Leonardo of Pisa wrote 5 mathematical works, 4 as books and one preserved as a letter: Liber Abbaci, 1202 but revised in 1228.
meaning The Book ofthe Abacus (or The Book ofCalcuating). One of the problems in this book was the problem about the rabbits in a field which introduced the series 1, 2, 3, 5, 8 It was much later (around 1870) that Lucas named this series of numbers after Fibonacci. Practica geometriae, 1220.
A book on geometry. Flos, 1225
Liber quadratorum, 1225
The Book of Squares, his largest book.
It was translated into English by L E Sigler and published as The Book of Squares in 1987, Academic Press. Another article about this book:
^^ Leonardo of Pisa and his Liber Quadratorum by R B McClenon in American Mathematical Monthly vol 26, pages 1-8. A letter to Master Theodorus, around 1225.
Theodorus was a philosopher at the court of the Holy Roman Emporer Frederick II. There is a very readable outline of the problems in the letter to Master Theodorus in:
^^ Fibonacci's Mathematical Letter to Master Theodorus A F Horodam, Fibonacci Quarterly 1991, vol 29, pages 103-107.
The most comprehensive translation of the manuscripts of the 5 works above is:
^^ Scritti di Leonardo Pisano B Boncompagni, 2 volumes, published in Rome in 1857 (vol 1) and
References to Fibonacci's Life and Times
^^ Leonardo of Pisa and the New Mathematics of the Middle Ages J Gies, F Gies, Crowell press, 1969.
^^ The Autobiography of Leonardo Pisano R E Grimm, in Fibonacci Quarterly, vol 11, 1973, pages 99-104 with corrections on pages 162 and 168.
^^ 800 Years young A F Horodam in Australian Mathematics Teacher vol 31, 1975, pages 123-134.
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©Dr Ron Knott [email protected] Created: 11 March 1998, Latest revision: 14 February
Pi and the Fibonacci Numbers
Surprisingly, there are several formulae that use the Fibonacci numbers to compute Pi ( 71). |
Here's a brief introduction from scratch to all that you need to know to appreciate these formulae.
Contents of this page
The I Bline means there is a Things to doinvestigation at the end of the section. How Pi is calculated o Measuring the steepness of a hill o The tangent of an angle o The arctan function o Gregory's Formula for arctan(t) o Radian measure o Gregory's series and pi o Using Gregory's Series to calculate pi o Machin's Formula o Another two-angle arctan formula for pi Pi and the Fibonacci Numbers I ■
o The General Formulae Some more formulae for two angles o Some Experimental Maths for you to try I ■ More links and References
Until very recently there were just two methods used to compute pi, one invented by the Greek mathematician Archimedes, and the other by the Scottish mathematician James Gregory. We'll just look at Gregory's method here.
Measuring the steepness of a hill
The steepness of a hill can be measured in different ways.
It is shown on road signs which indicate a hill and the measure of the steepness is indicated in differing ways from country to country. Some countries measure the steepness by a ratio (eg 1 in 3) and others by a percentage.
The ratio is converted to a decimal to get its percentage, so a slope of "1 in 5" means 1/5 or 20%.
The picture on the road-sign tells us if we are going up a hill or down. We could say that a 20% rise is a steepness measured as +20% and a 20% fall as a steepness of -20% too.
But what does "a slope of 1 in 5" mean?
Some people take "1 in 5" to mean the drop (or rise) of 1 (metres, miles or kilometers) for every 5 (metres, miles, kilometers) travelled along the road. In the diagram, the distances are shown in orange.
Others measure it as the drop or rise per unit distance travelled horizontally. A "1 in 5"
slope means that I would rise 1 metre for every 5 metres travelled horizontally. The same numbers apply if I measure distance in miles or centimeters or any other unit.
In the second interpretation it is easier to calculate the steepness from a map. On the map, take two points where contour lines cross the road. The contour lines give the rise or fall in height vertically between the two points. Using a ruler and the scale of the map you can find the horizontal distance between the points but make sure it is in the same units as the horizontal distance! Dividing one by the other gives the ratio measuring the steepness of the road between the two points.
But they look the same slope?
Yes, they do when the slope is "1 in 5" because the difference is very small - about 0-23° in fact.
Here is a slope of 1-01. The green line is 1-01 times as long as the blue height and the red line is too. You can see that they "measure" very different slopes (the green line and the black line are clearly different slopes now).
What do you think a slope of "1 in 1" means in the two interpretations? Only one interpretation slope of 45° - which one?
So we had better be clear about what we mean by slope of a line in mathematics!!
• The first interpretation is called the sine of the angle of the slope where we divide the change in height by the distance along the road (hypotenuse).
• The second interpretation is called the tangent of the angle of the slope where we divide the change in height by the horizontal distance.
The slope of a line in mathematics is ALWAYS taken to mean the tangent of the angle of slope.
So in mathematics, as on road-signs, we measure the slope by a a ratio which is just a number. The higher the number, the steeper the slope. A perfectly "flat" road will have slope 0 in both interpretations. Uphill roads will have a positive steepness and downhill roads will be negative in both interpretations.
In mathematics, a small incline upwards will have slope 0-1 (i.e.10% or 1/10 or a rise of1 in 10)
a road going slightly downhill had slope -0-2 (i.e. 20% or 1/5 or a fall of1 in 5); a fairly steep road uphill will have slope 0-4 (ie 40% or 2/5) and the same road travelled in the other direction (downhill) has the same number, but negative: -0-4 In mathematics, a "1 in 1 " slope will means a metre rise for every metre travelled "along", so the slope is 1:1 = 1/1 = 1 or 45 degrees (upward).
