FAMILIARITY IN FUNDAMENTAL RELATIONS AND UNITS
When someone, who is not a physicist, attempts to understand the relation of matter to energy,
one of the difficulties which encounters is what units should be used so as the correct result to they give. Also, the concept of energy not immediately
reveals clearly in our minds, what phenomenon summarizes exactly. But the original ambiguity, indeed should not prevent the proper use of this term and use of
calculations. We have an ability to we use a concept properly, which is hard to define it in the start or because it is general and not clearly defined.
This ability, we have with many words and concepts and not only with the energy. A provocative example and with history is the concepts of "truth" and "existence".
An example of calculation of energy (E) and rest mass (M) of the electron, with the most famous
equation: E = Μ_{ο}c^{2} If at this simple formula we put mass in kilograms (kg) and the speed of light (c^{2}) in
meters per second at the square (m^{2}/s^{2}), then the result will be the energy in Joule units. If a quantity of mass multiplied with
speed of light at square thus result an equivalent quantity of energy E=mc^{2}. If the quantity of energy is divided by speed of light at
square thus result an equivalent mass m=E/c^{2}
Remind:
1 Joule is equal to 1kgr m^{2}/s^{2}. The
1eV =1,602176462 ×10^{19} Joule and 1Joule =
0,62415097445 × 10^{19} eV (or 1,6 ×10^{12} erg).
That is, according to the units of Physics, 1Joule is the force of
1Newton × displacement of 1m. The 1Newton is defined as the force
where is should to be apply on a body of 1kg mass so as is accelerated one meter per second at squared (1 m/s^{2}).
EQUIVALENT ENERGY OF THE ELECTRON
E_{ο}=Μ_{ο}
c^{2} → E_{e}=Μ_{e}
c^{2}
E_{e}= (9,109389 ×10^{31} kg) × (8,987551 ×10^{16}
m^{2}/s^{2}) = 8,187110 ×10^{14} J
In eV: 8,187110 ×10^{14} J × 6,2415097445 ×10^{18}
eV = 510,9993 ×10^{3} eV/s (= 0,5109993 MeV/s) 
1eV =1,602176462 ×10^{19}
= 1,6021764 ×10^{12} erg → 1Joule = 6,2415097445
×10^{18} eV
MASS OF AN ELECTRON M_{e}
BY ITS EQUIVALENT OF ENERGY
M_{ο }
=E_{ο}/c^{2} → M_{e}=
8,18711 × 10^{14} J / 8,987551 ×10^{16} m^{2}/s^{2} = 9,109389 × 10^{31} kg
M_{e}= (510,9993 ×10^{3} eV/s) × (1,6021764 × 10^{19}
J) / 8,987551 ×10^{16} m^{2}/s^{2} 
HOW MANY TIMES "FIT" THE CONSTANT h IN THE ENERGY of an electron.
THIS CALCULATION GIVES FREQUENCY (f) That is, how
many times the energy of the numerator is greater than the constant h in the denominator (in time 1sec).
f=
E/h
f_{e}= E_{e} / h = 8,18711 × 10^{14} J /
6,626069 ×10^{34 }J·s = 1,2355909 × 10^{20} Hz
In units eV: E_{e}
/ h = 510,9993 ×10^{3} eV/s / 4,1356676 × 10^{15 }eV sec = 1,2355908 × 10^{20}
Hz 
EQUIVALENT
MASS OF ELECTRON BY FREQUENCY
h=E/f
=Μc^{2}/f →Μ_{e}
= h f_{e}/c^{2 }= 6,626069
×10^{34 }J s × 1,2355908 ×10^{20} Hz / 8,987551 ×10^{16} m^{2}/s^{2} = 8,187110 × 10^{14} J / 8,987551
×10^{16} m^{2}/s^{2} = 9,109389 ×10^{31} kg
* In physics, the formula c=f·λ for calculation of the frequency
and wavelength applies only for electromagnetic waves and photons, which are considered massless. 
EQUIVALENT LENGTH "WAVE" of
electron λ_{e} which called Compton's length.
Since frequency units we can find a length (corresponding)
wavelength or a radius (r=c/ω), according to the known relation: λ=c / f → λ_{e} = c / f_{e}
λ_{e}=2,997924 5 ×10^{8} m/s / 1,2355908 ×10^{20}
Hz = 0,2426308 ×10^{11}m (=2,426308 ×10^{12 }m)
The same result is found with formula: λ=h/Μc
(Compton's length)
λ_{e} = (6,626069 ×10^{34 }J·s) / M_{e} c
→
λ_{e} = (6,626069 ×10^{34 }J·s) / (2,730926 ×10^{22})
= 0,24263085 ×10^{11 }m
1m =10^{9 }nm  1nm =10^{9}
m
0,24263085 ×10^{11} (×10^{9} nm) = 0,24263085 ×10^{3}
nm
The visible light wavelength: 400 nm to 700 nm (or 4000 Å  7000
Å ≈ 4 ×10^{7 } 7 ×10^{7} m). A comparison Compton electron length to the wavelength of visible light:10^{7}
m/10^{12 }m. For the electron is about 10^{5} times less.
