He developed the Fibonacci summation series, which runs as
1 - 1 - 2 - 3 - 5 - 8 - 13 - 21 - 34 - 55 - 89 - 144 - ...
or, in mathematical terms,
an+1 = an-1 + an with a1 = a2 = 1
The mathematical series tends asymptotically (that is, approaching slower and slower) toward a constant ratio.
However, this ratio is irrational; it has a never-ending, unpredictable sequence ofdecimal values stringing after it. It can never be expressed exactly. Ifeach number, as part ofthe series, is divided by its preceding value (e.g., 13÷8 or 21÷13), the operation results in a ratio that oscillates around the irrational figure 1.61803398875 ..., being higher than the ratio one time, and lower the next. The precise ratio will never, into eternity (not even with the most powerful computers developed in our age), be known to the last digit. For the sake of brevity, we will refer to the Fibonacci ratio as 1.618 and ask readers to keep the margin oferror in mind.
This ratio had begun to gather special names even before another medieval mathematician, Luca Pacioli (1445–1514), named it “the divine proportion.” Among its contemporary names are “golden section” and “golden mean.” Johannes Kepler (1571–1630), a German astronomer, called the Fibonacci ratio one ofthe jewels in geometry. Algebraically, it is generally designated by the Greek letter PHI (ϕ), with
ϕ ≈ 1618
or, in a different mathematical form
ϕ = ½ (√5+1) ≈ 1.618
And it is not only PHI that is interesting to scientists (and traders, as we shall see). If we divide any number of the Fibonacci summation series by the number that follows it in the series (e.g., 8÷ 13 or 13÷21), we find that the series asymptotically gets closer to the ratio PHI′ with
ϕ′ ≈ 0618
being simply the reciprocal value to PHI with
ϕ′ = 1 ÷ ϕ = 1 ÷ 1.618 ≈ 0.618
or, in another form,
ϕ′ = ½ (√5-1) ≈ 0.618
This is a very unusual and remarkable phenomenon—and a useful one when it comes to designing trading tools, as we will learn in the course ofthe analysis. Because the original ratio PHI is irrational, the reciprocal value PHI′ to the ratio PHI necessarily turns out to be an irrational figure as well, which means that we again have to consider a slight margin oferror when calculating 0.618 in an approximated, shortened way.
