Pi and the Fibonacci Numbers

Surprisingly, there are several formulae that use the Fibonacci numbers to compute pi (π).
Here's a brief introduction from scratch to all that you need to know to appreciate these formulae.
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How Pi is calculated

Until very recently there were just two methods used to compute pi (π), 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.
Hill (USA) Steep Hill 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?

There are two interpretations.
sine 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.
tan 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 will mean a slope of 45° - which one?

So we had better be clear about what we mean by slope of a line in mathematics!!
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 of 1 in 10)
a road going slightly downhill had slope -0·2 (i.e. 20% or 1/5 or a fall of 1 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 both upwards and downwards. Such a line would have a slope of "infinity".

The tangent of an angle

Triangle with sides a b and hyp is h 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.
In the diagram here, the tangent of angle x is a/b, written:
tan(x) = a/b

45-45-90 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
30-60-90 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 √3.
So we have
tan(60°) = √3 and
tan(30°) = 1/√3

The arctan function

If we are given a slope as the tangent of an angle, we may want to find the angle itself. This would mean using the tangent function "backwards" which in mathematics is called finding the inverse function of the tangent.
The inverse function is called the arctangent function, denoted atan or arctan. Another notation is tan-1(x) to show that it is the inverse function, so long as this is not confused with 1/tan(x) which must then be written as (tan(x))-1. Because of this confusion, some mathematical authors prefer the atan or arctan notation, which is what we shall use on this page.
So 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):
arctan( t )  =  t  –  t3  +  t5  –  t7  +  t9  –  ...




One method of deriving and proving this theorem is given by the Math Forum's Ask Dr Math but it involves integration.
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.

Radian measure

radian diagram 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 π radians = 2 πr and halving this gives
180° = π radians = πr and
90° = π/2 radians = (π/2)r.
Since 60° is a sixth of a full turn (360°) then
60° = 2 π/ 6 = π/3 radians = (π/3)r and so
30° = π/6 radians = (π/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 π radians so
1° = 2 π/360 radians = π/180 radians
Similarly, 1 radian is 1/(2 π) of a full turn of 360 degrees so
1 radian = 360 / (2 π) degrees = 180 / π 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 Pi

We now have several angles whose tangents we know :-
tan 45° (or π/4 radians) =1, therefore
arctan(1)  =  π

and if we plug this into Gregory's Series: arctan(t) = t − t3/3 + t5/5 − t7/7 + t9/9 − ... we get the following surprisingly simple and beautiful formula for π:
arctan( 1 )  =  π = 1  –  1  +  1  –  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 π.

We can use other angles whose tangent we know too to get some more formulae for π. For instance, earlier we saw that tan 60° (orπ/3 radians) = √3 therefore

arctan( √3 )  =  π

So what formula do we get when we use this in Gregory's Series? But wait!!! √3 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 π/6 radians) = 1/√3 which is less than 1:
arctan(1)  =  π


The other angle whose tangent we mentioned above gives :
arctan(1)  =  π  =  1  –
1  + 
1  –  
1  +  ...






We can factor out the √3 and get
π  =  1 x (1  –  1  +  1  –  1  +  1  –  ...)



3 times 3 

5 times 32

7 times 33

9 times 34
π  =  2 √3 (1  –  1  +  1  –  1  +  1  –  ...)

3 times 3 

5 times 32

7 times 33

9 times 34

Using Gregory's Series to calculate π/4

If we try to work out the value of π/4 from the formula marked as (*) above, we find that the formula, although very pretty (or elegant as mathematicians like to say), it is not very useful or practical for calculating π:

  1   =   1·000000000000000000 -
  1/3 =   0·333333333333333333 +
  1/5 =   0·200000000000000000 -
  1/7 =   0·142857142857142857 +
  1/9 =   0·111111111111111111 -
  1/11=   0·090909090909090909 +
  1/13=   0·076923076923076923 -
In fact, the first 5 terms have to be used before we get to 1/11 which is less than 1/10, that is, before we get a term with a 0 in the first decimal place.
It takes 50 terms before we get to 1/101 which has 0s in the first two decimal places and
500 terms before we get terms with 3 initial zeros.
We would need to compute five million terms just to get π/4 to 6 (or 7) decimal places!

