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The Fibonacci Numbers: From Counting Rabbits to Counting Profits

"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pair of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?"

This is an English translation of the question that the great Italian mathematician Leonardo Pisano posed in his book Liber Abaci, a book on computations. The answer to this question is a number sequence that goes: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. Today, this set of numbers is more commonly known as the Fibonacci numbers, after Leonardo Pisano's nickname, Fibonacci.

It was not Leonardo Pisano, however, who first used the Fibonacci numbers in solving problems. In his "The Art of Programming", Donald Knuth documented that it was the Indian mathematicians Gopala and Hemachandra who first used the said number sequence way back in 1150. The number sequence resulted as their solution to the problem of exactly bin packing items of length 1 and 2. But Mediterranean and Asian mathematics was not known to the West until Leonardo Pisano introduced it in 1200 in his Liber Abaci. Through his book on computations, specifically on the problem of determining the idealized population growth of rabbits, he popularized the Fibonacci numbers.

But what's so important about the Fibonacci numbers? If you will notice, the Fibonacci numbers is not just a set of numbers but a sequence that follows a particular pattern. The next number in the series is determined by adding the previous two numbers (ex. 8 = 5 + 3). You will also notice that when you get the ratios of two succeeding numbers, you'll get nearer to the number 1.618, or its inverse which is 0.618. This ratio, which we now call the Golden Ratio or the Golden Mean, was often termed by the ancient Greeks as the divine proportion and was considered the perfect aesthetic proportion. Through the years, mathematicians have discovered a lot of applications for both the Fibonacci numbers and the Golden Mean. They have also discovered that many patterns of growth and nature is governed by the Fibonacci numbers or sequence.

Let's first take a look at how the Fibonacci numbers and the Golden Mean exist in nature. The nautilus shell, for example, exhibits a growth pattern that is somehow governed by the Fibonacci sequence, specifically by the shape of the Fibonacci rectangles or the Fibonacci spiral. The same is true with the opposing spiral seed pattern in sunflowers, with the number of spirals increasing like the Fibonacci numbers. Such pattern was also discovered in the branching patterns of leaves and grasses, as well as in most population growths and all natural growths.

Being a natural proportion, the Fibonacci numbers and the Golden Mean has found a lot of applications in various disciplines, including mathematics, science, music and the arts. In mathematics, the Fibonacci numbers was found useful in the run-time analysis of Euclid's algorithm in determining the greatest common divisor of two integers. The sequence also appears as the sum of the diagonals in the Pascal Triangle. The Fibonacci numbers are also used in tunings in the field of music, as well as in determining the length and size of content or formal elements in the field of arts.

Late in the 20th century, another important application of the Fibonacci numbers was discovered, and this time it was in the field of finance. Popularly known today as Fibonacci trading or Fibonacci analysis, this application uses the Fibonacci numbers and various ratios derived from it in projecting future market trends. The Fibonacci numbers and ratios are used as indicators in charting future market trends. The most common methods used in this type of technical analysis are the Fibonacci retracements, Fibonacci arcs, Fibonacci fans, and Fibonacci time projections. Many professional traders and businessmen have testified that these methods have given them advantages in trading, giving them more income and profit.

The numbers that Leonardo Fibonacci has introduced to the world has indeed found many applications through time. Who would have known that such simple and seemingly ordinary number sequence that was once used only for determining the idealized population of rabbits will find a lot of other applications. From counting rabbits, the Fibonacci numbers are now popularly used for counting profits.