Is my conception/ knowledge about calculus right?

[tarekatpf]

First I want to admit that I don't understand how calculus can give you a correct answer. But I do know that calculus is used in almost all fields of sciences. So, I think either what I know about calculus or my conception is not correct.

Here is what I know and what I think about calculus. Please let me if my knowledge or conception is wrong.

A friend of mine gave me ( I can't remember exactly, so there are chances that my memory may also be erroneous ) an example of using calculus. If I recall what he said correctly, he said to me, ''Suppose, you have a circle. Now you want to measure the circumference of this circle using calculus. What calculus does is breaking this circle in so many pieces that each piece can be considered a straight line. Now if you can measure the length of each individual straight line, you can multiply that length with the number of straight lines you got. And there you get the circumference of the circle.''

Now my conception on calculus is based on what he said ( or in case I remembered it wrong, what I think he said. ) So the rest of this post is based on what I remember him to say.

If you do that trick, how can you get a correct answer? For example, you break the circumference in 1,000,000 tiny equal pieces, and you might think each of them has become close to a straight line. Now, since those pieces are NOT actually straight lines, no matter how hard you try, you'll never get the exact length of that ( imagined ) straight lines. So, maybe you get the length of one of this broken down straight line as 1 mm, but in reality the length of that line ( which is in fact, still a curved line ) may be a little more: say, 1.000001 mm. While the difference is not much in case of a single straight line, the difference will be magnified when you get the total length of the circumference. If you multiply 1 mm with 1,000,000 the circumference is 1,000,000 mm ( or 1 km. ) However, if you multiply 1.000001 with 1,000,000 it is 1,000,001 mm ( or 1,000.001 m or 1.000001 km ).

Oops. The difference is not much. I thought the difference would be much more than. It's only by 1 mm. And 1 mm is negligible indeed when the actual length is above 1 km.

So, if that is how calculus is done, I assume calculus can give you pretty close answer indeed.

So, the question is, is it really how calculus works? Or my knowledge or conception is not correct?

 

[pbuk]

Yes that's pretty close The final essential step is to look at a limit.

If you took 10,000,000 pieces you will get closer to the actual circumference, perhaps only being 0.1 mm out. And if you take 100,000,000 pieces you will get closer still; the smaller the steps you take the closer you get to the actual circumference.

So by taking smaller and smaller intervals we can't just get "pretty close" to the correct answer, we can get however close we want.

When we can get arbitrarily close to a specific number by taking a sequence of smaller and smaller intervals, we say that the limit of the sequence is equal to that number.

 

[SteamKing]

The concept of a limit is key to how calculus works. The example given by your friend of calculating the circumference of a circle by using smaller and smaller straight line approximations is a very old technique, stretching back to the ancient Greeks when they developed their technique of 'exhaustion', which was their approach to defining a limit.

The Greeks knew from geometry that polygons could be inscribed and circumscribed for a circle of a given radius. They also knew how to calculate the perimeter of a polygon given the number of sides it contained. By increasing the number of sides of the inscribed and circumscribed polygons, the Greeks could establish a lower bound and an upper bound on the length of the circumference of the circle by 'wedging' it between the lengths of the perimeters of the inscribed and circumscribed polygons. By increasing the number of sides to a very large number, the lengths of the perimeters of the polygons and the circle would tend to one value.

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Limit_(mathematics)

 

[tarekatpf]

Yes that's pretty close The final essential step is to look at a limit.

If you took 10,000,000 pieces you will get closer to the actual circumference, perhaps only being 0.1 mm out. And if you take 100,000,000 pieces you will get closer still; the smaller the steps you take the closer you get to the actual circumference.

So by taking smaller and smaller intervals we can't just get "pretty close" to the correct answer, we can get however close we want.

When we can get arbitrarily close to a specific number by taking a sequence of smaller and smaller intervals, we say that the limit of the sequence is equal to that number.

Thank you very much. Yes, I missed that ''limit'' thing.

And sorry about the late reply.

The concept of a limit is key to how calculus works. The example given by your friend of calculating the circumference of a circle by using smaller and smaller straight line approximations is a very old technique, stretching back to the ancient Greeks when they developed their technique of 'exhaustion', which was their approach to defining a limit.

The Greeks knew from geometry that polygons could be inscribed and circumscribed for a circle of a given radius. They also knew how to calculate the perimeter of a polygon given the number of sides it contained. By increasing the number of sides of the inscribed and circumscribed polygons, the Greeks could establish a lower bound and an upper bound on the length of the circumference of the circle by 'wedging' it between the lengths of the perimeters of the inscribed and circumscribed polygons. By increasing the number of sides to a very large number, the lengths of the perimeters of the polygons and the circle would tend to one value.