This is called a slow rate of convergence. Can we improve Gregory's Formula by making it converge more quickly (i.e. use less terms to get a more accurate value of π)? Try the following questions to answer this (suggested by Robert H Douglass):

/ You Do The Maths... /

  1. If we combine the first two terms 1 − 1/3 we get 2/3.
    Combining the next two gives 1/5 − 1/7 = 2/35
    What single fraction results from the next two terms?
    1/9 − 1/11 = 2/99
  2. Can you find a formula for this new series of double-term fractions? Hint: try using denominators of (4n+1) and (4n+3)
    11 = 2
  3. Take two of your terms in this new double-term series and combine them into one fraction.
    Repeat with the next two terms.
    Can you find a formula for this new series which combines 4 of the Gergory's original terms?
  4. Do either of these series "converge faster" to π/4?
    No - with four terms combined into one we only need one quarter as many terms but it still takes one quarter of 5 million terms to get 6 or 7 dps of pi!

Using Gregory's Series to calculate π/6

The second formula above:
π  =  2 √3 (1  –  1  +  1  –  1  +  1  –  ...)

3 times 3 

5 times 32

7 times 33

9 times 34
was one that we derived from arctan(1/√3) and is a lot better:
  1        =  1·000000000000 
  1/9      = -0·111111111111 
  1/45     =  0·022222222222 
  1/189    = -0·005291005291 
  1/729    =  0·001371742112 
  1/2673   = -0·000374111485 
  1/9477   =  0·000105518624 
  1/32805  = -0·000030483158 
  1/111537 =  0·000008965634 
  1/373977 = -0·000002673961 
  1/1240029=  0·000000806432
After just 10 terms, we are getting zeros in the first 6 places and remember we would need at least half a million terms by the Leibnitz Formula!
Summing the above and multiplying by 2√3 gives
π = 3·14159 to 5 decimal places
The only problem with the faster formula above is that we need to use √3 and, before calculators were invented, this was tedious to compute.
Can we find some other formulae where there are some nice easy tangent values that we know but which don't involve computing square roots?
Yes! The next section shows one method.

Machin's Formula

In 1706, John Machin (1680-1752) found the following formula:
π  =  4 arctan(1) – arctan(1)



The 239 number is quite large, so we never need very many terms of arctan(1/239) before we've got lots of zeros in the initial decimal places. The other term, arctan(1/5) involves easy computations if you are computing terms by hand, since it involves finding reciprocals of powers of 5. In fact, that was just what Machin did, and computed 100 places by hand!

Here are the computations:

  All computations to 15 decimal places:
  arctan(1/5)                               arctan(1/239):
  1/5              = 0·200000000000000      1/239          = 0·004184100418410
  1/375            =-0·002666666666666      1/40955757     =-0·000000024416591
  1/15625          = 0·000064000000000      1/3899056325995= 0·000000000000256
  1/546875         =-0·000001828571428         
  1/17578125       = 0·000000056888889                          
  1/537109375      =-0·000000001861818                          
  1/15869140625    = 0·000000000063015
  1/457763671875   =-0·000000000002184
  1/12969970703125 = 0·000000000000077
  arctan(1/5)      = 0·1973955598498807  and arctan(1/239) = 0·004184076002074
  Putting these in the Machin's formula gives:
          π/4=  4 arctan(1/5 )        -   arctan( 1/239 )
   or     π  = 16 arctan(1/5 )        - 4 arctan( 1/239 )
              = 16×0·1973955598498807  - 4×0·004184076002074
              = 3·1415926535897922

Another two-angle arctan formula for Pi

Here's another beautifully simple formula which Euler (1707-1783) wrote about in 1738:
π = arctan(1) + arctan(1)



     Euler's Formula
It's even more elegant when we write π/4 as arctan(1):
arctan(1) = arctan(1) + arctan(1)



another proof Ko Hayashi writing in the Mathematics magazine in 2003 (Fibonacci Numbers and the Arctangent Function pages 214-5) gives the diagram shown here where, almost without words, we have a proof of the formula above since: He also finds similar relationships using the same kind of diagram where an angle whose tangent is 1/5 added to one with a tangent of 1/8 produces one with a tangent of 1/3, and various others. Why not experiment for yourself and see what you can find?
NOW we are ready for the formula using the Fibonacci Numbers to compute π!

Pi and the Fibonacci Numbers

Now we return to using the Fibonacci numbers to compute π. Euler's formula that we have just proved:
π = arctan(1) + arctan(1)



is good for computing π since 1/2 and 1/3 are smaller than 1. (The smaller the value of the tangent in Gregory's formula, the quicker the sum converges and the less work we have to do to find π!)

