4.11.2013

Division by zero in physics

Switch. Dividing by a zero in physics. Example of application of mathematics. Mathematics for blondes. Mathforblondes.
All laws can be divided into two groups - invented by us and the laws of mathematics. Laws invented by us may not work in spite of the fact that we invented them. The laws of mathematics, which are displaying the laws of nature, always work, regardless of whether we know them or do not know. That is the case with the law of multiplication and division by zero.

There is an old student's joke that the device that performs mathematical operations of multiplication and division by zero is an ordinary switch. Personally, I more trust the not fresh views of students, than the "scientific" opus of different "scientists". Usually, the first impression is correct.

All mathematical letups, that were written here, I deleted at first, because I believed that the harm from them will be more than good. Comments I cleaned similarly. But then changed my mind. If I will not tell about mathematical principles of work of electric switch, then others will not soon do it. And so, to begin a comment (to the page in Russian language):

"Idiot, Ohm's law correctly write down. Why is current is not equal to zero at zero voltage? And why does a resistance of burned out bulb is zero? You're either blind or can not to read. All is excellent divided by zero when you know the function of dependence. In most cases, the result is infinity. And about the real uncertainties you, probably, did not hear in general. I'm not going to talk about all the options, where it will not work. Even if not to pay attention to the serious errors in the examples. So I conclude you don't have a brain."

This is a typical reaction of a person to coach at zero, as a trained dog on a command "Attack!". It is thus needed to remember that the math is considering abstract concepts that aren't to need understood. In the result, we all turn into trained animals who think exactly as they were taught. I know from own experience how hard it is to get rid from the generally accepted stereotypes. So I explain that throughout the following discussion I will talk not about numerical values of physical quantities, and about the units, which are usually not considered in general mathematics.

Each of us in our daily lives every day many times uses multiplication and division by zero. Engineers, as well as we, suspecting nothing, created a special device that allows to multiply and divide by zero. And all of this is so firmly established in our lives, that without these devices is impossible to imagine our life around. But let's begin one after another, from the math.

All of you know the mathematical law of multiplication:

a*b = c

On the pages of this web-site I told about the mathematical rules of multiplication and division by zero. Take from this page these formulas, which we will use:

ab*0 = 0 and a or b

a/0 = ab

We will rewrite these formulas in a that kind, in what we will use them in our concrete example:

a*b*0 = (a*0)*b = a*(b*0) = c*0

c*0 = {c=0; a=0; b≠0} = {c=0; a≠0; b=0}

a/0 = b/0 = a*b = c


Now we'll check how these algebraic letups correspond to reality. For this, we represent the most ordinary domestic situation: you're sitting in the evening in the livingroom and suddenly light shuts off. What is your first thought? Correctly, either electricity disappeared or a bulb burned out. Or if trying is possible to think of two others variants: either you suddenly became blind or you suddenly died. Since the latter two options are more to biology, we do not consider they. But as far as exactly the first two variants can be described by our algebraic formulas, let's look.

Luminescence of bulb in physics is described by the Ohm's law that looks so:

I*R = U

In this kind the Ohm's law fully coincides the law of increase presented by us in an algebraic kind:

a*b = c

According to this, in the further reasoning, we can replace the algebraic elements of formulas by physical quantities:

a = I - it is a current that flows on wires, it is measured in Amperes;

b = R - it is a resistance to the electric current is in the spiral of bulb, it is measured in Ohms;

c = U - it is a voltage in an electrical circuit, that compels a bulb to shine, it is measured in Volts.

The first variant of the apocalyptic gloom involves to shutdown of knife-switch by some evil man (for to save energy but without our consent), as a result an electric current stops to enter on wires. Or broken wire in an accident on electrical networks.

U*0 = {U=0; I=0; R≠0}

As we see, this mathematical result reports us, that a bulb really left off to burn, as an electric current disappeared in wires, but with our bulb all is normal and she is ready again to begin to shine, as soon as a current will appear.

Now we will look at the second variant of total eclipse, when a bulb simply burned out for us, and with a current in electric networks all is normal:

U*0 = {U=0; I≠0; R=0}

As see, unlike traditional "any number increased on a zero equals a zero", we got establishment of fact of extinct bulb not only U=0, but also two possible reasons of this annoying incident : {I=0; R≠0} and {I≠0; R=0}.

It is here needed to mark that in traditional mathematics multiplying by a zero what or element of equality taken to one of basic mathematical equalities:

0=0

Usually all mathematics closes thereon. In the variant of multiplying offered by me by a zero this situation means disappearance of primary equality and passing to two inequalities - tension is not equal to strength of current and tension is not equal to resistance of electric chain :

U ≠ I

U ≠ R

In a general view for algebraic expression a*b = c it looks so:

c ≠ a

c ≠ b

For renewal of primary equality it is necessary to execute the mathematical operation of dividing by a zero. In our example it is necessary either to recover an electric current in wires or replace a burneout bulb. Thus there is the following:

I/0 = [I*(R*0)]/0 = I*(R*0/0) = I*(R*1) = I*R = U

R/0 = [(I*0)*R]/0 = (I*0/0)*R = (I*1)*R = I*R = U

Equality is used in our case 0/0=1, where as unit units come forward electric that or electric resistance. Introduction to the formula of any other unit of measuring will not result in a primary result, as electric tension ensues exceptionally co-operation of strength of current and resistance. you can go about in circles, winding meters long and почесывая itself in the back of head. You can get a purse and throw about money. Burneout bulb from it will not begin to shine:

I/0 = [I*(L*0)]/0 = I*(L*0/0) = I*(L*1) = I*L ≠ U

I/0 = [I*($*0)]/0 = I*($*0/0) = I*($*1) = I*$ ≠ U

As see, application of dividing by a zero supposes the presence of reason, but not dull implementation of mathematical actions.

In conclusion I want to say that engineers did switches allowing to execute multiplying and dividing by a zero in electric chains already a long ago. This device is basic custom control by electric chains. Switches are equip practically all electric devices: bulbs, engines, televisions, mobile telephones and other.