When is 1 1 equal to 1

One plus one doesn't have to be two

, one says when a conclusion is clearly imposed: 1 + 1 = 2 is a popular example of what can be taken for granted, similar to the question. But 1 + 1 doesn't have to be 2 if you leave classical arithmetic.

Example: water in the bathtub

We want to know how much water there is in a previously empty bathtub with a volume of 1.4 hectoliters if we plug the drain, then pour in one hectolitre of water and finally add another hectolitre of water. The corresponding invoice looks like this:

1 + 1 = 1,4

The calculation falls into the area of ​​saturation arithmetic, here with an upper limit of 1.4 hectoliters. At 1.4 hectoliters of water, the bathtub is saturated and does not take up a large amount, even if more is added. So the result and the answer to our question is 1.4 hectoliters.

Example: Circulating cable car for the horticultural show

We want to know how many stations we have removed from our starting point when we get on the Koblenz BUGA cable car, ride one station, get off, enjoy the plants of spring, get back on, travel one station further and get off again. The statement:

1 + 1 = 0

After driving two stops you are at the starting point! There are several names for this number art: remainder class or clock arithmetic, arithmetic with congruences, circular or modular arithmetic ... In Koblenz we do arithmetic modulo two. That is, we add up in the classic way and then take the remainder when dividing by two as the result.

Modular and saturation arithmetic in digital images

Digital cameras typically work with a saturation limit. If more light particles hit a sensor pixel than the associated counter can record, the maximum recordable value is recorded. The structures are lost in the affected parts of the image - one finds large, uniformly white areas.

In a modulo camera, on the other hand, when the counter has reached its maximum value, it would start again at zero and continue counting. This creates a seemingly psychedelic picture, but in which less information is lost. In the example, ripples can still be seen in the overexposed flow.

With reasonable assumptions about nature - for example, where continuous transitions rather than abrupt jumps occur - a lifelike representation can be reconstructed almost completely from the overexposed modulo image, which cannot be achieved with the saturated image.