The basic of mathematics is a number one - 1.

We can do a lot of operations with it. To plus, to multiply, to divide, etc. Of course we can replace the result with the symbol x or y or other. Or we can use some operators to define some mathematical process.

We have a number 1 or 2 or 3 until lots. What it means ?
  - number of entities, objects. We ask how many ?
  - number of character of entities: a weight, a force, an acceleration, etc. We ask how much.

We need some standards for the questions how many and how much. The standards exists and its name is Systeme Internationale (in French) - SI. There are basic units and other units which are derived from the first ones.



The SI base units and their physical quantities are:
the metre for measurement of length,
the kilogram for mass,
the second for time,
the ampere for electric current,
the kelvin for temperature,
the candela for luminous intensity,
the mole for amount of substance.

In this chapter we use only four basic units -  a time, a lenght, a mass, a number of amount of substancea.



unit of a the time is 1 second (s)

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

what is period I expect we know, f.e.: the period of the pendulum
and what is the hyperfine levels ? Try to find in WEB the following keywords - Zeeman efect, Stark effect and of course atomic clock.

         In the axis we have the time as T and the amplituda as A                              
 See the exact number of periods 9 192 631 770.  
 In the picture above you can see only the electric field without the magnetic one.    
 For more information you find electromagnetic field, Maxwell equations    


unit of a the lenght is 1 meter (m)

the meter is the path traveled by the light in vacuum during a time interval of 1/299, 792, 458 of a second.

vacuum is an empty space without some particles.

Notice: There is no ideal vacuum in a whole universe, f.e.: among stars the "vacuum" have probably three atoms to 1 cubiccentimeter. And today as we know, the vacuum contains quantum fluctuations. Theese fluctuations are unique at any kind of their changes.

 We can see in this picture how the lenght is equal to time.    

ΔT means a time, or a lenght
The time for some periods - the duration.

Some periods means the lenght


unit of a mass is 1 kilogram (kg)

the kilogram is equal to the mass of the international prototype of the kilogram, which is placed near Paris (France). This prototyp is made from alloy Platinum and Iridium. Its weight is roughly equal to 1 liter of water at a room conditions.
notice: this prototyp is "living", because its actualle mass is changing through the time. The reason of its changing is evaporate, rubbing, deposition, etc.

The scientist are thinking to replace the definition of 1 kg with a ball from Silicium with an accurate number of Si atoms. We use the mol.

Now we calculate how weighs 1 atom platinium

1 kg Pt contains approximately 3E24 atoms Pt.

1 atom Pt weighs 1/3E24 = 3.3E-25 kg. We know only a ratio ! The ratio how much is one atom heavier then other. One atom of Pt is equal to other atom Pt. But as we mentioned the important notice above, every atom of Pt is different. The reason is excitation with a quant of light or something else.

A one question - What is heavier ? A one kg of a iron or a one kg of feathers ? The answer is in the end of this chapter.


unit of a matter amount is 1 mol (mol)
the moll is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of Carbon 12.

When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. One mole contains exactly 6.02214076×1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol−1 and is called the Avogadro number.

See the number 6.022×1023 of atoms C12
in 12 gr.

The important notice : No two entities in the universe are alike. If one compares any two entities so closely, it will be found that they differ from each other.

I think, that the precission of the fact above is trivial in ordinary life. We can distinguish a apples from pears. And in details we can distinguish a kind apple from other kind one. But the importance of the notice above is grow up with the detail calculus. For example with the orbits of electrons.

Every atom, or ion or any other particle is alike. The same is valid for their velocity, mass, etc.
If we doubt about it, especially with the microparticles, then we have to realize, that every microparticle is surrounded by  quantum fluctuation (or vacuum state fluctuation or vacuum fluctuation).

We can say thees differences are extremely small. That's true. The infinitesimal is extremelly small either. And we use the infinitezimal more then plenty in calculus. And the results are different real numbers.

The end of the preview of fourth basis units.


What about frequency ?

The frequency is a time for which went some changes. Some regular changes - not occasional changes. The oscillations, vibrations etc. For example the pendulum, the oscillation of atoms or molecules in a heating state, or the mass with the spring. Where is the area of regular changes ?

How many changes went out for a real time, how much is the frequency.

Frequency 1 Hz is a 1 cycle (change) with a period 1 s. Thus the radiation of excited atom Cs 133 is equal to 1 because 9.2 bilions cycles per 9.2 bilions cycles is 1. The value is only the ratio , without Hz (Hertz)! If we take the 1 s and 1 m we obtain approximately 9 200 000 000 Hz for radiation of excited atom Cs133.

How many periods of caesium 133 have the meter ?
1 s is 9 192 631 770 period. And 1 m is the path of light for 1/299 792 458 s.

9 192 631 770 / 299 792 458 = 30.66331899

Then 1 m is equal to 30.66331899 periods. It is not a whole number.

How is long 1 period of the radiation of atomic clock, when we know the frequency 9.2 GHz ?
Repeat : 1 m has 30.66 period, then 1 period is equal = 0,03261225572 m.
The radiation of 9,2GHz atomic clock of caesium has the wavelenght cca 32,6 mm.
We are in the microwave radiation of electromagnetic spectrum.


What about the velocity?

It is the displacement of an object in a specific time. The displacement is the portion of the lenght through the object moved along for some portion of time.

Now we calculate the velocity of the light (c):

    c = 299 792 458 m/s  
    c 1  

The velocity of the light is exactly 1 (with the units SI). Which one ? Only 1 without unit !

The another velocities, including some objects will be less then 1. We will obtain only a ratio to c.

By the velocity of the light c = 1 don't flow the time, time is equal to the lenght. With velocities for objects which are less then c, the object went throuhg the path with x period Cs for y period Cs 133.  Then x << y.

Object moved along the path with 1 period of Cs133  for the time with 2 periods of Cs133 (see below).

S is the path


Then the velocity of object is equal to 0.5  or 1/2 ( a one period Cs133 for two periods Cs133) 

T is the time




The real "meter" is microwave periods of Cs133. We compare the time and the lenght.

What is the purpose of this chapter ? There is nothing new ! Of course, that is ! The purpose is only to remeber the source of basic units. The source is from our real world. From our real world with its so many changes.  Second, meter, etc. does not exist without our universe. And our measured values is only the ratio of measured values to basic units. The basic units there are not the same. The reason is  - a lot of changes of the real universe - in the end the vacuum oscillations.

The answer to the question: Of course a one kg of a iron is heavier then one kg of feathers. If you have doubts about it, let to fall the 1 kg iron to your left leg and to fall the 1 kg feathers to the other one. There are no doubts !

In next chapter we can use the relationship between the lenght and the time of moving objects when  c = 1 to each other.

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