geometric mean of the areas of a sheet of size AN and a sheet of size AN-1. The geometric mean of a set of k numbers is defined to be the kth root of their product. In this case there are only two numbers involved and we get: $$ Area(BN) = L(BN)W(BN) = \sqrt{Area(AN)Area(AN-1)},$$ where L(BN) and W(BN) are the length and width of the sheet. A sheet of size AN-1 only exists if N-1 is greater than or equal to zero, so the formula above works for N greater than or equal to 1.

As we worked out before, the area of an AN sheet is 2^-N square metres, so $$L(BN)W(BN) = (2^{-N} 2^{-(N-1)})^{1/2} = (2^{-2N+1})^{1/2} = 2^{-N+1/2}.$$

The aspect ratio of the BN sheet is still the square root of 2, so we also get $$L(BN)/W(BN) = 2^{1/2},$$ so $$L(BN) = 2^{1/2}W(BN).$$

Substituting this into the equation for the area we get: $$L(BN)W(BN) = 2^{1/2}(W(BN))^2 = 2^{-N+1/2},$$ so $$W(BN) = 2^{-N/2}\;\;\;\; and \;\;\;\; L(BN) = 2^{1/2}2^{-N/2} = 2^{-N/2+1/2}.$$

Although we developed this formula to only cover the cases where N is greater than or equal to 1, it turns out that a sheet of size B0 also follows its rule: it has area $$ L(B0) \times W(B0) = \sqrt{2} \times 1 = \sqrt{2}.$$

The C series of paper sizes is made from the geometric means of the same numbers in the A and B series, so C4 is the geometric mean between A4 and B4 etc.