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Plus Magazine

March 2007

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)=Area(AN)Area(AN1), 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)=(2N2(N1))1/2=(22N+1)1/2=2N+1/2.

The aspect ratio of the BN sheet is still the square root of 2, so we also get L(BN)/W(BN)=21/2, so L(BN)=21/2W(BN).

Substituting this into the equation for the area we get: L(BN)W(BN)=21/2(W(BN))2=2N+1/2, so W(BN)=2N/2andL(BN)=21/22N/2=2N/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)×W(B0)=2×1=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.

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