In mathematics, the silver ratio is a geometrical proportion with exact value 1 + √2, the positive solution of the equation x² = 2x + 1.

The name silver ratio is by analogy with the golden ratio, the positive solution of the equation x² = x + 1.

Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry.

If the ratio of two quantities a > b > 0 is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: $${\displaystyle {\frac {a}{b}}={\frac {2a+b}{a}}}$$ The ratio ${\displaystyle {\frac {a}{b}}}$ is here denoted ⁠${\displaystyle \sigma .}$⁠

Substituting ${\displaystyle a=\sigma b\,}$ in the second fraction, $${\displaystyle \sigma ={\frac {b(2\sigma +1)}{\sigma b}}.}$$ It follows that the silver ratio is the positive solution of quadratic equation ${\displaystyle \sigma ^{2}-2\sigma -1=0.}$ The quadratic formula gives the two solutions ${\displaystyle 1\pm {\sqrt {2}},}$ the decimal expansion of the positive root begins with 2.414213562373095... (sequence A014176 in the OEIS).

Using the tangent function  $${\displaystyle \sigma =\tan \left({\frac {3\pi }{8}}\right)=\cot \left({\frac {\pi }{8}}\right),}$$ or the hyperbolic sine $${\displaystyle \sigma =\exp(\operatorname {arsinh} (1)).}$$

⁠${\displaystyle \sigma }$⁠ and its algebraic conjugate can be written as sums of eighth roots of unity: $${\displaystyle {\begin{aligned}{\text{with }}\omega =&\ \exp(2\pi i/8)={\sqrt {i}},\\\sigma &=\omega -\omega ^{4}+\omega ^{-1}\\-\sigma ^{-1}&=\omega ^{3}-\omega ^{4}+\omega ^{-3},\end{aligned}}}$$ which is guaranteed by the Kronecker–Weber theorem.

⁠${\displaystyle \sigma }$⁠ is the superstable fixed point of the Newton iteration ${\displaystyle x\gets {\tfrac {1}{2}}(x^{2}+1)/(x-1),{\text{ with }}x_{0}\in [2,3]}$