45 replies to this topic

[galexcd]

Removed per request

[ghostrider]

A circle is simply an infinite-gon with all congruent sides. That proves it geometrically.

[husker]

Math is tricky. From what I was taught, a line is just a bunch of points. Now what is a point? It's hard to define. A circle is similar. It's easier if you just take math for what it is and don't think too deeply. But I do encourage thinking, and you bring up a good point. Hang on with me here as I give you my explanation, but remember I'm just in geometry. 179.99 etc. doesn't equal 180, and never will. It will come very close to 180, but it will never reach it. Therefore, a circle can't be a line because it never is 180. Just my ideas, you can think what you want.

[husker]

Quote

Ok I've got the equation: (lets say underlined numbers mean repeating)

x=.9 - Set x to .9 repeating
10x=9.9 - Multiply both sides by 10
10x-x=9.9-x - Subtract X
9x=9 - x is .9 repeating, so subtracting it from 9.9 repeating gives you 9
x=1 - divide by nine
.9=1 - Substitute

.9 times 10 equals 9 not 9.9

[TypoMage]

Why are you guys talking all mathematicly? A circle is the thing that goes on top of stick figures. Or this ( o ) That is a circle I do not really understand all these numbers and stuff?

[jlhaslip]

Quote

A circle is just an infinite number of angles on a line right?

I would say not...

A circle is a collection of contiguous 'points' equal distance from a 'centre' point. I didn't Google this definition, just based it on my understanding of what a circle is/is not. "An infinite number of angles on a straight line"... does not compute, Will Robinson...

[truefusion]

Quote

Well this keeps on going 179.9, 179.99999999, until it hits 179.9-repeating. Well it is mathematically proven that .9-repeating, equals 1.

It is not mathematically proven that .9-repeating equals 1. They tell you to round it off, which would equal 1. The only reason they tell you to round it off is so you won't have to write the number down.

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Ok I've got the equation: (lets say underlined numbers mean repeating)

x=.9 - Set x to .9 repeating
[1]10x=9.9 - Multiply both sides by 10
10x-x=9.9-x - Subtract X
9x=9 - x is .9 repeating, so subtracting it from 9.9 repeating gives you 9
x=1 - divide by nine
.9=1 - Substitute

I think you messed up somewhere in your equation... Like [1], don't you divide here instead of multiplying? Do the opposite?

[husker]

Well to continue my annoying streak , I'd like to point out that it is mathematical impossible for .9 repeating to equal 1. It's like saying in the equation 2^x=y that y could equal 0.

Anyways, it's an interesting idea, but still not going for it.

[RedAlert]

.9~ = 1, and anyone who doesn't think so should go back to school and take advanced math.

Here are some proofs...pick one you can understand:
Let us assume x = .99999~
Now we know that when we multiply something by 10, we
move the decimal one place to the right.
so 10x = 9.9999~ There is no 9 lost by doing this
since there are an infinite number of them.
Now we do simple arithmetic
10x - x = 9.999~ - .9999~
9x = 9 This is allowed because every 9 after the
decimal will cancel with another 9.
x = 1 and x = .999~ so 1 = .9999~

.9999~ = .9 + .09 + .009 + ....
here we represent .9~ as an infinite sum
sum[i:0->inf.](.9*.1^i)
We know how to solve infinite sums.
sum = .9/(1 - .1) = .9/.9 = 1
Since we said the sum was initially .9999~, we can
conclude that .9999~ = 1

1/3 = .33333~
This is true, and can be proven with an infinite sum
as above.
3*1/3 = 3*.33333~
1 = .99999~
We are allowed to multiply by 3 because no part is
going to carry over to the next part. Thus, every part
of the decimal will increase by factor of 3, making it
a 9.

The real numbers are defined as limits of Cauchy
sequences of rational numbers.

*A rational number is a fraction of two integers
*A cauchy sequence is a sequence x(1), x(2), ... such
that for every integer n there exists an integer m
such that |x(j) - x(k)| =< 1/n for all j,k >=m.

Two Cauchy sequences x(1), x(2),... and y(1), y(2),...
are considered equivalent if for every integer n there
exists an integer m such that |x(k) - y(k)| =< 1/n for
all k>=m.

Let x(j) = 1 - (1/10^j)
Let y(j) = 1.

I'll leave it to you to check these are Cauchy
sequences.
These two sequences are equivalent:
Given some integer n, |x(k) - y(k)| = |1-(1/10^k) - 1|
= |1/10^k| =< 1/n if 10^k >= n. So we'll set m =
{smallest integer greater than log(n)}.

