The Black Vault

at1with0 wrote:

The ability to use numbers is so basic to our existence that we rarely realize how sophisticated our number system is. It was not always so: early number systems were crude and cumbersome. Clawson takes us on a mathematical adventure that reveals the history of numbers as a reflection of the evolution of culture. He shows how this science was born out of necessity in agriculture and commerce, not out of virtue. As our technology progressed, so did our math, and from the Chinese, Mayans, and Greeks came new numerical concepts and increased abstractions. The views and discoveries of Pythagoras, Descartes, Gauss, and perhaps a dozen more heavy-hitters are discussed, with Clawson maintaining a sense of humor to keep it enjoyable. Even those with little knowledge of formal mathematics will find the first half of this book easy going; it gets a little sticky later, delving into irrational, infinite, and "really big" transfinite numbers. The text becomes pure philosophy at this point, but if the reader can stick with it, the experience will be worthwhile. David Siegfried --This text refers to an out of print or unavailable edition of this title.

Thanks at1

And that is certainly what my quasi-mathematical argument eventually boils down to. Philosophy!

I hope you all realize I haven't gone mad; I do realize it makes no sense that, for example, pi = infinity. Sure, it's an ever-growing number with no end, so technically, or at least philosophically it could be described as infinity.

Greeney mentioned there is no way infinity can exist between 2 whole numbers.

But just for fun let me postulate this: Infinity is stuck between 2 whole numbers. Infinity, just like any other number really, is a number that exists between: zero and (infinity + 1)

And orangetom, my brother with the same name... this stuffwe are discussing, because it does really boil down to a philosophical discussion about the nature of our reality, is actually relevant and belongs in the spiritual forum. So what's the big deal, kind sir.

And correct me if I'm wrong, but, I don't think the discussion has anything to do with public education. I can personally attest to the fact that I was never, ever taught anything like this in public school.

humphreys wrote:

at1with0 wrote:Between two points there are infinitely many points in between and that is what I think is causing Frosty to assert that the distance between two points is infinite.

If two points are in exactly the same position, they should not be considered two separate points, do you agree?

In which case, there must be a finite non-zero distance between every two points, yes?

Any non-zero number multiplied by infinity must equal infinity?

In which case, if there are an infinite number of points, the total distance must also be infinite. The total distance cannot actually be infinite, therefore there cannot be an infinite number of points, as far as I can see. Again, we have to accept the minimum size plank length to get around such paradoxes.

You say the total distance cannot be infinite. But, technically it is. It's been demonstrated.

I think you touched on the issue here. Perhaps our mathematical system just isn't quite good enough to tackle the problem. Alternatively, perhaps the system itself isn't flawed whatsoever; It's only when we try to physically follow through with a demonstration that the issue arises. As long as the math is done in our heads, we're good. Once we get a tape measure or a compass or a pencil and graph paper out, that's when we run into technical problems the naked eye isn't aware of... so they are dismissed.

Another question to ponder.

These types of discussions are generally thought to be held only when people are high. So I have another curiosity. Why is it that generally, when people are high, we start to think about and get caught up in technicalities like this but, when we are straight and sober, we ignore them completely?

It's as if the discussion here makes people uncomfortable and the use of drugs somehow diminishes our inhibitions. Me? I guess I'm on a permanent, natural high because I think about this stuff all the time with no drug use.

Pi is what is called an irrational real number. That implies that its decimal representation does not "end" in a repeated sequence of numbers. I think I see why you say Pi is infinite. Pi is approximately 3.141592654.

Every real number has a decimal representation. What does a decimal representation mean? Let's take an easy example of 0.5. 0.5 is actually a sum of 0 x 1 plus 5 x 1/10 which is 5/10 which reduces to 1/2.

A little harder of an example: 12.34. 12.34 is defined to be the sum 1 x10 + 2x1 + 3 x1/10 + 4 x1/100.

Notice that the positive powers of ten are 10, 100, 1000, 10,000, 100,000, etc.
The non-positive powers of ten are 1, 1/10, 1/100, 1/1000, 1/10,000, 1/100,000, etc. The non-positive powers of ten are reciprocals of the positive powers of ten.

In a decimal representation, there is always, by definition, a sum of terms in which each term is a number 0 through 9 times a power of ten.

Now the sticky issue is what happens with a real number that does not have a terminating representation. Let's take a simple example of 1/3. The decimal representation of 1/3 is 0.3333333333333333333333......

The dots at the end are my way of saying that the digit 3 is repeated. To write 1/3 as a sum you would have to have infinitely many terms:
1/3 = 3 x 1/10 + 3 x 1/100 + 3 x 1/1000 + 3 x 1/10,000 + 3 x 1/100,000 +....