/ You Do The Maths... /

  1. Use this formula to compute π to a few decimal places by hand

Are there any more formulae like it, that is, using two angles whose tangents we know and which add up to 45 degrees (ie π/4 radians whose tangent is 1)?

Yes, here are some (not proved here). Can you spot the pattern?

     π/4 = arctan(1)  and ...
     arctan(1)    = arctan(1/2)  + arctan(1/3)
arctan(1/3) = arctan(1/5) + arctan(1/8)
arctan(1/8) = arctan(1/13) + arctan(1/21)
arctan(1/21) = arctan(1/34) + arctan(1/55)
We can combine them by putting the second equation for arctan (1/3) into the first to get:

     π/4 = arctan(1) 
          = arctan(1/2) + arctan(1/3)
          = arctan(1/2) + arctan(1/5) + arctan(1/8)
and then combine this with the third equation for arctan(1/8) to get:

     π/4 = arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/21)

You'll have already noticed the Fibonacci numbers here. However, not all the Fibonacci numbers appear on the left hand sides. For instance, we have no expansion for arctan(1/5) nor for arctan(1/13).
Only the even numbered Fibonacci terms seem to be expanded (F2=1, F4=3, F6=8, F8=21, ...):

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More.. Calculator

The General Formulae

We have just seen that there are infinitely many formulae for π using the Fibonacci numbers! They are:
π/4 = arctan(1)
= arctan(1/2) + arctan(1/3)
= arctan(1/2) + arctan(1/5) + arctan(1/8)
= arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/21)
= arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/34) + arctan(1/55)
or, putting these in terms of the Fibonacci numbers:
π/4 = arctan(1/F1 )
= arctan(1/F3) + arctan(1/(F4)
= arctan(1/F3) + arctan(1/F5) + arctan(1/F6)
= arctan(1/F3) + arctan(1/F5) + arctan(1/F7) + arctan(1/F8)
= arctan(1/F3) + arctan(1/F5) + arctan(1/F7) + arctan(1/F9) + arctan(1/F10)
= ...

What is the general formula?
It is

arctan (
) = arctan ( 1 ) + arctan ( 1 )      (A)




What happens if we keep on expanding the last term as we have done above?
We get the infinite sum


arctan (


which only uses odd-indexed Fibonacci numbers.
arctan(1) = arctan(1/F3) + arctan(1/F5) + arctan(1/F7) + ...
= arctan(1/2) + arctan(1/5) + arctan(1/13)+...
which is a special case of the following when k is 1:
arctan (1/F2 k) =
arctan (1 / F2n + 1)

Wetherfield's notation

You will have noticed that we have been using the arctan of reciprocals (fractions of the form 1/N) quite a lot above. So, as is common in mathematics, it is worth finding some notation to save writing (mathematicains are always looking for notations to save space!) and it will also make our equations more readable.

Michael Wetherfield introduced such a notation in his article on The Enhancement of Machin's Formula by Todd's Process in Mathematical Gazette, Volume 80, No 488 of July 1996, pages 333-344. It also has lots of interesting formula like those above.

If the tangent of an angle is a/b then the cotangent of that angle is b/a. Tangent is usually abbreviated to tan and cotangent to cot.
So the angle whose tangent is a/b i.e. arctan(a/b) is the same as the angle whose cotangent is b/a which we write as arccot(b/a). So we can replace arctan(1/x) by arccot(x).

Euler's formula arctan(1)=arctan(1/2)+arctan(1/3) then becomes

arccot(1) = arccot(2) + arccot(3)

He further abbreviates arccot(A) to just {A} - note the curly brackets:

arctan(1)     will be written as     { T }

In this abbreviated notation, Euler's formula now becomes :
{1} = {2} + {3}        Euler's Formula

More formulae for two angles

There are many more angles which have tangents of the form 1/X which are the sum of two other angles with tangents of the same kind. Above we looked at such formulae which only involved the Fibonacci numbers. Here are some more examples, including the Fibonacci numbers ones where {2}={3}+{7} means arccot(2) = arccot(3)+arccot(7) or, alternatively, arctan(1/2) = arctan(1/ 3) + arctan(1/ 7)
{1} = {2} + {3} {7} = {12} + {17} {13} = {15} + {98}
{2} = {3} + {7} {8} = {9} + {73} {13} = {18} + {47}
{3} = {4} + {13} {8} = {13} + {21} {14} = {15} + {211}
{3} = {5} + {8} {9} = {10} + {91} {15} = {16} + {241}
{4} = {5} + {21} {9} = {11} + {50} {15} = {17} + {128}
{5} = {6} + {31} {10} = {11} + {111} {16} = {17} + {273}
{5} = {7} + {18} {11} = {12} + {133} {17} = {18} + {307}
{6} = {7} + {43) {11} = {13} + {72} {17} = {19} + {162}
{7} = {8} + {57} {12} = {13} + {157} {18} = {19} + {343}
{7} = {9} + {32} {13} = {14} + {183} {19} = {20} + {381}

/ You Do The Maths...

