# There is a mathematically beautiful reason why A4 size paper is widely used

Many people associate notations such as 'A4', 'A3', 'B1', and 'B2' when they hear 'paper size'. This notation used for the size of copy paper and notebooks is defined by the international standard

**ISO 216**that defines the dimensions of paper, and is adopted in almost all countries and regions except for some countries such as the United States. I'm here.

**Ben Sparks**, a math writer and YouTuber, explains why the 'A4' size is widely used.

**Why A4? – The Mathematical Beauty of Paper Size - Heidelberg Laureate Forum - SciLogs - Wissenschaftsblogs**

**https://scilogs.spektrum.de/hlf/why-a4-the-mathematical-beauty-of-paper-size/**

Many people must have touched A4 size paper and notebooks in various situations such as schoolwork and work, but there may not be so many people who have actually measured the exact dimensions. A4 size paper has a long side of 297 mm and a short side of 210 mm. Regardless of 210 mm, the number of 297 mm is quite incomplete, and in fact Mr. Sparks seems to remember when his classmate said '297 mm? Why not 300 mm?'

However, this ISO 216 sized paper has one important feature. That is, ``If you fold or cut it in half, it will be one size smaller with the same aspect ratio.'' If you fold A4 size paper in half, it will be A5 size (148mm x 210mm), and if you connect two A4 size sheets together, it will be A3 size (297mm x 420mm). All papers such as A1, A2, A3, A4 have the same aspect ratio, but if the size of A4 is '300 mm x 200 mm', folding it in half will change the aspect ratio.

If you cut it in half, it will be one size smaller with the same aspect ratio, so there is no waste when making 2 sheets of A2 size paper from 1 sheet of A1 size paper or 8 sheets of A4 size paper. I will not go out. Also, when printing or copying something, it is possible to ``print two sides on A3 size paper side by side, and then cut in half to make two A4 size prints''. On the other hand, letter size (215.9 x 279.4 mm), which is common in North America, cannot do this.

It is no coincidence that A4 size paper has become such a size. The idea of ``paper with the same aspect ratio even when cut in half'' was discussed in a letter written by the German scholar

**Georg Christoph Lichtenberg**in at least 1786, and had existed as a mathematical problem even before that. The possibility has also been suggested. In the early 20th century, this idea was standardized worldwide, and at the time of writing the article, it is defined as ISO 216.

The aspect ratio of paper such as A4 and B3 is 'long side: short side = √2: 1', which is the only ratio that satisfies the condition that 'the aspect ratio is the same even if it is cut in half'. Since √2 is an

**irrational number**that cannot be expressed as an integer ratio, an approximation is used in practice.

In addition, since the size of A0 '1189mm x 841mm' is determined to be 1m

^{2}, the area of A4 size is 1/16 of 1m

^{2}(10000cm

^{2}), which is 625cm

^{2}. increase. As for the weight of paper, if you know the weight of one size, you can easily calculate the weight of other sizes.

“Sometimes we have useful things today because someone did mathematics in the past,” said Sparks. 'It's easy to forget that we're in control of that nature, but we owe a lot to people who understand mathematics well.'

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