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Limit_(mathematics)

Thank you very much for information on ''exhaustion'' and a great explanation as well. And for those helpful links.

Sorry about being late to reply, though.

 

[arildno]

It should be said though, that the method of exhaustion is very exhausting, whereas (the fundamental theorem of) calculus is not.

 

[verty]

For example, you break the circumference in 1,000,000 tiny equal pieces, and you might think each of them has become close to a straight line. Now, since those pieces are NOT actually straight lines, no matter how hard you try, you'll never get the exact length of that ( imagined ) straight lines.

I believe this is the most useful way to think about it, that no matter how far one zooms in, the curve will always be curved. Another interesting example is a flight of stairs, if we make the stairs smaller, the length of the curve remains constant, call the length $L$, but what happens when the stairs become infinitely small? In the limit, the stairs become a ramp and the length of the ramp is $L \over \sqrt{2}$. BUT, the limit of the length of the curve is $L$, not $L \over \sqrt{2}$. So you don't want to be thinking that the circle becomes a polygon or the stairs become a ramp, it can lead to trouble.

 

[tarekatpf]

It should be said though, that the method of exhaustion is very exhausting, whereas (the fundamental theorem of) calculus is not.

God bless Newton!

I believe this is the most useful way to think about it, that no matter how far one zooms in, the curve will always be curved. Another interesting example is a flight of stairs, if we make the stairs smaller, the length of the curve remains constant, call the length $L$, but what happens when the stairs become infinitely small? In the limit, the stairs become a ramp and the length of the ramp is $L \over \sqrt{2}$. BUT, the limit of the length of the curve is $L$, not $L \over \sqrt{2}$. So you don't want to be thinking that the circle becomes a polygon or the stairs become a ramp, it can lead to trouble.

Yes, the curve is always a curve, until, maybe we zoom in up to the level of the ''fundamental elementary particles''.

 

[Integral]

Yes, the curve is always a curve, until, maybe we zoom in up to the level of the ''fundamental elementary particles''.

Calculus knows nothing about elementary particles.

 

[tarekatpf]

Calculus knows nothing about elementary particles.

Maybe elementary particles know nothing about calculus as well.

I think they should meet someday.

 

[mathwonk]

there are several ideas used in, say integral, calculus.

the first is the concept of a limit:

1) this is the idea that an infinite number of better and better approximations can uniquely determine a "limiting" value. I.e. if {an}, all n≥1, is an infinite sequence of approximations, and if L is a number such that the error |L-an| made by approximating L by an, can be made as small as desired, just by making n large enough, then L is the unique number that the sequence {an} is approximating.

then there is the problem of knowing which sequences do have limits:

2) if the sequence {an} has the property that their mutual differences |an-am| can be made as small as desired just by taking both n,m large enough, then the sequence }an} does have a unique limit L as in 1).

then one is challenged to actually find the limit of a given sequence:

3) I.e. after realizing that a given sequence {an} does have a limit L, there is the difficult problem of guessing what L is, since no finite part of the sequence is enough to reveal this. In the case of integral calculus, the limiting value of the sequence of Riemann sums computing an area, or the length of a circle, can often be guessed in simple cases, by applying the trick of the fundamental theorem of calculus. This gives, in cases where one can guess an appropriate antiderivative, an easy way to guess the limit of a very challenging sequence.


To actually use this to compute the length of a circle is somewhat illusory however, since the answer 2πr, must be given some meaning. In fact π is most naturally simply defined as the length of a semi circle of radius one. In order to use trig functions to "compute" the limit defining the length of a circle, requires one to first know about trig functions and the number π, before knowing about the length of a circle!

This artificial way of doing things can be done by defining trig functions by power series, but is a stretch that puts more sophisticated ideas ahead of simpler ones. Limit techniques do therefore allow one to given meaning to the idea that a circle of radius one has a length, but that length is most naturally just defined to be 2π.

My point is that the question of the length of a circle is a more basic notion than those used in calculus to compute it. Indeed the most natural definition of the sine (or cosine) function is as an inverse of the arclength function, which is thus more fundamental.