Thus the sequences .9, .99, .999, ... and 1, 1, 1...
are equivalent, so they have the same limit, namely,
.999~.

0.9~ = 0.9 + 0.09 + 0.009 + 0.0009 + ...
S = 0.9~
S = 0.9 + 0.09 + 0.009 + 0.0009 + ...
S = 0.9 + (1/10)[0.9 + 0.09 + 0.009 + ...]
S = 0.9 + (1/10)S
(9/10)S = 0.9
S = 1

Therefore, 0.9~ = 1.

If two numbers are not equal, there are an infinite
number of numbers between them. Give me a number
between .9999~ and 1.

All repeating and terminating decimals can be
represented as fractions. If .999~ is not represented
by 1, what fraction does represent it?

[Saint_Michael]

I like to point out that I lost all faith in the circle now with all this math and it's existence nonsense . So to help mess with people's mind's even more check out this wonderful info about the circle.
http://en.wikipedia.org/wiki/Circle

Of course through my researching this wonderful thought there are several other sites that would agree .9999~ = 1

here

http://digg.com/tech_news/.9999999=1

and here

http://polymathematics.typepad.com/polymat..._sorry_it_.html

But like I said you all ruin the beauty of the circle now I to rely on a triangle and a square to get me through life . but to stay on topic just a bit after reading these posts and those 3 websites I would have to agree especially with this part

Quote

Here are some proofs...pick one you can understand:
Let us assume x = .99999~
Now we know that when we multiply something by 10, we
move the decimal one place to the right.
so 10x = 9.9999~ There is no 9 lost by doing this
since there are an infinite number of them.
Now we do simple arithmetic
10x - x = 9.999~ - .9999~
9x = 9 This is allowed because every 9 after the
decimal will cancel with another 9.
x = 1 and x = .999~ so 1 = .9999~

The math teacher followed the same line of thinking, in which I conclude that .99999~ = 1 is both a true and false statement. Meaning that if you round up you will get 1; however, since the number is always repeating itself then it is not a true solid number (can't think of the word for it but you math geeks know what I am referring to). You call this a enigma in itself and odds are you would have to apply occam's razor to make the most sense out of this enigma.

Edited by Saint_Michael, 14 May 2007 - 10:40 AM.

[Sprnknwn]

Love this post. I go with the "smoking" answer, and it's winning so...
I can't discuss on this but I think a circle it's a circle and yes, it's also a polygon with so many angles that the eye can't see them and the final look is curved and sharpened. Both things, or one thing forms another, whatever. I need more inspiration to answer properly.

[RedAlert]

Saint_Michael, on May 14 2007, 02:39 AM, said:

But like I said you all ruin the beauty of the circle now I to rely on a triangle and a square to get me through life . but to stay on topic just a bit after reading these posts and those 3 websites I would have to agree especially with this part
The math teacher followed the same line of thinking, in which I conclude that .99999~ = 1 is both a true and false statement. Meaning that if you round up you will get 1; however, since the number is always repeating itself then it is not a true solid number (can't think of the word for it but you math geeks know what I am referring to). You call this a enigma in itself and odds are you would have to apply occam's razor to make the most sense out of this enigma.

If you and your math teacher really think that, then neither of you understand what infinity is. .9~ is not a process; it's not growing. It's a number that's exactly equal to 1 without rounding.

[salamangkero]

Whoah! That was some...whooh! I don't think I can add anymore to what was already said. It's kinda hard to stomach, 0.999... being equal to 1 and all, really, but I suppose advocates of the Ptolemaic model felt the same with Copernicus' heliocentric model.

Personally, I don't believe in the existence of circles. They are, for all we know, nothing more than constructs in the mind of sentient species, like humans, for example. Why is that? Because circles have no thickness! They have height and width, being in 2D but they have no thickness. Their z-dimension is zero. In our world, it would have to be a loop of wire in the shape of a circle but having an infinitesimally small, even non-existent, thickness.

Oh yeah, for that matter, I don't believe in squares and triangles too. As a matter of fact, I could be audacious enough to claim that polygons do not exist. So, sorry, Saint_Michael, I suppose the beauty of triangles and squares have also been dispelled

I do concede, however, that what we have here in the real world are approximations of a circle, or polygons, for that matter

[Saint_Michael]

RedAlert, on May 14 2007, 07:02 PM, said:

If you and your math teacher really think that, then neither of you understand what infinity is. .9~ is not a process; it's not growing. It's a number that's exactly equal to 1 without rounding.

