Non integer square roots and pi = irrational?

[|Orion's Thought|]

No, actually. This is a mathematics forum, and we do our best to keep it that way.

Cmon Hurkyl, you know what I mean, of course it should pertain to mathematics!

 

[Leonardo Sidis]

And the right answer is: there exist points on a Euclidean line separated by an irrational distance, which is what we've determined the OP meant by "plot". Even if you meant something else, such as "can I construct a line segment of irrational length with a compass and straightedge?", the answer is still yes.

Thank you for offering your answer to my question. I do still have some confusion as to how this is possible. I know that if I asked you this, you might say that I have shown how this is possible in my previous posts about drawing triangles. But my confusion lies in how anything can have an irrational length, because it would seem that if the decimal representation of a number didn't repeat or terminate, then another representation of it, such as a physical measurement, would not be possible. By posting this I do not mean to start an argument; it is only to say that I still don't understand. Of course it may be true that I never will understand, not because of my lack of intelligence or perception, but perhaps because of my possibly different viewpoint on the matter.

 

[ramsey2879]

Thank you for offering your answer to my question. I do still have some confusion as to how this is possible. I know that if I asked you this, you might say that I have shown how this is possible in my previous posts about drawing triangles. But my confusion lies in how anything can have an irrational length, because it would seem that if the decimal representation of a number didn't repeat or terminate, then another representation of it, such as a physical measurement, would not be possible. By posting this I do not mean to start an argument; it is only to say that I still don't understand. Of course it may be true that I never will understand, not because of my lack of intelligence or perception, but perhaps because of my possibly different viewpoint on the matter.

Just a note here that might help. As I tried to explain before mathematics is a precise science as it follows precise rules, but mathematics doesn't necessarily agree with the real world. One rule of mathematics is that any quantity can be divided in half no matter how small. Think about that. In mathematics, there is no such thing as a smallest element! In mathematics a point in space is just a point, it has no size. An infinite number of points exist on a line of any length because if you start with two end points, Call them 0 and 1. The length of the line does not matter for we just defined that length to be one. What is beautiful about mathematics is that you can express the following proceedure in simple mathematical terms. Locate and plot the point midway between the first two points on this line. If you can't believe that this can be done repeatedly without end, that is not a problem. All you need to know is that it is not necessary for the rules of mathematics to agree with the real world at all times. All you need to do is imagine that it can be done then try to find an expression for the length of the smallest segment after the nth time that the above procedure is preformed and then determine how to manipulate or use the expression to solve a problem. 1/2, 1/4, 1/8, 1/16 ... . Now each denominator is simply the previous denominator multiplied by two. Now comes the following which you may find hard to grasp, but it is a fact that no matter how many points that you plot on this line between 0 and 1, you still haven't covered billionth of the line, since the points that you plotted each have no real size. But let's place a small dose of reality on this proceedure and limit n to be a number such that $$2^n$$ is less than the number of atoms between the points and suppose that we know that there are about 2 trillion atoms between 0 and 1. Does the mathematical logic allow you to calculate the number of atoms completely in the smallest line segment? The answer is yes! Similarly still other rules can be set forth by using mathematical expressions and these expressions can be manipulated mathematically to determine solutions to any problem that has a mathematical solution. Once you can understand the logic of this, it should not be too hard to understand that both rational and irrational points exist between any two rational points on a number line. It might by helpful to note that I consider the distance between any rational point and its nearest irrational point to be zero and that this can be proven mathematically.

 

[Hurkyl]

Some nitpicks, for the sake of precision!

One rule of mathematics is that any quantity can be divided in half no matter how small.

Where quantity means something from a structure like the reals or rationals, but not the integers.

Think about that. In mathematics, there is no such thing as a smallest element!

When speaking about things like the reals, rationals, integers, positive reals, or positive rationals, but not the natural numbers or the nonnegative reals.

An infinite number of points exist on a line of any length

When speaking about things like Euclidean space, but not geometries over a finite field.

