# 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:

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

The aspect ratio of the BN sheet is still the square root of 2, so we also get

so

Substituting this into the equation for the area we get:

so

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

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.

Back to Outer space