That not my math teacher I am referring to, I was referring to the teacher in those links has follow the same procedure as stated by your previous post. Now just remember what I was referring to in my last post since .9999~ is not a whole number and not solid so a minor correction on that.

Well think about it though .9999~ is a fraction right? and if that 9 keeps repeating itself in just basic math then it will never equal 1. But I think it was mention somewhere you have to be high up there in math; like trig calculus and even physics. So I would say my state is true that depending on the situation that .9999~ is being used it will never be a true whole number, but it could be especially when you go into margin of error +/-. Taking this stats class last semester, .9999~ would become a whole number so to have none of those factions appear in statistics reports and thus the margin of error would go into effect. But hopefullly I can bump into this math teacher at college and ask him about it and see what he comes up with. I think he's been a math teacher since about 30 years so I would say he could give a good solid answer on this.

Quote

Whoah! That was some...whooh! I don't think I can add anymore to what was already said. It's kinda hard to stomach, 0.999... being equal to 1 and all, really, but I suppose advocates of the Ptolemaic model felt the same with Copernicus' heliocentric model.

Personally, I don't believe in the existence of circles. They are, for all we know, nothing more than constructs in the mind of sentient species, like humans, for example. Why is that? Because circles have no thickness! They have height and width, being in 2D but they have no thickness. Their z-dimension is zero. In our world, it would have to be a loop of wire in the shape of a circle but having an infinitesimally small, even non-existent, thickness.

Oh yeah, for that matter, I don't believe in squares and triangles too. As a matter of fact, I could be audacious enough to claim that polygons do not exist. So, sorry, Saint_Michael, I suppose the beauty of triangles and squares have also been dispelled happy.gif

I do concede, however, that what we have here in the real world are approximations of a circle, or polygons, for that matter happy.gif

This isn't the matrix lets try and stay in the physical plane . It reminds me of a Star Trek episode where they mention a dodecahedron, which I believe is like 26 sides shape or something like that, but of course now we would be going into physics about containing energy and what not.

[DeM0nFiRe]

A circle is really just an infinite amount of triangles. Technically circles actually do have sides. to make any regualar shape you use triangles. And as the number of triangles increases, the length of the base of each triangle decreases(assuming you are keeping the same area). this means that as the number of triangles approaches infinity, the length of the side approaches an infinitecimal.

[NNH1254]

I think they exist, I tried asking my pre-calc teacher about something a little like that once and he went off on a tangent (OMG that was the lamest joke on the face of the planet) that I didn't really follow. Math teachers are not people I like disagreeing with.
Circle definitions are confusing, it all depends on which one you use. If I put a compass point in the corner of a ceiling (so where the ceiling and two walls meet) and go all the way around, it still meets some textbook definitions of a circle. Since it's one three surfaces though, and covering 90 degrees for each, it's only 270 degrees, so is it a circle or not?
I don't know if this is what I hate about math or what I like about it.

[hitmanblood]

OK I'll begin waht the hell are you smoking Alex???

Joking, no you are not right people showed you mathematically that 0.9 is not equal to 1 and by the way circles do exist if not on our paper as geometrically they are infinite lines or even dots depends how you watch it.

Circles do exist in nature and it is light in fact that is when light leaves star like sun if we disregard star winds and sun spots then you can see that light is leaving star in perfect circles and this has been proven physically. However you have to take also in concern that light has dual nature like wave and particle and in this sense to have perfect circle we are taking into account wave nature of light.

So you see circles to exist.

I'll finish this with quote: "Perfection is strong message, and circles are perfection".

[DeM0nFiRe]

Ok, I'll prove right now that circles do not exist.
All regular flat shapes are maade of triangles right? This would mean that a circle is made up of an infinite amount of triangles right. In order for an infinite number of angles to not equal infinity when added, the measure of the inner most angle of each triangle is an infintecimal(or 0 repeating with a 1 at the end) right? this would mean that the remaining measures of the angles are 179.9 repeating.
now, 1/9 = .1 repeating, correct? let's do some algebra:
1/9=.1repeating <- now we'll multiply both sides by nine
9/9 = .9repeating
NOw, we all know that anything divided by itself =1 correct? well, using if A=B and C=B then C=A
we'll see that if .9repeating = 9/9 and 1=9/9 then .9 repeating =1
so that means that 179.9repeating is the same as 179+9/9, which means that the last angle, the one that is supposed to be an infintecimal, must be equal to the remaining ninths of 9/9 which means it's 0/9 which means that there are no other angles which means that a circle, by it's defintion of being closed regular shap, does not exist.