Note that with the other interpretation (using the sine of the angle) of 1 in 1 is a rise of 1 metre for every metre along the road. This would mean a vertical road (a cliff-face) which is not at all the same thing as a tangent of 1!
Similarly, in mathematics, a slope of -1 would be a hill going downwards at 45 degrees.
In maths, lines can have slopes much steeper than roads designed for vehicles, so our slopes can be anything up to vertical http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html (2 of 14) [12/06/2001 17:22:03]
There are two interpretations.
Pi and the Fibonacci Numbers both upwards and downwards. Such a line would have a slope of "infinity"
The tangent of an angle
So we can relate the angle of the slope to the ratio of the two sides of the (right-angled) triangle. This ratio is called the tangent of the angle. a In the diagram here, the tangent of angle x is a/b, written:
A 45° right-angled triangle has the two sides by the right-angle of equal size, so their ratio is 1, which we write as tan(45°) = a/a = 1
If we split an equilateral triangle (ie all sides and all angles are the same) in half, we get a 60°-30°-90° triangle as shown:
We can use Pythagoras' Theorem to find the length of the vertical red line. Pythagoras' Theorem says that, in any right-angled triangle with sides a, b and h (h being the hypotenuse which is the longest side - see the first triangle here) then a2 + b2 = h2
So, in our split-equilateral triangle with sides of length 2, its height squared must be 22-12=3, ie its height is V3.
The arctan function
If we are given a slope (a tangent of an angle) we may want to find the angle of that slope. This would mean using the tangent function "backwards" which in mathematics is called the inverse of the tangent function.
It is called the atan or arctan function so that arctan(t) takes a slope t (a tangent number) and returns the angle of a straight line with that slope.
Gregory's Formula for arctan(t)
In 1672, James Gregory (1638-1675) wrote about a formula for calculating the angle given the tangent t for angles up to 45° (i.e for tangents or slopes t of size up to 1):
t3 t5 t7 t9
Actually, it is not so much a formula as a series, since it goes on for ever.
So we could ask if it will it ever compute an actual value (an angle) if there are always terms to come?
Provided that t is less than 1 in size then the terms will get smaller and smaller as the powers of t get higher and higher. So we can stop after some point confident that the terms missed out contribute an amount too small to alter the amount we have already computed to a certain degree of accuracy. [The question now becomes: "How many terms do I need for a given degree of accuracy?"]
Why must the value of t not exceed 1?
Look at what happens when t is 2, say. t3 is then 8, the fifth power is 32, the seventh power 128 and so on. Even when we divide by 3,5,7 etc, the values of each term get bigger and bigger (called divergence). The only way that powers can get smaller and smaller (and so the series settles down to a single sum or the series converges) is when t<1.
For this series, it also gives a sum if t=1, but as soon as t>1, the series diverges.
Of course t may be negative too. The same applies: the series converges if t is greater than -1 (its size is less than 1 if we ignore the sign) and diverges if t is less than -1 (its size is greater than 1 if we ignore the sign). The neatest way to sum this up is to say that
Gregory's series converges if t does not exceed 1 in size (ignoring any minus sign) i.e. -1 < t < 1.
The error between what we compute for an arctan and what we leave out will be small if we take lots of terms.
The limiting angle that Gregory's Series can be used on has a tangent that is just 1, ie 45 degrees.
First, we note that the angle in Gregory's series is not returned in degrees, but in radians which turns out to be the "natural" measure of angles since formulae are much simpler if we use this rather than degrees. If we draw the angle at the center of a circle of unit radius, then the radian is the length of the arc cut off by the angle (hence the "arc" in "arctan": "the arc of an angle whose tangent is..."). So 360 degrees is the whole circumference, that is
360° = 2 Pi radians = 2 Pir and halving this gives 180° = Pi radians = Pir and 90° = Pi/2 radians = (Pi/2)r. Since 60° is a sixth of a full turn (360°) then 60° = 2 Pi/ 6 = Pi/3 radians = (Pi/3)r and so 30° = Pi/6 radians = (Pi/6)r.
Note that, when it does not cause confusion with "raising to the power r" then ar means "a radians". A single degree is 1/360 of a full turn of 2 Pi radians so
Similarly, 1 radian is 1/(2 Pi) of a full turn of 360 degrees so
1 radian = 360 / (2 Pi) degrees = 180 / Pi degrees.
Using radian measure explains why the inverse-tangent function is also called the ARCtan function - it returns the arc angle when given a tangent.
Gregory's series and tt
We now have several angles whose tangents we know :-tan 45° (or tt/4 radians) =1, therefore i.
and if we plug this into Gregory's Series: arctan(t) = t - t3/3 + t5/5 - t7/7 + t% - ... we get the following surprisingly simple and beautiful formula for Pi:
, N 0 1111 arctan( 1 ) = — = 1 - - + - - - + - - ...
Actually, Gregory never explicitly wrote down this formula but another famous mathematician of the time, Gottfried Leibnitz (1646-1716), mentioned it in print first in 1682, and so this special case of Gregory's series is usually called Leibnitz Formula for tt.
We can use other angles whose tangent we know too to get some more formulae for Pi. For instance, earlier we saw that tan 60° (onr/3 radians) = V3 therefore arctan( V3 ) = 3
So what formula do we get when we use this in Gregory's Series? But wait!!! V3 is bigger than 1, so Gregory's series cannot be used!! The series we would get is not useful since wherever we stop it, the terms left out will always contribute a much larger amount and swamp what we already have. In mathematics we would say that the sum diverges. Instead let's still use the 30-60-90 triangle, but consider the other angle of 30°. Since tan 30° (or tt/6 radians) = 1/V3 which is less than 1:
The other angle whose tangent we mentioned above gives :
We can factor out the V3 and get
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