### Square root of two

Also in an infinitely small square, the ratio between the infinitely small side and the infinitely small diagonal is 1 : √2.

##### Explanation

If you want to go from A to B you must first walk 1 km to the east and then 1 km to the north. You than travel 2 km.

B 1 km A 1 km

If you first walk a part to the east, than a part to the north, and so on, the distance you travelled remains 2 km.

B A

Even if you make the parts ever shorter, the distance you travelled is always 2 km.

B A

Only when you take the diagonal, the distance gets smaller. You do not walk to the east and to the north, but directly north-east. In that way you only travel 1.414213562… km.

B A

You might expect that if the parts become infinitesimal, the distance should stay 2 km. But that is a fallacy. The Pythagorean theorem also applies in an infinitely small square, and

Infinitely small does not exist and infinity is not a number. You may perform calculations with it, but then, of course, you should do this correctly.

##### Note

In Barcelona there is a road called the "Diagonal" that helps to shorten distances in that city, where all streets form square blocks with buildings.