Entertaining mathematics: paper sizes

Even now, in the digital age, there is hardly anyone who did not see a piece of paper. All of us dealt with the most common format - A4. Those who drew wall newspapers in the school or studied drawing at the institute, got acquainted with A3, A2 and A1. And the owners of sets of recording paper can sometimes find on the packaging information that the size of one sheet is presented in A6 or A7 format. Still others have heard that paper sizes can be represented in B and C format.

Try using the ruler to determine the size of sheets of A4 paper. We find that it is 297 mm long and 210 mm wide. What are the interesting figures? If you cut any standard sheet of paper in half, the ratio of the length of the sides of the new sheets will remain the same as that of the old large sheet. That is, a smaller paper is obtained from a larger paper by simply dividing it. Sheet A4 is half the sheet A3, and sheet A5 is half A4.

Maintaining proportions when reducing the size is very convenient for scaling images and many other operations wherever paper is used. For example, it is easy to calculate the weight of the mail, knowing that the area of the largest sheet in the A series is 1 square meter. Paper for printers has a density of 80 g / m2. Hence, A4 sheet (as 1/16 part of sheet A0) will weigh 5 grams. Calculate the weight of envelopes can also be guided by the above reasoning, because the sizes of paper formats B and C (used for envelopes) obey the same relationships as for A.

Each of the formats is used in printing for a specific purpose. A8 - business cards, A7 - labels, A6 - international size for postcards, A5 - flyers and flyers, A4 - documentation. The dimensions of the B-size paper are such that one side of the sheet has an integer number of centimeters. This is convenient for creating products with images, since it is easier for a designer and layout designer to create a layout. In format B, passports and other official papers and letterheads, as well as tight envelopes, are produced. Paper format C, in addition to making soft envelopes, is used for printing photos.

There are other formats. In the English-speaking countries, where the inch metric system is still preserved, the format is "Letter", and also "Legal" and "Tabloid". The relationship between the sides of the paper is made according to the principle described above, but the length of the sides is a multiple of an integer number of inches. In the A + format (otherwise RA), the paper sizes are slightly larger than those of A. So much so that A4 paper, for example, could make A4 folders. There is the SRA format, which is slightly more than RA.

You can also notice that the standard paper sizes look quite aesthetically pleasing. The ratio of the short and long side of the sheet is convenient not only in technical terms. It obeys the proportion of the "silver section", not so well known as the "golden section", but also built on irrational numbers. At the heart of the golden section (which, by the way, is found in the sizes of books) is the square root of five. Silver is built on the basis of the root of the two. And continuing to divide the sheet of paper in half to infinity, we could create a fractal sequence.