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Why π is an irrational number

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Why π is an irrational number

This is a question I've had for over 5 months and recently I think I've found the perfect solution for it. This is the first draft of my thesis. Be in mind that I'm only a 17-year-old engineering student and that English is my second language. Also, it is possible (and probable) that many other people have written about this, but I've come up with what I've written by myself.

According to the Euclidean geometry, a circle is a two-dimensional figure formed by the points equidistant from a center. This set of points is called a circumference and the distance between them and the center is called radio.

The ratio between the circumference and the radius of any circle, according to Euclid, is 2π, π being a number close to 3.14.

Throughout history, people tried to figure out the exact value of π, until Lambert proved using tangent theory that π was irrational number, meaning that it can not be defined as a fraction of a whole and, more importantly, does not correspond to anything that exists (hence the term irrational).

But how is this true? How can a figure that we supposedly see every day have an irrational measurement, i.e., a nonexistent one?

The answer to this intriguing question is that Euclidean circles do not exist. It is impossible to find more than four points that have the same physical distance from a central point.

You can see it when trying to find points equidistant from another point on a Cartesian plane in R2 (x, y). This is true regardless of the size of the plane.

(Note: only natural numbers can be used in this plan because there are only natural numbers in the universe. There aren't two atoms and a half, and even without knowing what is the basic unit of the universe, we know it has a specific x, y, z dimension.)

For example, the point (10,10) only has four points which are 5 away from it: (5,10), (15,10), (10,5) and (10,15). There are, however, points which have a distance close to 5, as the point 14,11) which has a distance of square root of 17.

It is easy to see on paper the difference between 5 and 17 ^ 0.5 cm, but not between 5 x 10 ^ -10 and 0.5 x 17 ^ 10 ^ -10, something closer to reality when you look at a circle drawn with a modern computer. Or between 5 x 10^-googolplex and 17^0.5 x 10^-googolplex, closer to the circle used when super-computer software tries to estimate estimate π.

For this reason, we are faced with (approximate) "circles" in our day to day lives, but in reality there are no perfect circles as Euclid described them.

Edited by JacobGalt

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First thought: the right triangle of sides 3-4 and hypotenuse of 5 units will create four additional points of distance exactly 5 away from (10,10): (7, 6) (7, 14) (13,6) (13, 14)

Second: the irrational numbers are an artifact of the method used to count and so it is entirely unjustified to claim that an irrational ratio such as pi does not exist. In a base pi numbering system the numerical value assigned to pi would be 1 and other different values would become irrational. Irrationality per se is interesting because it exists and is not a problem to be explained away but rather understood.

Third: I would think a proof (such as Lambert's) should constitute reason enough to explain why something is so without needing a further "proof of the proof".

Throughout history, people tried to figure out the exact value of π, until Lambert proved using tangent theory that π was irrational number, meaning that it can not be defined as a fraction of a whole and, more importantly, does not correspond to anything that exists (hence the term irrational).

It is particularly erroneous to claim that there is an absence of correspondence in irrationality. Pi corresponds to the ratio of the circumference of a circle with its diameter. That this turns out to never be a ratio of whole numbers says nothing about whether circles exist or their diameters exist or the legitimacy of dividing those numbers.

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Why π is an irrational number

This is a question I've had for over 5 months and recently I think I've found the perfect solution for it. This is the first draft of my thesis. Be in mind that I'm only a 17-year-old engineering student and that English is my second language. Also, it is possible (and probable) that many other people have written about this, but I've come up with what I've written by myself.

According to the Euclidean geometry, a circle is a two-dimensional figure formed by the points equidistant from a center. This set of points is called a circumference and the distance between them and the center is called radio.

The ratio between the circumference and the radius of any circle, according to Euclid, is 2π, π being a number close to 3.14.

Throughout history, people tried to figure out the exact value of π, until Lambert proved using tangent theory that π was irrational number, meaning that it can not be defined as a fraction of a whole and, more importantly, does not correspond to anything that exists (hence the term irrational).

But how is this true? How can a figure that we supposedly see every day have an irrational measurement, i.e., a nonexistent one?

The answer to this intriguing question is that Euclidean circles do not exist. It is impossible to find more than four points that have the same physical distance from a central point.

You can see it when trying to find points equidistant from another point on a Cartesian plane in R2 (x, y). This is true regardless of the size of the plane.

(Note: only natural numbers can be used in this plan because there are only natural numbers in the universe. There aren't two atoms and a half, and even without knowing what is the basic unit of the universe, we know it has a specific x, y, z dimension.)

