Where do you reside?”

“In front of Post office.”

“Where is then post office?”

“It is in front of my house.” The answer has baffled all at one time or the other as the reference is related and is not fixed. In fact, both locations are the reference for one another. That makes the location not only unlocatable but puts one into never ending loop of hopping from one location to the other. Such loops are called closed and never ending.

But where is mathematics involved in it? Per Se there appears no mathematics in it but in mathematics, there are abundant such situations where there are such unending loops. I recollect vividly as a teacher of computer programming way back in eighties, I used to make programme involving loops based on mathematics questions, language at that time was simple ‘basic’ and commands were go to that line and from that back to original to puzzle the calculating machine. But machines are beyond nervousness and confusion. To my amusement, Computer when asked to display the status used to indicate, “Still calculating.” Hardly the machine knew, it had been put in never ending task.

Coming to my initial question of location of John’s house, if John says, his house is situated in front of Post Office in Peter Lane, then reference as Post Office is fixed but one when goes to Post Office, there one finds, Post Office is comprised of a huge building and there are a number of houses in front of that building and John’s house can not be found. Again John will be contacted and he informs, it is a grey coloured house in front of telegraph section of Post Office. But there are a number of grey houses in front of that section. On again inquiring, John replies, it is in front of cashier’s window. In situation highlighted here, there is a loop but it is localising on each cycle thus reaching nearby its destination. Such loops in mathematics are called convergent loops. Relationship of Post Office and house is called recursive relation.

Hope, I could clarify the concept of recursive relation. Such recursive relations have charismatic effects and can land mathematicians in new unexplored land of mathematics. Continued fractions, continued Ramanujan nested radicals, continued products are gifts given by such relations.

I take up some examples to bring home further the concept of recursive relation. Let us consider a right angled triangle ABC with base BC, perpendicular AC and right angle at C, ratio of perpendicular AC to hypotenuse BC is called sin B where B is angle at point B. This angle B can be halved by formula,

sin B = ½.sin B/2.cos B/2. ………………………………………………………….(1)

Examination of this formula reveals that sin of an angle appears in both left hand side LHS and right hand side RHS. This sine on right hand side RHS will reflect it back to the formula of sine written in left hand side LHS. This reflection of sine formula from LHS to RHS and then from RHS to LHS continues indefinitely. But in the process angle B goes on halving from B to B/2, from B/2 to B/4 and so on infinitely. This is analogous to reflection from Post Office to opposite building and opposite building to post office and continuous halving of angle is analogous to localising the house to smaller area.

Equation (1) is recursive relation and applying this formula successively, the angle B will be reduced to infinitesimal value B/2^n. How this angle will be reduced by application of recursive relation is explained in steps below.

sin B = ½.sin B/2.cos B/2.

Therefore, sin B/2 on RHS can be written as 2.sin B/4.cos B/4. That makes

sin B = 2.cos B/2. 2.sin B/4.cos B/4 = 2^2. cos B/2.cos B/4.sin B/4.

Again, sin B/4 on RHS can be written as 2.sin B/8.cos B/8. That makes

sin B = 2^3. cos B/2.cos B/4.2.sin B/8.cos B/8 = 2^3 cos B/2.cos B/4.cos B/8.sin B/8.

In this way, it can written,

sin B = 2^n cos B/2.cos B/4.cos B/8…………………………….sin B/2^n. …………………………….(2)

Angle B will go on reducing but for derivation of formula, it has been aborted at n terms.

Let us see what happens to this equation if a little mathematics is applied to it. Sin A = A is true when angle A is very very small tending to zero. In equation (2), B/2^n also tends to zero on successive halving particularly when n tends to infinitely large, therefore, sin B/2^n equals B/2^n. Applying it to equation (2), it transforms to

sin B = 2^n cos B/2.cos B/4.cos B/8…………………………….B/2^n.