[dau1970]

i would like to retort with another question: is a straight line a particular case of a circle?
imagine what happens if you take a circle and you start increasing its radius towards infinity... the curvature of the circle will start to flatten until, in the end, it's nothing but a straight line

Edited by dau1970, 24 May 2007 - 12:46 AM.

[DeM0nFiRe]

No, a straight line is not a circle. The definition of a circle includes that it is a closed shape. A straight line holdds all the properties of a circle except the one. Circles are purely imaginary, as the definition is an impossibility.

[Plenoptic]

Well I've decided yes I see what you are saying, and in a perfect world they might not exist, but to me they do You see you can still draw them and have equations so to me they exist I am all confused right now and I guess you can say that they technically don't exist, or maybe we need a new definition for a circle, that way we don't have this problem. My mind will still be thinking about this for days now of how .999999999~ = 1 All the different ways to prove it so if you get in trouble with your teacher for "rounding" you can whip out a proof.

[matak]

Here's my 2 cents..

Circle is geometry construct, like dot, line triangle, and square, cube, sphere. When you look at geometry (Euclid 3D space) first you start with dot. Dot is definition of 1 dimension. Then you add, 2 dimensional space by adding plane. Plane is defined by one dot and 1 line that doesn't lie on that dot. Imagine it like paper without thickness. If you have dot on 1 place, and line on another, and that dot and line have no crossings or don't lie on each other, when you put that paper so that line lies on the paper and that paper touches dot, you got 2d plane.

Circle is 2 dimensional geometry construct, and can be drawn on that plane we got from line and a dot. It's defined by center and radius. if you define center as one dot and radius like one line of defined lenght eg 3cm, and you put one end of the line on dot, and rotate the other end of the line for 360°, trail which that end leaves is called a circle.

Most interesting number which you get from a circle is called Pi. (link)

Pi (π) from wiki said:

Pi or π is a mathematical constant and a transcendental (and therefore irrational) real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with an Archimedes number) and as Ludolph's number.

Pi is used to calculate circles area.

Maybe you are confused with circles, beacouse you don't understand that number, and for further reading on mathematics, i would suggest you that you read about it, so that you don't start with some "crazy" inventions of your own

As for the 3d space, maybe later...

Edited by matak, 27 May 2007 - 08:47 AM.

[RedAlert]

For all that writing, matak sure doesn't do much reading. Try reading the posts in the thread before you add your "2 cents", or you might as well be responding to a thread on what you ate for breakfast.

[Carson]

Well, I think that there are no Circles in nature. Because they have to be perfect, they can only be formed by a computer.

[salamangkero]

Wokies, I think I saw where the fallacy is. (I already did a long time ago but it took me a while to get back to this thread)

Quote

Well if a circle has an infinite number of sides, that means each angle gets infinitely close to 180°.

No conflict here. I'll just repeat, each angle gets infinitely closer (but does not actually become) 180°.

Quote

Well this keeps on going 179.9, 179.99999999, until it hits 179.9-repeating.

This is impossible. It never hits 179.9-repeating. It just keeps on going from 179.9, 179.99, 179.999, etc but it is perfectly impossible to actually hit 179.9-repeating, that is an infinite number of 9's. How so? Well, how do you reach infinity? How do you count up to infinity? Infinity is, by nature, an exceedingly large and unattainable number. Besides, you said it yourself.

Quote

Well it is mathematically proven that .9-repeating, equals 1.

Therefore, 179.9-repeating is exactly 180, right? And, as we said before, each angle gets infinitely closer (but does not actually become) 180°. In other words, it will never hit 180, 179.9-repeating, 179+1 or any other representation of the same number.

Quote

Well if that is true, that means that circles are just strait lines.

If it were true, which, as we have seen now, is not the case. Circles cannot be equal to lines.

Quote

I told my calculus teacher this, and he just told me that that is similar to the theory that disproves motion: In order to move, you have to pass through an infinite number of points, and that of course is impossible. But that theory is obviously wrong because we can all move still.

I believe your teacher must be thinking of Zeno's paradoxes of motion. Yes, it is quite similar