If you keep up this sort of nonsense I predict (and hope!) that you will never become an engineer. You've made two gross errors:

1) You claimed that irrational numbers do not exist. ("How can a figure that we supposedly see every day have an irrational measurement, i.e., a nonexistent one?")

2) You've claimed that only natural numbers exist. ("Note: only natural numbers can be used in this plan because there are only natural numbers in the universe.")

I suggest you reread Lambert's proof and study it until you understand it. You might also want to study other proofs which you can find here:

http://en.wikipedia....;_is_irrational I might also suggest that you discuss what you've written with a professor of mathematics, but be prepared for an earful unless you find someone with the patience of a saint.

John Link

P.S. I said you made two gross errors, but that was only up to the point where I stopped quoting.

Edited by John Link

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Lambert proved using tangent theory that π was irrational number, meaning that it can not be defined as a fraction of a whole and, more importantly, does not correspond to anything that exists

Where are you getting this second part, about irrational numbers not corresponding to anything that exists? Or about Lambert proving that π does not correspond to anything that exists?

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For this reason, we are faced with (approximate) "circles" in our day to day lives, but in reality there are no perfect circles as Euclid described them.

This is true in the sense that perfect circles don't exist in the physical world, they're a mathematical abstraction. But when you say "there are only natural numbers in the universe", that would also mean that the number 1/3 doesn't exist in the universe (what is 1/3 of a hydrogen atom?), but that doesn't mean that it is irrational! In the same way an exact square doesn't exist in the physical universe.

Mathematical terms like "irrational", "imaginary", "transcendental" have nothing to do with the supposed "existence" or "non-existence" of such numbers. In that sense a "real" number is certainly not more "real" than an "irrational" number! However, those terms don't have any physical meaning, they are abstract concepts that have an exact meaning on the basis of mathematical axioms. In that sense they all do exist, just like geometrical constructs like squares and circles. You shouldn't confuse mathematics with physics, even if the first is used extensively in the second.

Edited by Tensorman

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<p>You had the meaning of &quot;irrational&quot; right the first time: can&#39;t be expressed as a ratio. It&#39;s not an insult. The utility of pi (and other irrational numbers) is that they allow for an arbitrary level of precision in calculating what you&#39;re trying to calculate. Just as &quot;real&quot; infinities don&#39;t exist in reality, neither do &quot;complete&quot; irrational numbers - but you can go to as many decimal places as you need.</p>

<p> </p>

<p>Given what you&#39;re trying to do, you might be interested in my comments on maths and reality here: http://www.monorealism.com/reflections/maths.html (especially part B under the heading &quot;Abstract Maths&quot;, where I discuss how the different types of numbers relate to what&#39;s in reality given their increasing levels of abstraction). For example, it is interesting that you only need 30 decimal places of pi to calculate the upper and lower bounds of the circumference of a circle drawn with the width of a hydrogen atom and the diameter of our galaxy (ignoring the physical impossibility of a &quot;perfect circle&quot; in space-time)!</p>

<div id="myEventWatcherDiv" style="display:none;"> </div>

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Let's try again using the full editor.

You had the meaning of "irrational" right the first time: can't be expressed as a ratio. It's not an insult. The utility of pi (and other irrational numbers) is that they allow for an arbitrary level of precision in calculating what you're trying to calculate. Just as "real" infinities don't exist in reality, neither do "complete" irrational numbers - but you can go to as many decimal places as you need.

Given what you're trying to do, you might be interested in my comments on maths and reality here: http://www.monoreali...ions/maths.html (especially part B under the heading "Abstract Maths", where I discuss how the different types of numbers relate to what's in reality given their increasing levels of abstraction). For example, it is interesting that you'd only need 30 decimal places of pi to calculate the upper and lower bounds of the circumference of a circle drawn with a line the width of a hydrogen atom at the diameter of our galaxy (ignoring the physical impossibility of a "perfect circle" in space-time)!

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Mathematical terms like "irrational", "imaginary", "transcendental" have nothing to do with the supposed "existence" or "non-existence" of such numbers. In that sense a "real" number is certainly not more "real" than an "irrational" number!

Furthermore, let's remember that the real numbers include the irrational numbers, which in turn include transcendental numbers (but not all of them, since complex [or imaginary] numbers may be transcendental).

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Many irrational numbers are constructible by straightedge and compass. Pi cannot but that doesn't mean that it cannot be constructed with additional tools. Even Zeno of Elea would have known that the square root of two is constructible.

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