  1. One pattern in the table above is {n} = {n+1} + {?}. What is the formula for the missing value?
  2. Another pattern is {n} = {n+2} + {?}.
    1. For which n does this pattern apply?
    2. What is the missing value?
  3. Find a proof for your results above.

Lucas Numbers and a formula for Pi

Shane Findley from Dover NH (USA) emailed me with a formula for π which uses the Lucas Numbers, a number series that has the same rule as the Fibonacci series (add the last two to get the next) but starts with 2 and 1 instead of 0 and 1:
n: 0 1 2 3 4 5 6 7 8 9 10 ...
Fn: 0 1 1 2 3 5 8 13 21 34 55 ...
Ln: 2 1 3 4 7 11 18 29 47 76 123 ...
Shane only gave a hint at a reason why this is true and I will expand on this to give a full proof and justification here.

Using the Collected Formulae page, (in particular, Vajda-5, Vajda 17a and Vajda-23) we can prove the following arctan relationship:

arctan(1) = arctan(1) + arctan(1)    (B)



If we put (B) in the abbreviated form together the with Euler's formula (A), we have

{ F 2n } = { F2n+1 } + { F2n+2 }       (A)
{ F2n+1 } = { L 2n } + { L2n+2 }       (B)

We can use equations (A) and (B) together now to expand Euler's formula in a different way, replacing all the Fibonacci numbers by Lucas numbers.
There are two steps:
  1. When we find an arctan of a reciprocal of an even-indexed Fibonacci number, we can use (A) to replace it by a sum of two terms, one an odd-indexed Fibonacci number and another even-indexed Fibonacci number.
  2. Now we can use (B) on the odd-indexed Fibonacci number to replace it with two Lucas numbers instead.
The other Fibonacci number was even-indexed so we can perform these two steps on it, introducing two more Lucas numbers and yet another even-indexed Fibonacci number. We can keep on repeating this to get an infinite series involving only Lucas numbers.

So, starting from

{ 1 } = { 2 } + { 3 } = { F3 } + { F4 }
we can replace the odd-indexed Fibonacci number F3 with a sum of two Lucas number reciprocals as follows:
{ 1 } = { 2 } + { 3 }
= { F3 } + { F4 }
First we use (A) on {F3}:-
= { L2 } + { L4 } + { F4 }
Now we keep using steps (a) and (b) to expand { F4 }:-
= { L2 } + { L4 } + { F5 } + { F6 } by step (a) on {F4}
= { L2 } + { L4 } + { L4 } + { L6 } + { F6 } by step (b) on {F5}
= { L2 } + { L4 } + { L4 } + { L6 } + { F7 } + { F8 } by step (a) on {F6}
= { L2 } + { L4 } + { L4 } + { L6 } + { L6 } + { L8 } + { F8 } by step (b) on {F7}
= { L2 } + 2{ L4 } + 2{ L6 } + 2{ L8 } + 2{ L10 } + ...
What we end up with, when we translate our Wetherfield abbreviated notation back into ordinary mathematical notation is:
π  = arctan ( 1 )  + 2



and we can also replace 3 by L2 and so only use even-indexed Lucas numbers.

More series?

A R Guillot published three formulae in the Fibonacci Quarterly in 1977 (vol 15, pages 232 and 257):
(2 F(2n+1))
( F(2n) F(2n+2))
( 2 F(2n+1))
Are there any more Fibonacci-type series that we can use apart from Fibonacci itself and Lucas Numbers?
The reason lies in the formula called Vajda-20a (for the Fibonacci numbers themselves) and its generalization to formula Vajda-18 on the Collected Formulae page at this site.

Some Experimental Maths for you to try

Here are some suggestions to see if we can find some reasons for the above results, and some order in the numbers.

You can use a computer to do the hard work, then you have the fun job of looking for patterns in its results! This is called Experimental Mathematics since we are using the computer as a microscope is used in biology or like a telescope for astronomy. We can find some results that we then have to find a theory or explanation for, except that what we look at is the World of Numbers, not plants or stars.

