A mathematical coincidence is said to occur when two expressions show a near-equality which has no theoretical explanation.
Contents
- Introduction
- Rational approximants
- Concerning
- Concerning e
- Concerning base 2
- Concerning musical intervals
- Concerning powers of
- Containing both and e
- Containing or e and 163
- Concerning logarithms
- Other numerical curiosities
- Decimal coincidences
- Length of six weeks
- Speed of light
- Earths diameter
- Gravitational acceleration
- Rydberg constant
- Cubic miles and kilometers
- Fine structure constant
- References
For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:
Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.
Introduction
A mathematical coincidence often involves an integer, and the surprising (or "coincidental") feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'.
Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance. Beyond this, some sense of mathematical aesthetics could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance (see Ramanujan's constant below, which made it into print some years ago as a scientific April Fools' joke). All in all, though, they are generally to be considered for their curiosity value or, perhaps, to encourage new mathematical learners at an elementary level.
Rational approximants
Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why such improbably large terms occur is often not available.
Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers.
Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
Concerning π
Concerning e
Concerning base 2
Concerning musical intervals
Concerning powers of π
Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is
The two sides of this expression only differ after the 42nd decimal place.
Containing both π and e
Containing π or e and 163
Concerning logarithms
Other numerical curiosities
Decimal coincidences
Length of six weeks
The number of seconds in six weeks, or 42 days, is exactly 10! (ten factorial) seconds (as
Speed of light
The speed of light is (by definition) exactly 299,792,458 m/s, very close to 300,000,000 m/s. This is a pure coincidence, as the meter was originally defined as 1/10,000,000 of the distance between the Earth's pole and equator along the surface at sea level, and the Earth's circumference just happens to be about 2/15 of a light-second. It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).
Earth's diameter
The polar diameter of the Earth is equal to half a billion inches, to within 0.1%.
Gravitational acceleration
While not constant but varying depending on latitude and altitude, the numerical value of the acceleration caused by Earth's gravity on the surface lies between 9.74 and 9.87, which is quite close to 10. This means that as a result of Newton's second law, the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 newtons of force exerted on an object.
This is actually related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the meter was the length of a pendulum whose half swing had a period equal to one second. Since the period of the full swing of a pendulum is approximated by the equation below, algebra shows that if this definition was maintained, gravitational acceleration measured in meters per second per second would be exactly equal to the square of pi.
When it was discovered that the circumference of the earth was very close to 40,000,000 times this value, the meter was redefined to reflect this, as it was a more objective standard (because the gravitational acceleration varies over the surface of the Earth). This had the effect of increasing the length of the meter by less than 1%, which was within the experimental error of the time.
Another coincidence related to the gravitational acceleration g is that its value of approximately 9.8 m/s2 is equal to 1.03 light-year/year2, which numerical value is close to 1. This is related to the fact that g is close to 10 in SI units (m/s2), as mentioned above, combined with the fact that the number of seconds per year happens to be close to the numerical value of c/10, with c the speed of light in m/s.
Rydberg constant
The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to
Cubic miles and kilometers
A cubic mile is close to
Fine-structure constant
The fine-structure constant
Although this coincidence is not as strong as some of the others in this section, it is notable that