I consider the distance between any rational point and its nearest irrational point to be zero and that this can be proven mathematically.

And this... is just wrong. I can't think of any context where this makes any sense. (I.E. I cannot think of any context in which there is a rational point that has a "nearest irrational point")

Theorem: Let a and b be any two distinct real numbers. Then there exists a rational number c and an irrational number d such that a < c < b and a < d < b.

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But the spirit of the post is correct; mathematics is just mathematics, and mathematicians make no attempt to capture reality. It is the physicists who are attempting to capture reality, and they are the ones that pick and choose which mathematical structures they want to use, and how those particular structures will apply to reality.

Of course, sometimes physicists say "I need a structure that looks like this!" and the mathematicians will try to oblige, because it's fun to devise new structures. Sometimes we've already created the thing and can tell them how it works. Other times, the physicists just have faith that it will all work out in the end and start using the structure before we've invented it. (I wish I could say they had faith in mathematicians )

 

[Leonardo Sidis]

This brings to mind one of Zeno's Paradoxes. "The Dichotomy Paradox"

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as: ...1/16, 1/8, 1/4, 1/2, 1

This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time -- which is to say, it can never be completed. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even be begun. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox.

(Above is from Wikipedia)

I had a discussion with my math teacher about this and she said that it is possible to travel from point A to point B because we have mass. Besides this explanation, and without calculus, I can't think of any other way to "disprove" it. Is it wrong because: in order to travel a distance you must travel in time, and therefore to divide distance, you must divide time. But time is a continuum and cannot be so precisely divided? Is this wrong? How else can it be disproven?

 

[matt grime]

By going from A to B. That would seem to disprove the rationale behind the argument most succintly. Of course the mathematical explanation is that the sum of that GP is finite, and all the paradox is saying is that if you need to travel for n seconds/miles to reach a place you don't get there in fewer than n seconds/miles.

 

[Leonardo Sidis]

By going from A to B. That would seem to disprove the rationale behind the argument most succintly.

I know that one can travel from A to B, but this doesn't prove why. How can one travel from A to B? Are any of the reasons I said above right?

all the paradox is saying is that if you need to travel for n seconds/miles to reach a place you don't get there in fewer than n seconds/miles.

I think the paradox is saying that motion is impossible, since it says one cannot begin to move, or complete a journey.

 

[matt grime]

I know that one can travel from A to B, but this doesn't prove why. How can one travel from A to B?

By catching a bus? perhaps walking, tube, car, taxi, bike might be better.

 

[Leonardo Sidis]

By catching a bus? perhaps walking, tube, car, taxi, bike might be better.

C'mon matt...

 

[ramsey2879]

Some nitpicks, for the sake of precision!

"In mathematics, any Quantity can be divided in half no matter how small" ...
Where quantity means something from a structure like the reals or rationals, but not the integers.

Only if you maintain that the quantity must keep its identity

"There is no such thing as a smallest element" ...
When speaking about things like the reals, rationals, integers, positive reals, or positive rationals, but not the natural numbers or the nonnegative reals.

I meant to say that there is no smallest non-negative real > 0

"I consider the distance between any rational point and its nearest irrational point to be zero and that this can be proven mathematically" And this... is just wrong. I can't think of any context where this makes any sense. (I.E. I cannot think of any context in which there is a rational point that has a "nearest irrational point")

I should have said that an $$irrational number$$ can be calculated to such a point that the difference between it and and the approximated rational number can be deemed to be zero without loss of significance. In the calculation of the approximate decimal value of an irrational point one will arrived at a point in which there is no useful purpose in continuing the calculation.
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The key to parodoxes of infinite series etc is to understand how to determine the limit of a convergent infinite series, especially, where the series is a series of partial sums. See http://mathworld.wolfram.com/ConvergentSeries.html

 

[matt grime]

C'mon matt...

I'm giving the question exactly as much seriousness as it merits, mathematically.

