---
title: "Mathematical coincidence"
chunk: 5/5
source: "https://en.wikipedia.org/wiki/Mathematical_coincidence"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T07:23:33.671306+00:00"
instance: "kb-cron"
---

The ratio of a mile to a kilometre is also approximately the golden ratio 
  
    
      
        φ
        =
        
          
            
              1
              +
              
                
                  5
                
              
            
            2
          
        
        ≈
        1.618
        
        .
      
    
    {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618\,.}
  
  As a consequence, a Fibonacci number of miles is approximately the next Fibonacci number of kilometres.
The ratio of a mile to a kilometre is also very close to 
  
    
      
        ln
        ⁡
        (
        5
        )
        ≈
        1.6094379
        ,
      
    
    {\displaystyle \ln(5)\approx 1.6094379,}
  
 within 0.006%. This means that 
  
    
      
        
          5
          
            m
          
        
        ≈
        
          e
          
            k
          
        
        ,
      
    
    {\displaystyle 5^{m}\approx e^{k},}
  
 where 
  
    
      
        m
      
    
    {\displaystyle m}
  
 is the number of miles, 
  
    
      
        k
      
    
    {\displaystyle k}
  
 is the number of kilometres and 
  
    
      
        e
      
    
    {\displaystyle e}
  
 is Euler's number.
A density of one ounce per cubic foot is very close to one kilogram per cubic metre: 1 oz / ft3 = 0.028349523125 kg / (0.3048 m)3 ≈ 1.0012 kg/m3.
The conversion factor between troy ounces and grams (1 troy ounce = 31.1034768 g) is approximately equal to 
  
    
      
        10
        π
        −
        
          
            π
            10
          
        
        =
        
          
            99
            10
          
        
        π
        ≈
        31.1018
        
        .
      
    
    {\displaystyle 10\pi -{\frac {\pi }{10}}={\frac {99}{10}}\pi \approx 31.1018\,.}
  

==== Fine-structure constant ====
The fine-structure constant 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is close to, and was once conjectured to be precisely equal to ⁠1/137⁠.  Its CODATA recommended value is

  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 = ⁠1/137.035999177(21)⁠

  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is a dimensionless physical constant, so this coincidence is not an artifact of the system of units being used.

==== Earth's solar orbit ====
The number of seconds in one year, based on the Gregorian calendar, can be calculated by: 
  
    
      
        365.2425
        
          (
          
            
              days
              year
            
          
          )
        
        ×
        24
        
          (
          
            
              hours
              day
            
          
          )
        
        ×
        60
        
          (
          
            
              minutes
              hour
            
          
          )
        
        ×
        60
        
          (
          
            
              seconds
              minute
            
          
          )
        
        =
        31
        ,
        556
        ,
        952
        
          (
          
            
              seconds
              year
            
          
          )
        
      
    
    {\displaystyle 365.2425\left({\frac {\text{days}}{\text{year}}}\right)\times 24\left({\frac {\text{hours}}{\text{day}}}\right)\times 60\left({\frac {\text{minutes}}{\text{hour}}}\right)\times 60\left({\frac {\text{seconds}}{\text{minute}}}\right)=31,556,952\left({\frac {\text{seconds}}{\text{year}}}\right)}
  

This value can be approximated by 
  
    
      
        π
        ×
        
          10
          
            7
          
        
      
    
    {\displaystyle \pi \times 10^{7}}
  
 or 31,415,926.54 with less than one percent of an error: 
  
    
      
        
          [
          
            1
            −
            
              (
              
                
                  
                    31
                    ,
                    415
                    ,
                    926.54
                  
                  
                    31
                    ,
                    556
                    ,
                    952
                  
                
              
              )
            
          
          ]
        
        ×
        100
        =
        0.4489
        %
      
    
    {\displaystyle \left[1-\left({\frac {31,415,926.54}{31,556,952}}\right)\right]\times 100=0.4489\%}
  

==== Proton-to-electron mass ratio ====

  
    
      
        6
        
          π
          
            5
          
        
        ≈
        1836.12
      
    
    {\displaystyle 6\pi ^{5}\approx 1836.12}
  
 is very close to the proton-to-electron mass ratio 
  
    
      
        μ
        =
        
          m
          
            p
          
        
        
          /
        
        
          m
          
            e
          
        
        ≈
        1836.153
      
    
    {\displaystyle \mu =m_{p}/m_{e}\approx 1836.153}
  
 (a dimensionless constant), within 0.002%. When this was first pointed out in 1951, the most exact known value for 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 was 1836.12, which differs from 
  
    
      
        6
        
          π
          
            5
          
        
      
    
    {\displaystyle 6\pi ^{5}}
  
 by just 0.0001%.

== See also ==
Almost integer
Anthropic principle
Birthday problem
Exceptional isomorphism
Experimental mathematics
Koide formula
Narcissistic number
Sophomore's dream
Strong law of small numbers

== References ==

== External links ==
(in Russian) В. Левшин. – Магистр рассеянных наук. – Москва, Детская Литература 1970, 256 с.
Davis, Philip J. - Are There Coincidences in Mathematics - American Mathematical Monthly, vol. 84 no. 5, 1981.
Hardy, G. H. – A Mathematician's Apology. – New York: Cambridge University Press, 1993,  (ISBN 0-521-42706-1)
Weisstein, Eric W. "Almost Integer". MathWorld.
Various mathematical coincidences in the "Science & Math" section of futilitycloset.com
Press, W. H., "Seemingly Remarkable Mathematical Coincidences Are Easy to Generate"