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## Plus Magazine

## 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:

*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

Substituting this into the equation for the area we get:

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.

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