Every irrational number 
can be expanded in a unique continued fraction expansion
![x=b_0+(e_1)/(b_1+(e_2)/(b_2+(e_3)/(b_3+...)))=[b_0;e_1b_1,e_2b_2,e_3b_3,...]](/images/equations/NearestIntegerContinuedFraction/NumberedEquation1.svg) |
such that 
, 
, 
, 
and 
for 
. This continued fraction expansion is known as the nearest
integer continued fraction expansion of 
.
For example, the nearest integer continued fraction expansion of e is given by
![[3;-4,-2,2k+5^_]_(k=0)^infty.](/images/equations/NearestIntegerContinuedFraction/NumberedEquation2.svg) |