Finding more arctan relations for 1/N

Is there a formula of the kind
arctan(1/N) = arctan(1/Y) + arctan(1/Z)

for all positive integers N (Y and Z also positive integers)? So if I give you an N can you always find a Y and a Z?

How would you go about doing a computer search for numerical values that look as if they might be true (ie searching through some small values of N, Y and Z and seeing where the value of the left hand side is almost equal to the value of the right hand side? [ Remember, it could just be that the numbers are really almost equal but not exactly equal. However, you have to allow for small errors in your computer's tan and arctan functions, so you almost certainly will not get zero exactly even for results which we can prove are true mathematically. This is the central problem of Experimental Maths and show that it never avoids the need for proving your results.]

/ You Do The Maths...

  1. Can you spot any patterns in the numerical results of your computer search?
  2. Can you prove that your patterns are always true?
    Try a different approach to the proofs. Since we have a proof for the first result (we used the dark blue and light blue triangles in the diagram earlier in this page), can we extend or generalize the proof method?
  3. Once you have a list of pairs of angles which sum to another, you can use it to generate three angles that sum to another (as we did for 3 then 4 and an infinite number for the arctan(1) series for π above). Eg:
              arctan(1/4) = arctan(1/5) + arctan(1/21) 
         and                arctan(1/5) = arctan(1/6) + arctan(1/31) 
         and substituting gives
              arctan(1/4) = arctan(1/6) + arctan(1/21) + arctan(1/31)
    Perhaps there are sums of three angles that are NOT generated in this way (ie where any two of the angles do not sum to one with a tangent of the form 1/N)? It looks like:
    arctan(1/2) = arctan(1/4) + arctan(1/5) + arctan(1/47)

    might be one (if, indeed, it is exactly true). If so, how would you go about searching for them numerically?

Arctan relations for M/N

We've only looked at angles whose tangents are of the form 1/N. Perhaps there are some nice formula for expressing angles of the form arctan(M/N) as the sum of angles of the form arctan(1/X)? or even as a sum of other such "rational" tangents, not just reciprocals. What patterns are there here?

To start you off:
One such pattern looks like having Y=X+1, that is,

arctan(1/X) = arctan(1/(X+1)) + arctan(1/Z)
Often mathematicians and some programming languages use atan instead of arctan.
Here are some results from a computer search :

atan(1/2) = atan(1/3) + atan( 1/7)
atan(1/3) = atan(1/4) + atan(1/13)
atan(1/4) = atan(1/5) + atan(1/21)
atan(1/5) = atan(1/6) + atan(1/31)
atan(1/6) = atan(1/7) + atan(1/43)
atan(1/7) = atan(1/8) + atan(1/57)
atan(1/8) = atan(1/9) + atan(1/73)
What is the pattern here?

Reducing two arctans to one

Tadaaki Ohno, a mathematics student at the University of Tokyo , Japan, (July 1999) mentioned a nice method of looking for arctangent relations which depends on factoring numbers (although Hwang Chien-lih says that Lewis Carroll (C L Dodgson) knew this method also). Using the following formula for the tangent of the sum of two angles, a and b:
tan(a + b) = tan(a) + tan(b)
1 − tan(a) tan(b)

He transforms it into the problem of finding integers x, y and z which satisfy:
(x – z)(y – z)  =  z2 + 1

(You can derive this expression from the tan(a+b) formula as follows:
Let tan a = 1/x i.e arctan(1/x) is angle a and let tan b = 1/y so arctan(1/y) is angle b.
Then a+b = arctan(1/x) + arctan(1/y) = arctan(1/z) so that tan(a+b) = 1/z
Put these values in the tan(a+b) formula above and then simplify the right hand side by multiplying top and bottom by xy. After rearranging you will then need to add z2 to both sides and then Tadaaki Ohno's formula appears.)
So, for instance, if arctan(1/z)= π/4 and therefore z is 1 then we can find values x and y by solving
(x – 1)(y – 1)  =  12 + 1 = 2

The important thing is that x and y are integers so we only need to look for integer factors of 2 and there are only two factors of 2, namely 1 and 2:
x – 1 = 1 and y – 1 = 2 which gives x = 2 and y = 3
This is the first two-angle formula that we mentioned earlier that Euler found in 1738:
π/4 = arctan( 1/2 ) + arctan( 1/3 )
The important other part of Tad's proof is that
all two-angle values satisfy this formula.

