In mathematics, a mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation. One of the expressions may be an integer and the surprising 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 so-called law of small numbers. Although mathematical coincidences may be useful, they are mainly notable for their curiosity value.
- ; correct to about 3%
- ; correct to about 3%. This coincidence was used in the design of slide rules, where the "folded" scales are folded on π rather than , because it is a more useful number and has the effect of folding the scales in about the same place.
- ; correct to about 0.03%; , correct to six places or 0.000008%. (The theory of continued fractions gives a systematic treatment of this type of coincidence; and also such coincidences as (ie ).
- ; leading to Donald Knuth's observation that, to within about 5%, log2(x) = log(x) + log10(x).
- ; correct to 2.4%; implies that log102 = 0.3; actual value about 0.30103. Engineers make extensive use of the approximation that 3 dB corresponds to doubling of power level
- ; correct to about 0.004%
- is close to an integer for many values of n, most notably n = 163; this one has roots in algebraic number theory.
- π seconds is a nanocentury (ie 10 - 7 years); 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).
- ; 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 indeed most known systems of music.
accurate to 9 decimal places (due to Ramanujan).