the hAnDbook oF teChnICAl AnAlysIsNow, in the limit, that is, as n approaches infinity, both:(^) F F/nn− 1 =Φ (2)
(^) F F/nn+1 =Φ (3)
Therefore, substituting (2) and (3) above into (1) we get:
ΦΦ=+1 1/( )
Rearranging to solve, we get:
ΦΦ^2 −−= 1 0
With the roots of the equation being:
Φ=±( )1 5 /2
Selecting the positive value, this will give us the value of Φ =1.618033 (to six
decimal places).
10.2.3 Other Φ‐related ratios based on the
Fibonacci iterative series
We already know that Φ = 1.618 (to three decimal places). Here are some other
important ratios related to Φ:
■ (^) ( ) .1/ 0 382Φ=
■ (^) ΦΦ×=2 618.
■ (^) ( )Φ−= .2/ 1 0 236
■ (^) ( ) .1/ =0 786Φ
■ (^) Φ=1 272.
The items in this list of Φ‐related ratios are regarded as significant ratios in
technical analysis and are used widely by technical traders and analysts. Please
note that higher‐order magnitudes of 1.618 are also employed in technical analy-
sis, especially when calculating Fibonacci projection levels, for example, 1.618,
2.618, 3.618, and 4.618. It is customary for charting packages and platforms to
also include the ratio 0.5, 1.0, 2.0, 3.0, and 4.0 within Fibonacci retracements,
extensions, expansions, and projections. Some analysts contend that these ratios
are not true Fibonacci ratios while others argue that 0.5 is a legitimate Fibonacci‐
related ratio since we can always derive 0.5 by dividing 1 by 2, which are both
numbers from the Fibonacci series. By extension, multiples of 0.5 give rise to the
ratios 1.0, 2.0, 3.0, and 4.0, respectively.
10.2.4 Converting Fibonacci ratios to percentages
These ratios may also be expressed as percentages by simply multiplying the ratio
by 100. For example, the Fibonacci ratio 0.618 may be expressed as (0.618 × 100)