7の平方根
64705...,\quad {\frac {82}{31}}=2.64516...,\quad {\frac {127}{48}}=2.645833...,\quad \ldots }
Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the Pell's equation
x 2 − 7 y 2 = 1. {\displaystyle x^{2}-7y^{2}=1.}
When 7 {\displaystyle {\sqrt {7}}} is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2th convergent of the continued fraction:
x 1 = 3 , x 2 = 8 3 = 2.66... , x 3 = 127 48 = 2.6458... , x 4 = 32257 12192 = 2.645751312... , x 5 = 2081028097 786554688 = 2.645751311064591... , ... {\displaystyle x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.
平面幾何学において、 7 {\displaystyle {\sqrt {7}}\,} は一連の動的な長方形により、すなわち上図の長方形の最大の対角線として表される。
辺の長さが2の正三角形に外接する最小の長方形は長さ 7 {\displaystyle {\sqrt {7}}\,} の対角線を持つ。