To be more explicit, trying to use the FTC to compute the arclength of a circle directly would involve guessing an antiderivative of the function (1-x^2)^(-1/2). That antiderivative is of course arcsin (or arccosin), but that creates a circular problem since the most natural definition of sin or cosin is as inverse to arclength. Hence the actual natural definition of the arcsin and arccos functions is in terms of arc length on a circle. I.e. one is thus led to deduce only that the length of the circle is its length!

 

[SteamKing]

I don't think that sines and cosines are defined in terms of arclength of a circle. If you can provide details, please do so. Sines and cosines are properties of triangles inscribed within circles of unit radius. Perhaps you are confusing the definition of radian measure with trig function definitions.

 

[mathwonk]

please think about what I wrote. I am tired now, or i would explain it more clearly to you, (I actually tried but the browser trashed what I wrote, and I will not try again tonight.)

for starters, think about what it means to measure angles in radians, i.e. arc length.

i.e. if angles are measured in radians, then one needs to know what arc length means before even knowing what a radian is, much less what cos of a radian means.

 

[SteamKing]

That's all very fine, but radians are not required to calculate trig functions. Radians make it convenient for series expansions of trig functions and what not. But, think about what I said. Trig functions can be calculated purely from drawing triangles within a unit circle. Trig functions were calculated using degree measure long before anyone had the concept of a 'radian'.

 

[mathwonk]

You raise an interesting question. Of course radian measure is standard in calculus of trig functions since even the basic fact that sin’ = cos depends on using radians. To be honest, I recall now the feeling I had when I first realized that measuring arc length on a circle was equivalent to measuring angles – it was eye opening and made me wonder why no one had made this explicit before. So the key point is whether one can measure angles without measuring arc length along a circle, I say no.

We are taught that angles can be measured in degrees, but we are not always told how to do this. E.g. how does one measure 20 degrees, in order to compute say cosine of 20 degrees? It requires computing the point on the unit circle whose radius makes an angle of 20 degrees with the x axis. This means that this angle subtends 1/18 of a full circle.

But how do we find such an angle? We can easily find an angle of 60 degrees, but then we are required to find 1/3 of that angle of 60 degrees. Unfortunately there is no ruler and compass construction which will trisect an angle of 60 degrees. I.e. it is not possible to “draw” a triangle having angles 20-90-70 in a unit circle.

But even if it were, that would be equivalent to finding an arc of length 2π/18. I.e. measuring the number of degrees in an angle is completely equivalent to measuring the arc length of a sector of the unit circle. I.e. if an angle centered at the origin can be determined to have t degrees, then the arc length subtended by that angle has length 2πt/360. So if you know the degree measure then you also know the arc length or radian measure.

Thus knowing the function arcsine or arccosine is the same as knowing the function of arclength. What do you think of this?

 

[SteamKing]

You can always measure a certain angle by inscribing a polygon of a certain number of sides into a circle of known radius. The ancient Greeks knew this as did Ptolemy. By using geometric arguments, the Greeks and Ptolemy were able to divide circles into 360 parts, which were later called 'degrees' and further to subdivide each degree into 60 smaller parts. Their clever geometry was based on constructing chords on circles of arbitrarily large radius, and Ptolemy was able to construct tables of sines of angles as small as 0.5 degree, which tables were published in the Almagest. Ptolemy found that the sin(30') = 0.0087268, which is correct to 6 decimal places.

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html

Similarly, Archimedes and other Greeks, including Ptolemy, were able to use polygons to estimate the value of π to a reasonable accuracy (Ptolemy's value was equivalent to 3.1416). Polygons were ultimately used to calculate π to at least 38 digits by the early 17th century AD, after which infinite series were used to vastly expand the number of digits of this constant.

http://en.wikipedia.org/wiki/Pi

 

[mathwonk]

if you read your own references you will see that the only chords that these ancients could construct are the classical ones, which are for n = 2,3,5,6, 10, and halves of known chords, but not thirds. That is why I gave as an impossible example 20degrees, which would require taking 1/3 of a constructible 60degree angle. This does however allow for approximations of values of sin and cos, but not for a rigorous definition of these functions at all real numbers, hence not a theory of their calculus. If you are only concerned with tables of approximations you are quite right. I did not think that was the issue here. Moreover your examples of approximating pi illustrate the principle that angle measure is equivalent to arc length measure.

 

[SteamKing]

Still, Ptolemy constructed his table of chords to angles as small as 30 arcminutes.

You are correct that an arbitrary angle cannot be trisected using a plain straightedge and compass. However, clever Archimedes sidestepped this problem by using a marked straightedge:

http://www.geom.uiuc.edu/docs/forum/angtri/