Welcome to NRICH.

 
The formula for pi
By Zhidong Leong (P3545) on Wednesday, January 17, 2001 - 12:27 pm:

May I know the formula for pi?

By James Myatt (Jem50) on Wednesday, January 17, 2001 - 06:32 pm:

pi (p) is the ratio of the diameter of any circle to its diameter.

However if you want to know the value of pi, then that's a different matter. The value of pi can be found by taking sufficient terms of an infinite series. It is possible to obtain a Taylor series expansion of many functions such as tan x, sin x, arctan x, etc.

arctan x = x - x3/3 + x5/5 - x7/7 ... for -1< x £1

It just so happens that tan p/4 = 1, hence p/4 = arctan 1 = 1 - 1/3 + 1/5 - 1/7 + ...
By taking a large number of terms then this will converge on p/4 hence you can calculate the value to as much accuracy as you need. However this particular expansion converges very slowly.

Having just done 10000 terms (on my computer), you can be sure of 1 dp after 25 terms, 2dp after 1255, 3dp after 2464, and after 10000 you still don't have 4 dp of accuracy. So as you can see this is a particular series is very inefficient, as the difference between successive terms gets very small very quickly, that is 1/n - 1/(n+1).

There are many much faster algorithms, but I can't remember them, but are often work along similar lines.

If you want to know the value of pi, it's best looked up, and there are a variety of online resources, which can probably be located by sticking "pi" and "value" into any good search engine, such as this one.

Of course, what is pi? I would say pi is the ratio of the diameter of any circle to its diameter, someone else might say the sum of an infinite series, but an engineer or scientist might say pi is 3.14, or a school child might say it's a number that helps them calculate the area of a circle.

James

By Hal 2001 (P3046) on Thursday, January 18, 2001 - 10:38 am:

Since pi is very much geometrically understood by people, how is the infinite series to get pi related to its geometrical side?

By Brad Rodgers (P1930) on Friday, January 19, 2001 - 12:42 am:

I think that the formula for pi deals with the fact that there are 2p radians in a revolution. Not sure if that entirely answers your question though. On a related note, though, is there any reason why d/dx(tan-1x)¹1/(1+x2) for x in degrees rather than radians? If not, can't we say that 180°=3.14...?

Thanks,

Brad

By Hal 2001 (P3046) on Friday, January 19, 2001 - 02:35 pm:

Thank Brad. I think that makes sense, but not fully got the point how this gets the geometric relationship to Pi.

By James Myatt (Jem50) on Friday, January 19, 2001 - 06:21 pm:

The infinite series is directly related to the geometry.
We use tan p/4 = 1 to give p = 4 arctan 1 and using the power series for arctan x, we have an infinite series that evaluates to pi. I don't see how this isn't geometrical.

James

By Olof Sisask (P3033) on Friday, February 2, 2001 - 05:20 pm:

How do you prove that pi is the ratio of the circumference of any circle to its diameter, just out of interest? Might be a silly question, haven't looked into it yet :).

By Anonymous on Friday, February 2, 2001 - 05:33 pm:

Pi is defined as the smallest positive real number x so that sin(x) = 0. Once you have this and the geometric result that opp/hyp = sin(x). A little integration reveals all.

However I would like to see a proof that sin(x) = opp/hyp

By James Myatt (Jem50) on Friday, February 2, 2001 - 05:41 pm:

Olof,

pi is the ratio of the circumference of a circle to its diameter. It just so happens that pi is also very useful in other places, e.g. integration, power series, etc. but these are all just consequences of the fact that it is this ratio.

I certainly wouldn't say that it's the smallest real number that satisfies sin(x) = 0. As I said this is just a consequence of this definition. Similarly, sin(x) is the ratio of the opposite side to the hypotenuse of a right-angled triangle. If you think through the dependencies and derivations then I'm sure it'll be clear. If there's demand I'll go through some examples, but I won't unless someone wants me to.

James

By Anonymous on Friday, February 2, 2001 - 05:45 pm:

I have Hardy's course of pure mathematics and he defines pi as the smallest positive real so that sin(x) = 0. sin(x) is defined in terms of a power series.