The length measuring unit Angstrom:
Å = 0,1nm = 1 ×10^{10} m and micron µ = 1
×10^{−6} m 
MASS ELECTRON SINCE RESPECTIVE WAVELENGTH
λ=h/Μc
→Μ=h/cλ =
6,626069
×10^{34} /
(2,997924 ×10^{8}
) (0,2426308 ×10^{11}
)→
M_{e}=
6,626069 ×10^{34}
/ 0,7273886 ×10^{3}
= 9,1094 ×10^{31}
kg 
THE CONSTANT h IN eV UNITS
6,6260693 ×10^{34 }J sec
×
0,62415097445 × 10^{19} eV = 4,1356676 × 10^{15 }eV sec (or eV/Hz) 
THE CONSTANT h
EQUIVALENT TO FREQUENCY IN Hz
If 4,1356676 × 10^{15 }eV corresponds per Hz (or for sec) then for 1 eV :
1Hz ·1eV / 4,1356676 ×10^{15 }
eV = 2,41798929 ×10^{14} Hz (times the quantity h).
1 eV = 2,41798929 ×10^{14}
Hz (times the quantity h)
1eV correspond at frequency n=241,8 THz (where 1THz
=10^{12} Hz)
h =4,1356 ×10^{15} eV/Hz = 1eV / 241,8 THz
10^{12} Hz=1THz and 1Hz=10^{12} THz

Frequency fe
of electron in THz: 1,2355908 × 10^{20} Hz × 10^{12}
THz =
123559,08 ×10^{3} THz
Equivalent mass of energy 1eV
1Joule = 0,62415097445 × 10^{19} eV
→ 150,919037 × 10^{31} Hz
1eV = 1,602176462 ×10^{19} J
→ 241,798929 ×10^{12} Hz
(or 241,798929 THz)
m=E/c^{2} → 1ev × (1,602176 × 10^{19} J) /c^{2} =
0,178266131 ×10^{35}
kg
The minimal quantity h multiplied by a frequency f is equivalent with a quantity of
energy (energy that are bring the electromagnetic waves with the " null " mass of the photons). So as, mass on the square in speed of light
amounts also with energy, that is to say:
h f = Μ c^{2} = E → Μ = h f / c^{2}
= h /cλ = h / f λ^{2} or Μ = Mmin f 
With mass M we can find a frequency f and reversely. When we multiply the Planck's
constant h by a minimal frequency 1Hz then results a minimal quantity of energy E_{min} = h 1Hz. The minimal quantity h on
each other frequency f gives us multiples quantity of energy Ε=hf. When on the contrary, we divide the quantity of energy with a certain
frequency, then results the minimal quantity of energy h=E/f .
Because the reverse of the frequency is the (periodic) time
t = 1/f is also in effect t = h/E.
(!) We do not forget, that in the microscopic dimensions,
a mass M does not exist in (stable) situation of rest and we use the measurement in kg for reasons of comprehension and simplification. In
physics, it is known and we should take in our consideration, the formula c= f λ is valid for the calculation of frequency and length for
the electromagnetic waves only and for the photons, that are considered without mass. However, this aspect does not prevent to we speak with term of
frequency, when we observe phenomena of repetition and similar quantities that change multiple or submultiple.
We observe, the first useful mathematic relations for the description of structure of matter and shaping of mass.
A minimal quantity of energy E_{min} = h 1Hz is connected with a minimal frequency f_{min} and the largest length λ_{max}=c/f_{min}.
As long as the frequency f increases, also increases the quantity of energy h f and it minimizes the length λ
(since c/f and λ=h/M c). In increased frequency and quantity of energy amounts more mass (h f_{max} / c^{2}
= M_{max}). In more much quantity of mass corresponds more quantity of energy and mass is connected with higher frequency and smaller
length. Thus in the theoretical mass of unification (M_{planck} =√(h c/G) we find a maximum high frequency f_{max},
a most quantity of energy E_{max} and a minimal length λ_{min}.
Without well knowledge of physics we observe:
E_{max} / c^{2}
= h f_{max} / c^{2} = h / c λ_{min}
= M_{max}
The minimum quantity of energy that express by constant h
(×1Hz) can be considered as a product of a minimum mass by c^{2}^{ }(that is Mmin c^{2}). Therefore, if the
minimum quantity of energy of the constant h is divided by speed of light at square c^{2} then result a minimum quantity of mass h/c^{2} =M_{min}
sec. (If we want a mathematical result for clear mass in unit of kg then will multiply the constant h in numerator by the unit of
the minimum frequency f =1Hz).
That is, if energy E = h*1Hz then :
h
1Hz/c^{2} =Mmin→6,6260693 ×10^{34 }
J / 8,9875513 × 10^{16} m^{2}/sec^{2} =0,737249 ×10^{50}^{ }
kg
► As multiple quantities of energy (E=hf) result from the minimum quantity h,
equivalently multiple quantities of mass (M=Mmin f) result from the minimum quantity Mmin.
Therefore, if we divide a quantity of mass m by the minimum quantity Mmin results a frequency of mass, as correspondingly if we divide a
quantity of energy E by the minimum quantity h results a frequency of energy quantity.
f =Μ / M_{min}
or Μ / (h/c^{2} ) and f
=E/h = Μc^{2 }/ h
For example f =Μ_{e}/M_{min}→9,1093897
×10^{31}
kg / 0,73724967 ×10^{50}kg =12,355908 ×10^{19}
Hz We find the same result as from the formula f = E/h
The h in multiplication with a frequency f give us the same result as much as mass m
in multiplication with the c^{2} , that is the equal quantity of energy:
Μ
c^{2} = h f = E
→
Μ = h f / c^{2}
→
Μ = h /cλ
or Μ =
M_{min}
· f

* The mark "→" in formulas is used as "συνεπάγεται".
* Attention to the decimal point. This is a different mark for many countries
