Mathematical Coincidence

In Mathematics , a mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation.

INTRODUCTION

A mathematical coincidence often comprises an Integer , and the surprising (or "coincidental") feature is the fact that a Real Number is close to a small integer; or, more generally, to a Rational Number with a small Denominator .

Given the large number of ways of combining mathematical expressions, one might expect a large number of coincidences to occur; this is one aspect of the Law Of Small Numbers . Although mathematical coincidences may be useful, they are mainly notable for their curiosity value.

SOME EXAMPLES

Numeric expressions

Concerning the constant "e"

$e^\pi - \pi\approx 19.9990999$ is very close to 20. (Conway, Sloane, Plouffe, 1988).

$e^{\pi\sqrt{n}}$ is close to an integer for many values of $n$ , most notably $n=163$ $(e^{\pi\sqrt{163}} \approx 640320^3+744-7.5 imes10^{-13});$ this one is explained by Algebraic Number Theory ; see Heegner Number .

$\sum_{k=1}^8 rac{1}{k} = 1 + rac{1}{2} + rac{1}{3} + \cdots + rac{1}{7} + rac{1}{8} \approx e$ , within 0.016%.

For any integer n, the early decimal digits of $n-e$ are composed of 2,7,1, and 8, the first digits of ''e''

$\log_e (x) \approx \log_2 (x) - \log_{10} (x)$ to within 1%

$\exp(-\Psi(\sqrt{3}/4+1/2))) \approx 2$ to one part in ten million.

$e+\pi+\Phi \approx 7.5$ , within 0.29%.

Concerning pi

$\pi\approx 22/7,$ correct to about 0.04%; $\pi\approx 355/113,$ correct to six decimal places or about 85 parts in a billion. ( $\pi$ has unusually large terms in its Continued Fraction representation very early: $\pi$ = 7, 15, 1, 292, ... ).

$\pi^2\approx10;$ correct to about 1.3%. This coincidence was used in the design of Slide Rule s, where the "folded" scales are folded on $\pi$ rather than $\sqrt{10},$ because it is a more useful number and has the effect of folding the scales in about the same place;

$\pi^2\approx 227/23,$ correct to 0.0004% (note 2 , 227 , and 23 are Chen Prime s).

$\pi^3\approx 31$ (actually 31.0062...); $\pi^5\approx 306$ (actually 306.0196...).

$\pi\approx\left(9^2+rac{19^2}{22}
ight)^{1/4},$ or $\pi^4\approx 2143/22;$ accurate to 8 decimal places (due to Ramanujan : Quarterly Journal of Mathematics, XLV, 1914, pp350-372). Ramanujan states that this "curious approximation" to $\pi$ was "obtained empirically" and has no connection with the theory developed in the remainder of the paper. This can be told humorously as: ''Take the number "1234", transpose the first ''two'' digits and the last ''two'' digits, so the number becomes "2143". Divide that number by "''two-two''" (22, so 2143/22 = 97.40909...). Take the ''two''-''squared''th root (4th root) of this number. The final outcome is remarkably close to π'' (within about one part in a billion).

$\pi^4+\pi^5\approx e^6;$ correct to about 44 parts in a billion.

Concerning base 2

$2^{10}\approx 10^3;$ correct to 2.4%, see Binary Prefix

--- implies that $\log_{10}2 \approx 0.3;$ actual value about 0.30103; engineers make extensive use of the approximation that 3 DB corresponds to doubling of power level.

--- This is also very useful for estimating the size of a binary number, for example a 512 bit number is $2^{512}= 2^2 \cdot (2^{10})^{51}\approx 4 \cdot (10^3)^{51}=4 \cdot 10^{153}$

--- Using this approximate value of $\log_{10}2,$ one can derive the following approximations for logs of other numbers:

-- $3^4\approx 10\cdot 2^3,$ leading to $\log_{10}3=(1+3\log_{10}2)/4\approx 0.475;$ compare the true value of about 0.4771

-- $7^2\approx 10^2/2,$ leading to $\log_{10}7\approx 1-\log_{10}2/2,$ or about 0.85 (compare 0.8451)

-- $2^7\approx 5^3,$ leading to $5\approx 2^{7/3}=2^{28/12},$ i.e. $5/4\approx 2^{1/3}=2^{4/12}.$ The .