I would like to see geometric interpretations please James....

By Sean Hartnoll (Sah40) on Friday, February 2, 2001 - 05:51 pm:

There are many ways of defining pi. They are all of course equivalent, so it doesn't matter which one you take as a starting point.

I tend to think of the definition of sin as opp/hyp, where a simple similar triangles argument shows that this is well defined. From here you find the derivative of sin and hence the power expansion. But there is nothing wrong with starting with the power expansion and working backwards to the triangular properties (which you would get to via the standard properties like sin2 + cos2 = 1).

There is no one definition of pi or sin, you can start wherever you like as long as you shows that everything is equivalent.

Sean

By Olof Sisask (P3033) on Friday, February 2, 2001 - 05:58 pm:

How would you prove that there is a ratio of the circumference of a circle to its diameter in the first place? Still haven't looked into it in any depth, so I'm sorry if it's a silly question again :).

By Anonymous on Friday, February 2, 2001 - 05:58 pm:

Ok, but how do you show the equivalence. I have a problem with non set-theoretic definitions.

By James Myatt (Jem50) on Friday, February 2, 2001 - 05:59 pm:

I think that starting points and definitions are by definition the most simple way of defining something. This means that you start using things that everyone can understand, i.e. axioms, and derive more and more intricate things. For example, when you learnt about spelling, you learn what the letters were first; when you learnt subtraction, you learnt addition first; square roots require principles of multiplication. When you learnt trigonometry, you knew lots of useful things about triangles and circles and rectangles and other more simple things before hand.

I'd continue, but I'll be late for a supervision, but I shall continue when I return.

James

By Olof Sisask (P3033) on Friday, February 2, 2001 - 06:04 pm:

Just speculating, perhaps you can use the formula x2 + y2 = r2 and try to find the length of this curve, divide it by r and maybe it'll be a constant. Maybe it won't work :). Going to try...

By Sean Hartnoll (Sah40) on Friday, February 2, 2001 - 06:13 pm:

It will work...but in doing so you will have to use the way pi appears in relation to inverse sine, which is the definition anonymous gave.

Sean

By Sean Hartnoll (Sah40) on Friday, February 2, 2001 - 06:15 pm:

The way to show that pi is well defined as a ratio of circumference and radius is to use geometry a la Euclid. I'm not particularly good at that sort of thing, but I know it can be done.

Sean

By Sean Hartnoll (Sah40) on Friday, February 2, 2001 - 06:18 pm:

So once you have a geometric construction of pi, and a geometric definition of sine, it is completely obvious that sin(pi)=0.

Or equivalently start with a series definition of sine, and deduce the geometric definition (to do this express sine and cosine in terms of imaginary exponentials, from here deduce that cos2 + sin2 = 1, and this is essentially it by comparison with with Pythagoras for a triangle of hypotenuse 1)

Sean

By Olof Sisask (P3033) on Friday, February 2, 2001 - 06:27 pm:

I kinda see. How do you think Euclid did it (if it was him), and would there be any websites with the information on?

Thanks,
Olof.

By Olof Sisask (P3033) on Friday, February 2, 2001 - 07:07 pm:

Unless the method you described is the 'classical' method Sean?

/Olof.

By Michael Doré (Md285) on Friday, February 2, 2001 - 07:36 pm:

I think Olof's method will work. It is not necessary to get the integral into a closed form involving arcsin - you can just leave the result as an integral and show via a linear substitution that the length of circumference is proportional to r. This would show that defining p as the ratio of circumference:diameter is a sensible definition (i.e. it would be independent of the circle you pick).

The only problem is that by writing x2 + y2 = r2 you are implicitly assuming Pythagoras. Pythagoras' theorem in turn rests on the similar triangle theorem. So it is necessary to prove the theorem of similar triangles from the axioms of geometry. I'm not exactly sure how you'd do this (partly because I don't know what any of the axioms are apart from the famous parallel postulate one). Perhaps similar triangle "theorem" is an axiom of geometry - I'm not sure.

