Occurence of fundamental mathematical constants

In summary, the interval 0 to 5 contains many fundamental, dimensionless mathematical constants such as 0, 1, phi (the Golden ratio), e, pi, delta (Feigenbaum's constant), and others. This confluence may exist because small constants are easier to discover and work with, and the infinite sums that involve small numbers often yield interesting and general constants. Additionally, our concept of "small" may be tied to the size of these constants.
  • #1
Loren Booda
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0, 1, phi (the Golden ratio), e, pi, delta (Feigenbaum's constant) and comparatively many other fundamental, dimensionless mathematical constants occur on the interval 0 to 5. With a potential infinity of numbers to choose from, why does such a confluence exist?
 
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  • #2
For the physical constant.. perhaps they lie on [0,5] due to the strength of the interactions (weak, strong) in many cases using 'Natural units' you can set them equal to value '1'
 
  • #3
Sangoku said:
For the physical constant.. perhaps they lie on [0,5] due to the strength of the interactions (weak, strong) in many cases using 'Natural units' you can set them equal to value '1'

But physical constants are in no way limited to [0,5] like mathematical constants appear to be. Just consider Planck's constant and Avogadro's number in SI units.
 
  • #4
Loren Booda said:
0, 1, phi (the Golden ratio), e, pi, delta (Feigenbaum's constant) and comparatively many other fundamental, dimensionless mathematical constants occur on the interval 0 to 5. With a potential infinity of numbers to choose from, why does such a confluence exist?

Maybe small ones are easy to discover. Maybe our idea of 'small' is tied to the size of these constants (no, really -- 1 is the measure by which we count, so if it were 'larger' so would be our concept of 'small').

Maybe it's just easier to find and work with small constants -- perhaps the order of the Monster group is just as fundamental, but less has been done with it since it's so large.
 
  • #5
Maybe it's because the infinite sums that start off involving small numbers like 1, 2, 3, .. or 1!, 2!, 3!, etc. yield the most interesting and general constants:

e=1+1+1/2+1/6+1/24+1/120+... approx = 2 + 1/2 = 2.5
pi= 4 - 4/3 + 4/5 - 4/7 +... approx = 4 - 4/3 = 2.66666...
phi = (1 + 1/phi) = (1 + 1/(1+1/phi)) = ... approx = 1 + 1 / (1 + 1) = 1.5
euler mascheroni gamma = (1 - ln 2/1) + (1/2 - ln 3/2) + (1/3 - ln 4/3) + ...
 
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1. What are fundamental mathematical constants?

Fundamental mathematical constants are numbers that have a fixed value and are used in various mathematical equations and formulas. These constants are usually denoted by symbols and are considered fundamental because they play a crucial role in many mathematical concepts and theories.

2. How do we determine the value of fundamental mathematical constants?

The value of fundamental mathematical constants is determined through various methods, such as experimental measurements, mathematical calculations, and theoretical models. These values are constantly being refined and updated as new techniques and technologies are developed.

3. What are some examples of fundamental mathematical constants?

Some examples of fundamental mathematical constants include π (pi), e (Euler's number), Φ (golden ratio), γ (Euler-Mascheroni constant), and √2 (square root of 2). These constants have been studied and used for centuries and have important applications in fields such as geometry, calculus, and number theory.

4. Are fundamental mathematical constants universal?

Yes, fundamental mathematical constants are considered universal because they have the same value regardless of the units or systems of measurement used. This means that their value is consistent across different branches of mathematics and can be applied in various contexts.

5. Why are fundamental mathematical constants important?

Fundamental mathematical constants are important because they provide a foundation for many mathematical principles and equations. They also have practical applications in fields such as science, engineering, and technology. Additionally, the study of these constants helps us gain a deeper understanding of the fundamental laws and patterns of the universe.

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