Silver ratioThis article is about a constant also termed the silver number. For a different constant sometimes called the silver number, see plastic number. For the gold-to-silver ratio, see Silver as an investment.

List of numbers Irrational and suspected irrational numbers

γ ζ(3) √2 √3 √5 φ ρ δS e π δBinary 10.0110101000001001111...Decimal 2.4142135623730950488...Hexadecimal 2.6A09E667F3BCC908B2F...

Continued fraction

Algebraic form

In mathematics, two quantities are in the silver ratio if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its covergents, square triangular numbers, Pell numbers, octagons and the like.The relation described above can be expressed algebraically:

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

The convergents of this continued fraction (2/1, 5/2, 12/5, 29/12, 70/29, ...) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

PropertiesNumber-theoretic propertiesThe silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 = −1δS ≈ −0.41 has absolute value less than 1. In fact it's the second smallest quadratic PV number after the golden ratio. This means the distance from δnS to the nearest integer is 1δnS ≈ 0.41n. Thus, the sequence of fractional parts of δnS, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

PowersThe lower powers of the silver ratio are

The powers continue in the pattern

where

For example, using this property:

Using K0 = 1 and K1 = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation

which becomes

Trigonometric propertiesThe silver ratio is intimately connected to trigonometric ratios for π8 = 22.5°.

So the area of a regular octagon with side length a is given by

Silver meansThe more general simple continued fraction expressions

are known as the silver means or metallic means[1] of the successive natural numbers. The golden ratio is the silver mean

between 1 and 2, while the silver ratio is the silver mean between 2 and 3. The term "bronze ratio", or terms using other names of metals, are occasionally used to name subsequent silver means. The values of the first ten silver means are shown at right. Notice that each silver mean is a root of the simple quadratic equation

where n is any positive natural number.

Silver means

0: (0 + √4)/2 1

1: (1 + √5)/2 1.618033989

2: (2 + √8)/2 2.414213562

3: (3 + √13)/2 3.302775638

4: (4 + √20)/2 4.236067978

5: (5 + √29)/2 5.192582404

6: (6 + √40)/2 6.162277660

7: (7 + √53)/2 7.140054945

8: (8 + √68)/2 8.123105626

9: (9 + √85)/2 9.109772229

...

n: (n+√(4+n^2))/2

PropertiesThese properties are valid only for integers m, for nonintegers the properties are similar but slightly different.The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means. For the silver mean S of m, the property can be generalized as

where

Using the initial conditions K0 = 1 and K1 = m, this recurrence relation becomes

The powers of silver means have other interesting properties:

If n is a positive even integer:

Additionally,

Also,

In general:

The silver mean S of m also has the property that

meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean.

where a is the integer part of S and b is the decimal part of S, then the following property is true:

Because (for all m greater than 0), the integer part of Sm = m, a = m. For m > 1, we then have

Therefore the silver mean of m is a solution of the equation

It may also be useful to note that the silver mean S of −m is the inverse of the silver mean S of m

Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number

then the following properties are true:

 if c is real,

 if c is a multiple of i.The silver mean of m is also given by the integral