$2^{7/12}\approx 3/2;$ correct to about 0.1%. In music, this coincidence means that the Chromatic Scale of twelve pitches includes, for each note (in a system of Equal Temperament , which depends on this coincidence), a note related by the 3/2 ratio. This 3/2 ratio of frequencies is the Musical Interval of a fifth and lies at the basis of Pythagorean Tuning , Just Intonation , and most known systems of music. From the consequent approximation ${(3/2)}^{12}\approx 2^7,$ it follows that the Circle Of Fifths terminates seven Octave s higher than the origin.

Coincidences of units

$\pi$ Second s is a nanocentury (ie $10^{-7}$ Year s); correct to within about 0.5%

one Attoparsec per microfortnight approximately equals 1 Inch per Second (the actual figure is about 1.0043 inch per second).

one Furlong per fortnight is approximately equal to one centimetre per minute

a cubic Attoparsec (a cube where each edge is one attoparsec) is within 1% of a fluid ounce.

one mile is about $\phi$ kilometers (correct to about 0.5%), where $\phi={1+\sqrt 5\over 2}$ is the Golden Ratio . Since this is the limit of the ratio of successive terms of the Fibonacci Sequence , this gives a sequence of approximations $F_n$ mi = $F_{n+1}$ km, e.g. 5 mi = 8 km, 8 mi = 13 km. Another good approximation is 1 mile = ln(5) km, 1 mile = 1.609344 km and ln(5) = 1.6094379124341...

''N_A'' ≈ 2⁷⁹, where ''N_A'' is Avogadro's Number ; correct to about 0.4%. This means that a Yobibyte is slightly more than two Moles of bytes. It also means that 1 mole of any material (e.g. 12 g Carbon , or 25 l of gas at room temperatuer and normal pressure) cannot be halved more than 79 times.

The Speed Of Light in a vacuum is about one Foot per Nanosecond (accurate to 2%) or 3×10⁸ m/s (accurate to about 0.07%) or 1 billion km/h (accurate to 7.93%)

Other numeric curiosities

$10! = 6! \cdot 7!$ .

Golden ratio $\phi = -2 \cdot \sin(666^\circ)$ (an amusing equality with an angle expressed in degrees)

$\,5^2=25$ and $3^3=27\,$ are the only square and cube that differ by 2.

$\,3 + 2 = \log_2{32}$ .

$\,4^2 = 2^4$ is the only integer solution of $a^b = b^a, a$

Lambert's W Function

Not only $\,3^2 + 4^2 = 5^2$ , but also $\,3^3 + 4^3 + 5^3 = 6^3$ .

$\!\ \sin 9 - \cos 9$ is equal to the Plastic Constant to within 0.111%

1 · 9 + 2 = 11, 12 · 9 + 3 = 111, 123 · 9 + 4 = 1111, ... up to 12345678 · 9 + 9 = 111111111

31, 331, 3331 etc. up to 33333331 are all Prime Number s, but then 333333331 is not. See also Formula For Primes .

Decimal coincidences

$2^5 \cdot 9^2 = 2592$ .

$rac {16} {64} = rac {1\!\!\!$

ot6 4} = rac {1} {4},

rac {26} {65} = rac {2\!\!\!
ot6} {
ot6  5} = rac {2} {5}

rac {19} {95} = rac {1\!\!\!
ot9} {
ot9  5} = rac {1} {5}

$\,(4 + 9 + 1 + 3)^3 = 4,913$ and $\,(1 + 9 + 6 + 8 + 3)^3=19,683$ .

$\,1^3 + 5^3 + 3^3 = 153$ ; $\,3^3 + 7^3 + 0^3 = 370$ ; $\,3^3 + 7^3 +1^3 = 371$ ; $\,4^3 + 0^3 +7^3 = 407$

$3^2 + 7^2 - 3 \cdot 7 = (3^3 + 7^3)/(3 + 7) = 37$ .

$\,(3 + 4)^3 = 343$ (important in the numerological symbolism of St. Stephen's Cathedral, Vienna )

$\,35 - 3^2 - 5^2 = 75 - 7^2 - 5^2$ .

$\,588^2+2353^2 = 5882353$ and also $\, 1/17 = 0.0588235294117647...$ when rounded to 8 digits is 0.05882353

The number which equals the sum of its digits in consecutive powers: $\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7.$

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