On transposing B/2^n to RHS,

(sin B)/(B/2^n = 2^n cos B/2.cos B/4.cos B/8……………………………

Or (sin B)/(B/2^n = 2^n cos B/2.cos B/4.cos B/8……………………………

Or 2^n.(sin B)/(B) = 2^n cos B/2.cos B/4.cos B/8…………………………

On cancelling 2^n on LHS and RHS,

sin B/B = cos B/2.cos B/4.cos B/8…………………………………. (3)

This is a beautiful formula, first derived by great mathematician Euler and also indirectly used by another great mathematician François Viète. The formula derived here is the outcome of application of recursive relation and was widely used particularly for determining the value of π. Euler used it to factorise sin B in terms of angle whereas François Viète used it as it is.

Another example of recursive relation is angle halving formula for Cosine of an angle. For that I refer to right angled triangle as mentioned before wherein cos B is the ratio of base BC to hypotenuse AC and it can be written as

cos 2.B = 2.cos^2 B – 1 and it can be rearranged as

cos B = {(1+ cos 2.B)/2}^2 = ½.(2+ 2.cos 2.B)^2. …………………………………………………(4)

Examination of equation (2) reveals that it contains cosine of an angle in LHS and also in RHS, therefore, it can form a never ending loop. Cos 2.B in right hand side can be replaced by ½.(2+ 2.cos 4.B)^2 and again cos 4.B again replaced by ½.(2+ 2.cos 8.B)^2 and so on. Therefore, cos B can be written as

I take another example of angle tripling formula whereby sin 3.B is written as

sin 3.B = 3.sin B – 4.sin^3 B = sin B.(3- 4.sin^2 B) or

sin B = sin 3.B/(3- 4.sin^2 B) …………………………………………………………………………………..(6)

Inspection of equation (6) reveals that its RHS has sine of angles in numerator and denominator and LHS also has sine of an angle. In RHS, sin 3.B in numerator will be considered for recursive relation and will be replaced by sin 9.B/(3- 4.sin^2 3.B) and sin 9.B by sin 3^3.B/(3- 4.sin^2 3^2.B) and so on till sin 3^n.B = sin B. At that stage, sin B on LHS and RHS cancels and denominator which is product of terms (3- 4.sin^2 B).(3- 4.sin^2 3.B), …………………{3- 4.sin^2 3^(n-1) B} equals 1, giving rise to an identity.

For illustration, let us take the example of sin π/10.

Here angle B = π/10 which can be written as

sin π/10 = sin 3π/10/(3- 4.sin^2 π/10) = sin 9 π/10/{(3- 4.sin^2 π/10).(3- 4.sin^2 3π/10)}. That is

But sin 9.π/10 = sin π/10, therefore,

Here recursive relation stops when sin^n B equals sin B or – sin B. From this, it can be concluded

n

Π(3-4.sin^2 3^(j-1) = 1 …………………………………………………………………………..(7)

j=0

when sin (3^k B)= sin B and

k

Π(3-4.sin^2 3^(j-1) = – 1 …………………………………………………………………………(8)

j=0

when sin (3^k B)= – sin B.

In this way, a new trigonometric identity is established using recursive relation.

Coming to equation (5) which is an equation that contains roots of root, such continuous nested radicals were first used by François Viète when he derived formula for pi first time from entire Europe. Recently, these have been abundantly used by Indian mathematician Ramanujan. As such, use of recursive relation gave unexpected and astonishing results that mesmerised English mathematician Hardy.

I also experimented on recursive relations and results were encouraging. If you feel a bit interest in it, don’t shy away. Take a note book and jot down a formula. Find all possibilities for its recursive nature, try differently and look for different result. You will find one I am sure and that would be blissful. Do you feel interested in recursive relation, you may go through this article at http://www.ijma.info/index.php/ijma/article/download/5227/3074

End

Notes:

1. A part of this article is excerpts from my paper written on “MORE TRIGONOMETRIC IDENTITIES ,” available at http://www.ijma.info/index.php/ijma/article/download/5227/3074

2. Title image courtesy brewbooks at https://www.flickr.com/photos/brewbooks/184343329/

About author

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.