 

[ramsey2879]

This brings to mind one of Zeno's Paradoxes. "The Dichotomy Paradox"

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

The resulting sequence can be represented as: ...1/16, 1/8, 1/4, 1/2, 1

This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time -- which is to say, it can never be completed. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even be begun. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox.

(Above is from Wikipedia)

I had a discussion with my math teacher about this and she said that it is possible to travel from point A to point B because we have mass. Besides this explanation, and without calculus, I can't think of any other way to "disprove" it. Is it wrong because: in order to travel a distance you must travel in time, and therefore to divide distance, you must divide time. But time is a continuum and cannot be so precisely divided? Is this wrong? How else can it be disproven?

I believe that you can get an answer to these parodoxes by searching the internet and I rather that you do that than brother us with the question. As you indicated above the answer is to understand calculus , etc; so forums on those topics would be more appropriate. Try for instance http://www.artofproblemsolving.com/Forum/index.php?f=149

 

[Hurkyl]

This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time

Really? I've never seen the point argued... merely assumed.

 

[ramsey2879]

This brings to mind one of Zeno's Paradoxes. "The Dichotomy Paradox"

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal." Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

I had a discussion with my math teacher about this and she said that it is possible to travel from point A to point B because we have mass. Besides this explanation, and without calculus, I can't think of any other way to "disprove" it. Is it wrong because: in order to travel a distance you must travel in time, and therefore to divide distance, you must divide time. But time is a continuum and cannot be so precisely divided? Is this wrong? How else can it be disproven?

If you add all these partial distances as well as the time to reach them, the sum of these infinite series is still a finite number, so there is in fact no paradox. It is morover merely an imaginary illusion to say that you are prevented from moving that is worth no comment. The mathematics is valid, you just need to study the theory of convergent series to understand it. This is more appropriate in another forum.

 

[Leonardo Sidis]

If you add all these partial distances as well as the time to reach them, the sum of these infinite series is still a finite number, so there is in fact no paradox.

Actually, your previous post made this relate specifically to this thread. One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach. This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

 

[Hurkyl]

"In mathematics, any Quantity can be divided in half no matter how small" ...
Where quantity means something from a structure like the reals or rationals, but not the integers.

Only if you maintain that the quantity must keep its identity

I only maintain that "x/2" be defined for one to be able to say that the quantity x can be divided in half.

1/2 is not defined, when we're working over the integers.

Let me reiterate. When working over the integers, it's not that 1/2 is something that isn't an integer: it's that 1/2 has no meaning at all.


Or, if you prefer that "divided in half" means that there exists a thing y, such that y + y = x, then 1 still cannot be "divided in half", because there does not exist a thing y such that y + y = 1. (the domain of + is the integers -- anything that is not an integer cannot be plugged into +)

One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach.

Of course one can -- it's a standard exercise in calculus classes, and is often done even earlier with things like geometric series.

The sum of an infinite series is exactly the limit of the sequence of partial sums. Nothing more, nothing less.

This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

The problem with the reasoning is that it's always presented as a non sequitor -- there is nothing that even resembles a deductive chain of reasoning from "there are an infinite amount of finite distances" to "you cannot travel".

 

[HallsofIvy]

Actually, your previous post made this relate specifically to this thread. One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach. This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

If that is the reasoning behind the paradox, it's no wonder you get a paradox! One certainly can find the exact sum of an infinite series of numbers. For example the sum 1+ 1/3+ 1/9+ 1/27+ ... is exactly 1.5. That is not a "finite number that it may approach". I'm not certain what you mean by "it" here. If you mean the sum, it is not approaching anything, it is 1.5. If you mean the sequence of partial sums (which is what many people mean when they talk about something like this), there is no "may" that sequence is approaching that number and so sum is, by definition, 1.5.

 

[matt grime]

Actually, your previous post made this relate specifically to this thread. One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach. This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

 

[Leonardo Sidis]

I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.

-Leonardo da Vinci

 

[HallsofIvy]

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.

-Leonardo da Vinci

Would you mind pointing out where, in this thread, there was an "appeal to authority"?

(as opposed to stating standard definitions.)