# Mathematical coincidence

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.

## Some examples

- ; 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 rather , 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%, .
- ; correct to 2.4%; implies that ; actual value about 0.30103.
- ; correct to about 0.004%
- is close to an integer for many values of , most notably ; this one has roots in algebraic number theory.
- seconds is a nanocentury (ie years); correct to within about 0.5%

## See also