So we now know that there is only one way to write arctan(1) as the sum of two angles of the form arctan(1/x) + arctan(1/y).

/ You Do The Maths... /

  1. Prove that
    tan(a + b) = tan(a) + tan(b)
    1 − tan(a) tan(b)
    sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and
    cos(a+b) = cos(a)cos(b) − sin(a)sin(b) in
    tan(a+b) = sin(a+b)

    then divide the denominator and the numerator by cos(a)cos(b) and simplify each
  2. Find all the two-angle sums (x and y) for z from 1 to 12.
  3. Research Problem Can you find a similar formula for x, y and z when
    arctan(1/z) = 2 arctan(1/x) + arctan(1/y)

    What about
    arctan(1/z) = 3 arctan(1/x) + arctan(1/y)

    arctan(1/z) = 4 arctan(1/x) + arctan(1/y)

    and, in general,
    arctan(1/z) = k arctan(1/x) + arctan(1/y)

    Tad says he has proved that Machin's formula (which has z=1, x=239 and y=5) is the only solution for k=4.

Research Problems

Two-term arctans

Hwang Chien-lih of Taiwan told me that Størmer proved that there are only four 2-term formulae for arctan(1), that is Euler's and Machin's and only two more:
arctan(1) = 4 arctan(1/5) – arctan(1/239) discovered by Machin in 1706.
arctan(1) =   arctan(1/2) + arctan(1/3) discovered by Euler in 1738
arctan(1) = 2 arctan(1/2) – arctan(1/7) discovered by Hermann in 1706?
arctan(1) = 2 arctan(1/3) + arctan(1/7) discovered by Hutton in 1776?
Leroy Quet of Denver, Colorado, has found a simple proof (here it is) of the real pattern.

Other two-term arctans

Pridon Davlianidze of the Republic of Georgia told me of these interesting arctans (July 2020):
arctan(28/29) + arctan(1/57) = arctan(1)
arctan(58/59) + arctan(1/117) = arctan(1)
Andrew Scheller (July 2020) provides the general equation here, found using the formula for the sum of two angles whose tangents we know (proved above) ...
arctan(x) + arctan(y) = arctan x + y
1 − x y
... then we let x be n/(n+1) and y be 1/(2n+1) which lets the arctan on the right reduce to 1.
So the general formula is
arctan n + arctan 1 = arctan 1


Hwang Chien-lih also says the same Størmer found 103 three-term formulae, J W Wrench had found 2 more and Hwang Chien-lih has found another. One example was given earlier on this page:
arctan(1) = arctan(1/2) + arctan(1/5) + arctan(1/8) (Daze)
and some others are
arctan(1) = 4 arctan(1/5) – arctan(1/70) + arctan(1/99) (Euler, Rutherford)
arctan(1) = 8 arctan(1/10) – arctan(1/239) – 4 arctan(1/515) (Klingenstierna)
arctan(1) = 12 arctan(1/18) + 8 arctan(1/57) – 5 arctan(1/239) (Gauss)
all of which are given in Lehmer's 1938 article referenced at the foor of this page.
How many are there in total?

You might find this formula useful:

arctan x + arctan y + arctan z = arctan x + y + z − x y z
1 − x y − y z − z x

/ You Do The Maths... /

  1. Prove the formula for the sum of THREE angles whose tangents we know:
    arctan x + arctan y + arctan z = arctan x + y + z − x y z
    1 − x y − y z − z x
  2. Use the formula above to find A and B where
    arctan 1 + arctan 1 + arctan 1 = arctan A
    arctan 1 + arctan 1 + arctan 1 = arctan 1 − X Y − Y Z − Z X
    XYZ X + Y + Z − X Y Z

    Can you explain why the fraction on the right is just the reciprocal of that in the previous question?

If you get some results from these problems, please send them to me - I'd be interested to see what you come up with so I can put your name and your results on this page too. Perhaps you can find some results in the Journals in your University library (not so easy!)? Even if the results you discover for yourself are already published in books and papers you will have done some real maths in the meantime. And your results my really be new or your proofs much simpler than those known and we need to let the world know so have a go!

Pi, Fibonacci and Phi

Here is a formula connecting the Fibonacci numbers and the golden ratio (Phi = Φ = 1.618...) in a formula for π:
 = √5
(–1)n F(2n+1)

More links and References



Robert Erra of E.S.I-E-A (Ecole Supérieure d'Informatique- Electronique- Automatique), Paris, has contributed the following references:

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