Now this is no ordinary sum, it is what's called an infinite series.

At this point it might be good to consult
http://en.wikipedia.org/wiki/Decimal_expansion



That is the notation for a generic infinite series corresponding to the real number r whose digits are {a0, a1, a2, a3, ...}.

In a decimal representation, there is always, by definition, an infinite series of terms in which each term is a number 0 through 9 times a power of ten. For a terminating real number like 12.34, it is still an infinite series it's just that after the 4, all digits are zero.

Now the reason I think you are saying Pi is infinite is that Pi has a decimal representation which is an infinite series. You're probably thinking that an infinite series must have an infinite result. Am I right? That's not a bad intuition. I will stop here and make sure we're on the same page.

As a slight clue, what is the "sum" of an infinite series in which all terms are zero? If you "add" infinitely many zeros, what is the total? 0 + 0 + 0 + 0 +.... = ? This example is meant to indicate that an infinite series doesn't necessarily sum to infinity. Where we need to look at is an infinite series of the form given in the picture I included above.

at1with0 wrote:As a slight clue, what is the "sum" of an infinite series in which all terms are zero? If you "add" infinitely many zeros, what is the total? 0 + 0 + 0 + 0 +.... = ? This example is meant to indicate that an infinite series doesn't necessarily sum to infinity. Where we need to look at is an infinite series of the form given in the picture I included above.

The problem with that example is, it doesn't represent, nor could it possibly represent, a number that grows infinitely.

The only way to conclude that 1.33333333333...... is not infinity is to admit that there is an infinite amount of "numerical space" between 1.33 and 1.34 ... which again means infinity exists between 2 (whole, or rational, it doesn't matter) numbers.

Frrosty I think you are confusing yourself with the decimal point. To the right is the fraction of only one whole number, forinstance Pi the whole number is 3.xxxxxxxxxx, the x's being what part of the next whole number, between 3 and 4. That fractional decimal may continue into exceedingly smaller parts, but in will never achieve the next whole number. So pie is greater than 3 and lesss than 4, which can not be infinity. Infinity must be expressed to the left of the decimal point, becasue infinity is the highest whole number + 1. Pi can not be infinity, becasue we all know there is a whole number called 4. Math also has the rules of rounding and rules of repeating the same decimal like .333 or .666. What is zero? 0.0000000000000000000000000000 does that mean nothing = infinity?

I watched about 20 minutes of the video which is 1 1/2 hours, and it got late, but seemed interesting. Only saw the first 2 speakers so far.

frrostedman wrote:

at1with0 wrote:As a slight clue, what is the "sum" of an infinite series in which all terms are zero? If you "add" infinitely many zeros, what is the total? 0 + 0 + 0 + 0 +.... = ? This example is meant to indicate that an infinite series doesn't necessarily sum to infinity. Where we need to look at is an infinite series of the form given in the picture I included above.

The problem with that example is, it doesn't represent, nor could it possibly represent, a number that grows infinitely.

The only way to conclude that 1.33333333333...... is not infinity is to admit that there is an infinite amount of "numerical space" between 1.33 and 1.34 ... which again means infinity exists between 2 (whole, or rational, it doesn't matter) numbers.

That's an odd way to put it because Pi doesn't grow at all. Successive approximations to Pi do grow as they become more refined.

Greeney had a good point 0 = 0.000000000000000000000......(repeating) does that equal infinity?

There are infinitely many numbers between 1.33 and 1.34. In fact, no one can provide the first number after 1.33. But 1.33333333333... is not infinite mainly because the infinite series the notation represents converges rather than diverges.

What might be the most important point is that while many infinite series diverge to infinity, many also converge to a finite number..The intuition being that if what is being added is miniscule enough, then the successive terms contribute almost nothing. For example, 1.3333333... is what is known as a limit of a sequence of successive approximations {1, 1.3, 1.33, 1.333, 1.3333, 1.33333, ...}. As every term in this sequence is less than 1.34, there is a finite limit that won't exceed 1.34. The limit of that sequence is 1.3333333...

This kind of stuff was done by Archimedes who used limits to find the area of a circle by taking a limit of better and better approximations to a circle with inscribed polygons whose formulas were known from Euclidean geometry. He didn't call them limits of course.

Believe it or not, this stuff is normally taught in second or third semester Calculus... So secretly, we've been talking a lot about calculus.

You're the expert so I will bow out on the point, but, technically speaking if a number increases in value and never stops, that means it grows infinitely and that can equal nothing but infinity.

Ok I'll stop. You win. I begrudgingly hand you the trophy.