A Math Problem

I think the Taylor Series is supposed to represent polynomials as a series. Not sure if his name is on other stuff.

No, the Taylor expansion can be used to write any function as a series of powers. The elements of a taylor series ARE polynomials, so we would not ned to use a Taylor series t represent a polynomial.  They are more used to represent transcendental functions like sine cosine, e^x, etc.

Those who said that .9999....= 1  are, of course correct.  Intuitively here is one way to look at it:

What happens if you subtract:

1.00000000

-.99999999

You will borrow from the 1, and all of the decimal places in the first number will b 9 except the last place, which is 10.  And so the answer is a series of zeros with a 1 in the last place.

When we now shift to the repeating .99... , if you think about doing the same thing what happens is that you never get to the place value that would get the 10, and hence there is no place value that can have the 1.  So the answer is just a series of zeros.

More formally, it is almost trivially easy to do a delta-epsilon limit proof that the limit of the sum of 9*10^-n for positive integer n's as n approaches infinity is 1.  Because no matter how small a non-zero number you can name, I can name a place value which would make the difference between 1 and the sum smaller than your number.  And that is the essential definition of a limit.

People sometimes get hung up on the fact that the difference between the limit and 1 gets as close to 0 as we want but never "gets there".  But that is a limitation on our ability to grasp something abstractly that never occurs in nature, i.e, infinity.

There is something called the Hilbert Hotel "paradox" about a hotel with an countably infinite series of hotel rooms that is full.  A countably infinite number of new guests arrive.  All full up, right?  Nope.  Move the guest in #1 to #2.  Move the guest in #2 to #4, Move the guest in #3 to #6.  Continue moving each original guest from room n to room 2n.  Now all of the odd numbered rooms are empty.  Since there are a countably infinite number of them and there are a countably infinite number of new guest we can accommodate everyone.

Another weird infinity quirk, this time with an infinity that is not countable, is that we can prove (trivially, once you know how) that every pair of line segments has exactly the same number of points in each segment.  Yup, this "-" has exactly the same number of points on it as a 93,000,000 mile line going from the Earth to the Sun.  This is another one that is both completely non-intuitive yet so easy to demonstrate.  Intuition is a bad guide in the world of infinities.  Like who would have imagined that there are the same number of integers as there are rational numbers, relying on intuition?

Yeah, that is the way we taught the kids to convert between infinitely repeating decimals and fractions.  The problem is that while it demonstrates something that is true, it is not really a proof.  It is not a proof because it intrinsically assumes that you can extend concepts and operations to infinite things that you have really only proven for finite things.  [generic you, not Erik you]

In your demonstration the unproven point is that the .9999... at the end of the 99.999...  is equal to the .999... you started with.  It is intuitively clear but you really are assuming that arithmetic operation on infinite decimals work the same as on finite numbers.  It turns out that largely they do, but that is something that technically needs to be proven, not assumed.  But of course it still works wonderfully as a demonstration, and only a math geek would quibble even the slightest amount.

The study of infinities and infinite sets was an area of math I particularly enjoyed back in the day.

Even many mathematically inclined people have a bad intuition for the infinite. In my favorite class to teach, one of my favorite lectures to give is where we introduce infinite cardinal numbers. Showing that the set of natural numbers is infinite doesn't surprise students. Showing that integers are equal in size to that surprises some. Then showing that the set of rationals is equal in size surprises some, many of whom want to object (and are welcome to do so during lecture, of course): it seems so obvious that the naturals are strictly smaller than the rationals, but they aren't. And yet, they're equal in size. And then showing that the real numbers are strictly larger than the naturals... I love looking out at the classroom during the proof and seeing the looks on their faces. The material isn't tough; they all follow the lecture. But the consequence of the proof gets them.

Yeah. I lost interest when I realized that this had nothing to do with a quick way of adding numbers in relation to par. Besides, I'm only working on a few hours of sleep after watching OSU pull off a major upset against #1 ranked Bama. Sorry. I had to slip that in.

Slip noted and added to my math for next time!