# Fibonacci

Fibonacci, or more correctly his number sequence, figures throughout our lives, even though we seldom know and recognize it. Fibonacci was an Italian mathematician who lived centuries ago, and the number sequence has his name because he brought it to prominence, even though it was known to earlier civilizations such as the Greeks.

The sequence simply answers the question, “If you put two rabbits together, and rabbits breed every month, how many total rabbits do you have each month?” – at least, that’s one way of expressing the problem. The sequence of numbers is made by adding the previous two numbers together each time to make the next, like so –

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc. for as long as you want.

Mathematicians have written lots of things about this sequence, and you’ll find you can get lost in the stories if you do an Internet search. It is closely related to the number is called the “golden ratio”, or sometimes golden section or golden mean. Two numbers are in the golden ratio to each other if the ratio of the sum of the numbers to the larger of the numbers is equal to the ratio between the large number and the smaller number. The golden ratio can’t be definitively stated (in mathematical terms it is called irrational), but is approximately equal to 1.6180340:1 or 1:0.6180340.

So the definition is fulfilled by noticing that the sum of the numbers 1 and 0.618 (1.618) in ratio to 1 is the same as the ratio of 1 to 0.618.

What has this to do with Fibonacci? Well, as the Fibonacci sequence progresses, the ratio between each two adjacent numbers becomes closer to the golden ratio – 144 divided by 89 equals 1.61798, 233 divided by 144 becomes 1.61801, 377 divided by 233 equals 1.61803, etc.

As with Elliott Waves, the Fibonacci numbers have their fans in trading circles, and there are many books and articles about their use. They are more immediately understandable and usable (in my opinion) however. There are many ways that they can be applied, and we’ll just look at the main ones here.

Firstly, in their involvement with Elliott Waves, there are a number of other rules or guidelines that you can be looking for –

1. The length of wave 3 is often 1.618 times the length of wave 1.
2. The target for the top of wave 5 (in an up-trend) is the length of wave 1 times 1.618, doubled. This price length can be added to the top and bottom of wave 1 to give the range of minimum and maximum target.
3. For the correction ABC, you will recall that A and C are often equal. The bottom of C is often 0.618 below the bottom of A.

These are not coincidental observations – these were made by Elliott, who claimed in his book Nature’s Law that his wave theories were based on the Fibonacci sequence, which recurs as you drill down into smaller and smaller subdivisions of the waves. But apart from the merits or otherwise of the Elliott Wave Theory, the Fibonacci numbers figure in charting, to the extent that most charting software will assist you in plotting lines or angles divided in the Fibonacci ratios.

In an earlier section we started looking at ways to figure how far retracements can go – remember even in a strong trend you’re likely to see retracements along the way – and I mentioned the figures of one third (33%), one half (50%), and two thirds (67%), and also 38.2% and 61.8%. Perhaps you can see now that the 61.8% comes from the Fibonacci ratio, with the 38.2% being the remainder. So one of the most basic uses of Fibonacci numbers is to set targets for retracements. The 50% level is not usually considered a Fibonacci number, although it is if you count all the ratios of adjacent numbers in the series (1:2), and you will find it is used in most Fibonacci calculations as it seems to work.

But Fibonacci numbers can be used for more than just price targets . Elliott also considered time in his work, and one way he did this was by counting forward from a significant peak or trough in Fibonacci numbers, looking for example at the eighth, 13th, and 21st days in expectation that something like a reversal may happen there. This idea is not seen much nowadays, but can be applied to all periods of charts.

The time relationship is difficult to use, so many current day analysts ignore it. It is supposed to hold true on all time scales, so you can apply it to weekly and monthly charts as well as the daily.

There are some other ways of looking at Fibonacci numbers, and if your software includes them you might want to try them out. Some traders draw fan lines, where the lines are at Fibonacci angles to each other – 38 degrees and 62 degrees for example – and the lines are supposed to represent support or resistance levels, or some similar thing. As an ex-engineer, I can’t get excited about stretching the use of these ratios to such applications. The angle that any line makes on the chart is dependent on the relative scales of the axes, so although the lines may seem to be significant on the commonly drawn chart, I cannot see that as a “magical” fact.