The square root of two is an irrational number because it cannot be represented by the ratio of any two numbers. It is the number which when multiplied by itself is two. The square root of two is approximated by 1.41421 with a continuing series of non-repeating digits after that.
The square root of two makes photography rational.
If you look at one of the older 35 mm cameras, you will see a scale called the F/stop. This was usually located around the outside of the lens and would have numbers like 5.6, 8, 11, and so on. Those numbers represent the aperture of the lens. The F/stop was reciprocal of the diameter of the lens to the focal length, and also an indication of how much light the lens let in. A circle that has its diameter reduced by the square root of two will have half as much area as the original. Start with one, and multiply each number by 1.41421, then round off the result, and you will get the progression of F/stops.
Turns out you can use the square root of two to determine flash exposures as well. The light from a flash drops off at the rate of the square of the distance. So something that is 1.41421 times the distance from the flash to the subject will receive exactly half as much light as the subject.
How does this relate to macro flash photography? Let’s show some examples. We are going to assume that digital photography can capture about eight zones of exposure, and that the subject is exposed to a neutral gray. The point at which the light drops off to black is at about four zones. That would be four times the square root of two times the distance from the flash to the subject, or twice the flash to subject distance.
So I took my camera with a macro lens fully extended. At this length, images on the sensor are about life size. The image is in focus about 5 1/2 inches in front of the lens, represented by the pin closest to the camera. Since the flash is about 4 1/2 inches from the subject, the background is black about 9 inches behind the subject.
Using the on-camera flash the distance is about 11 inches, making the zone where the background fades to black about 22 inches behind the subject.
Adding extension tubes makes the subject closer and the difference even more dramatic.
So in this photo, taken in an apple tree, the leaves in the background still show up as dark green, rather than totally black.
With the ring flash, the background would have been totally black.