By definition, to measure is to compare with an etalon (AKA standard) unit.

Now, the tale, it all started with a stupid idea, how to get a 1V reference with a multiplier and a comparator:

- take an analog multiplier, and tie together both inputs, as a result, Vout = V

^{2}in

- note that for Vin = 1V, Vout would be 1V, too, because 1*1=1

- now, hook a voltage comparator between the input and output of the analog multiplier

- for any input less than 1V (we are talking only positive voltages for now) the output of the multiplier will be less than 1V

- for any input bigger than 1V the output of the multiplier will be bigger than 1V

- if we browse the Vin around the 1V point (e.g. swipe a voltage with a potentiometer), the comparator will switch when we pass over the 1V input point

- in conclusion, we can detect where the 1V is, and ideally, we can get exactly 1V without having a voltage reference.

Obviously, this doesn't work in real life, because there is no such thing like an ideal multiplicator.

A multiplier

*must to be calibrated*, so in a way, the multiplier itself contains a voltage reference inside. It's even more clear if we look at the measuring units, because Volt*Volt should give Volt

^{2}, which is not the same as Volt. The output of an analog multiplier produces Volts, not square Volts.

Going a little into the math, for why 1*1=1, this property is called the

*identity element*, let's call it e.

By definition, depending on the type of the operation (natural numbers addition, matrices multiplication, etc.) the identity element can be:

- valid only as a left side identity

(e some_operation a) = a

- valid only as a right side identity

(a some_operation e) = a

- valid on both sides, sometimes called neutral element

(e some_operation a) = (a some_operation e) = a

Let's take some examples

- Zero is the identity element for addition: 0 + a = a + 0 = a

- One is the identity element for multiplication: 1 * a = a * 1 = a

If we want to find out a zero, it's simple, and it does not need us to pick an arbitrary ethalon unit. We can simply use a comparator and symmetry, so by switching the comparator's inputs between them, we can tell if something is "0" without having a Zero Etalon unit.

However, we can not use the same techniques (with a multiplier) in order to identify a "1", a One Etalon Unit, because there is no such thing as a multiplier that does not require calibration.

Why a multiplier requires calibration, while a comparator doesn't?

Other said, why 1 unit (e.g. a Volt, an Ampere, or whatever other "1" unit) requires calibration (so an etalon/standard unit), while 0 doesn't require calibration, and doesn't requires a "0" etalon?

It's some bamboozling between math and physics here, which I can not pinpoint yet.