Another point is that I'm not sure how to prove rigorously that the length of an arc is the integral of sqrt(1 + (dy/dx)2). Perhaps we should define it like that, and then show that it has all the properties you'd expect - i.e. if you rotate, translate or reflect a curve, the length is unchanged and if you join two curves together then the length of the new curve is the sum of the lengths of the old curve. I admit this is a rather unsatisfactory definition intuitively but I can't really think of a better one.

Regarding the definition of p I agree with James that it is nice to define things in the most intuitive way possible. However the benefit of defining p via a non-geometric approach is that you are not allowing the definition to be dependent on the axioms of geometry. Suppose geometry were mathematically flawed? (i.e. the axioms were inconsistent). Then the geometric definition of p would make no sense at all. However if we define p via a series or something then if it were the case that Euclidean geometry were flawed but arithmetic was OK, then we would still have our definition of p and would be able to use it in a lot of mathematics. Now it is impossible that arithmetic is flawed and geometry is OK because geometry is dependent on arithmetic. So a definition of p in terms of arithmetic means we are using the minimal amount of assumptions possible. From there we can prove that if Euclidean geometry is OK then p is the ratio of circumference to diameter.

By Brad Rodgers (P1930) on Friday, February 2, 2001 - 08:02 pm:

My schoolbook says that triangle similarity is a postulate, but it has a way of making far to many assumptions. Most of these are obvious but take a real clever proof (like commutability). Anyways, pythagoras can be proved without similar trianges too. Try drawing a square of sides (a+b) and make 4 triangles inside of length a and b.

Perhaps I don't understand Olof's question properly, but wouldn't it be a very quick exercise of similarity to prove that a circle's diameter and its circumference have some set ratio? As all circle's are similar, this would seem to be the case...

Brad

By Michael Doré (Md285) on Friday, February 2, 2001 - 08:07 pm:

The point is, how do you prove all circles are similar?

In the proof of Pythagoras you refer to, presumably you are adding up areas? In which case we have to prove that if you divide up shapes into little bits then the sum of the areas of the bits is equal to the area of the shape. (Perhaps this is one of Euclid's axioms.) If we can add up areas like this then I think we can prove similar triangles from that premise.

By Brad Rodgers (P1930) on Friday, February 2, 2001 - 08:21 pm:

Well, I think my book could call it a postulate, but the proof is fairly simple so:

We know that one definition of a circle is that it is a regular polygon as the number of sides tends to infinity. We also know that if a regular polygon has n sides, it will be similar to another regular polygon with n sides. See this by deconstructing it into triangles aroung its circumcenter. Since this relationship holds even as the polygon's number of sides tends to infinity, it holds for a circle, this completes the proof.

Also, you can prove that the ratio is constant using integral calculus, and you find that the ratio is 360°r, at last proving that 360°=2p.

Brad

By Sean Hartnoll (Sah40) on Friday, February 2, 2001 - 08:28 pm:

Although I can't do it off the top of my head, I am completely positive that from the five postulates of Euclidean geometry you deduce the similarity theorems for triangles (which imply that sine is well defined) and also that the ratio of radius to circumference is constant for circles (hence pi is well defined). So neither of these is a postulate. A circle is defined as a set of point which are equidistant from a given point.

Sean

By Brad Rodgers (P1930) on Friday, February 2, 2001 - 08:34 pm:

Hmm, not sure about the area thing,

Here are Euclid's postulates-

1) A straight line may be drawn joining any two points

2) Any straight line segment can be extended indefinitely into a straight line

3) Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center

4) All right angles are congruent

5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

I remember seeing a volume of The Elements somewhere on the internet. When I find it, I'll post a link and try to see where it proves or assumes similarity.

By Brad Rodgers (P1930) on Friday, February 2, 2001 - 08:55 pm:

The Elements

I haven't had a chance to look through much of it yet.

By Brad Rodgers (P1930) on Friday, February 2, 2001 - 09:08 pm:

You'll find similarity in Book VI Propositions 5 through 7.

By Dan Goodman (Dfmg2) on Saturday, February 3, 2001 - 01:58 am:

I think the best (ie pure mathsy) way to deal with it all would be to define sin and cos on C using their power series expansions, prove the opp / hyp rules from this using their expansions in terms of exponentials and complex numbers (easy). Then, define p to be the smallest positive real root of sin(x)=0. From this, and using the integrals talked about above you should be able to prove it has the required properties.

Outline of proof:

After defining the power series of sin and cos and proving eix=cos(x)+isin(x). Now, a right angle triangle is basically just a point in the complex plane if 0 is one of the vertices of the triangle, from the expansion above the opp / hyp rules drop out easily. If the angle of a right angle triangle with vertices 0,z,Re(z) at 0 is q, and the hypoteneuse is r, then z=reiq, so Re(z)=rcos(q). So the adjacent distance is rcosq, the opposite distance is rsinq and the hypoteneuse is r. So opp / hyp = rsinq/r=sinq as desired, etc. If you do the integrals, you get the circumference of a circle of diameter 2r to be 2sin-1(0)r (in fact, you need to be a bit careful about this bit, but if you choose a consistent function sin-1 then this will work out, even though sin(0)=0 and sin(2p)=0 as well, this is because the integral is actually the difference of two values of sin-1, a technical matter...), which by the definition of p is 2pr, so the ratio is p as required.

Final comments: I think it's better to define things "unintuitively" and then demonstrate their equivalence with intuitive things, because it usually takes a bit of work to get to the point of being able to demonstrate their equivalence, and it's useful to have definitions as simple as possible. Defining p to be the ratio of the circumference and diameter takes a lot of work to even prove it is meaningful, whereas defining it as the smallest positive zero of sin is comparitively elementary. As to defining the length to be the integral of sqrt(1+(dy/dx)2), I think you have to take it as a definition of length. You can make it more intuitive to define length for straight line segments, the length of the straight line between a and b is just |a-b| in the Euclidean plane, extend this to polygonal arcs (made up of straight line segments, if the polygonal arc is a0 to a1, then a1 to a2 and so on until an, define the length to be åi=0n -1 |ai - ai+1| ), and then extend this to general curves by continuity, in other words the length of a limit curve should be the limit of the lengths of the curves. So if cn is a sequence of polygonal arcs with lengths Ln which tend uniformly (technical matter) to a curve c, we define the length of c to be L=limn®¥cn, or ¥ if the sequence cn diverges. So if we agree on the lengths of polygonal arcs, and we agree that the length function should be continuous, then we have to agree on this definition of lengths. This, with a bit of work, leads to the formula L(y)=ò sqrt(1+(dy/dx)2). Phew. My thoughts on the matter.

By Olof Sisask (P3033) on Saturday, February 3, 2001 - 02:07 am:

I find that it's usually quite interesting proving the things we take for granted. Thanks for the link Brad.

By James Myatt (Jem50) on Saturday, February 3, 2001 - 02:40 am:

OK, I see there is serious mileage in defining thing in a more complicated way and then proving that these statements are equivalent to more intuitive statements, or starting points as I would define them. But the starting points/axioms must be intuitive/obvious/taken-for-granted statements as far as I see it, but need not necessarily be proved directly, but can be proved from more complicated independent premises, and showing equivalence, as proposed.

I can not see an answer to this debate however, as different people choose to take different things as their starting points.

James

By Olof Sisask (P3033) on Saturday, February 3, 2001 - 11:28 am:

Thanks for your input James, Sean, Michael, Brad and Dan.

By Kerwin Hui (Kwkh2) on Saturday, February 3, 2001 - 06:19 pm:

Actually, Euclid takes more assumptions than the 5 postulates, for example, he 'cheated' in Elements I, Proposition 4, about SAS. He just said put one on top of another and it fits. This assumes that you can do this is not a direct consequence of the 5 postulates and the 5 common notions.

The way to 'find' the circumference, in classical Greek geometry, involves the method of exhausion. I am not very familiar with this method, but this method is, in a sense, a pre-Renaissance integral calculus. It involves dividing the object into bits and an over-estimate and an under-estimate so that they agree when you 'take the limit'.

I'm sure geometry looks "pure mathsy", whatever the phrase means.

Kerwin