How can this (math) be?

[tw]

Originally Posted by glatt

since a=b, then (a-b) = (a-a) = 0

It is called the trivial solution.
a * c = b * c
has a unique solution (for real numbers) EXCEPT when c = 0. We tend to forget the other part of that algebraic relationship: that c cannot equal 0. Once c becomes zero, then any number can equal any other.

Naive will then proclaim that math can be manipulated - another interpretation of "lie, damn lies, and statistics". Instead, by providing only a half fact (by forgetting the part where c cannot equal zero), then a mistake occurs.

c=0 is called the trivial solution; an overlooked mathematical error that glatt has successfully identifed.

[glatt]

Originally Posted by tw

an overlooked mathematical error that glatt has successfully identifed.

To be fair, Happy Monkey beat me to it. I just didn't see his solution because he hid it behind a white font color. It wasn't until Beestie quoted him later that I saw his entire post.

[BigV]

juuuussst catching up here, as my cellar reception in the mountains is also equal to zero... but I can honestly say I did observe the division by zero misdirection when trying to remove the factor "(a-b)". I'm not as articulate as Happy Monkey though. Nicely done. Good puzzle tw.

[Shocker]

Here's another for you all to ponder... from http://www.evilmadscientist.com/article.php/SumTrick



Here is a cool math trick that shows that the sum of an infinite number of positive integers is equal to negative one.

Show that the infinite sum S = (1 + 2 + 4 + 8 + 16 + . . .) adds up to S = -1.
Given that S = (1 + 2 + 4 + 8 + 16 + . . .), if you multiply both sides by two, you get
2S = (2+ 4 + 8 + 16 + 32 + . . .).

Then, add one to both sides:
2S + 1 = 1 + (2 + 4 + 8 + 16 + 32 + . . .)
= 1 + 2 + 4 + 8 + 16 + 32 + . . . = S.

Thus, 2S + 1 = S.

To solve for S, subtract 1 from both sides:
2S = S - 1.

Finally, subtract S from both sides:
S = -1.

Isn't just amazing that you can add up so many positive numbers and get a negative answer?


Yes, it's a trick. I found it in the book Mathematical Methods in the Physical Sciences, by Mary L. Boas. Can you figure out why this actually doesn't work?

[Spexxvet]

Originally Posted by Shocker

Wow, those are really cool symbols.

[tw]

Originally Posted by Shocker

Given that S = (1 + 2 + 4 + 8 + 16 + . . .), if you multiply both sides by two, you get
2S = (2+ 4 + 8 + 16 + 32 + . . .).

Then, add one to both sides:
2S + 1 = 1 + (2 + 4 + 8 + 16 + 32 + . . .)
= 1 + 2 + 4 + 8 + 16 + 32 + . . . = S.

Thus, 2S + 1 = S.

At this point, we have an impossible inequality.

2∞ + 1 = ∞

Clearly that is not possible. However I fail to grasp the algebraic rule that was violated.

[Flint]

Infinity behaves strangely, though. :::goes digging through stack of science mags for neat infinity trick I read the other day:::

[Clodfobble]

Originally Posted by tw

2∞ + 1 = ∞

Clearly that is not possible.

It is by definition possible. Infinity is not a number, infinity plus one (or multiplied by two, or whatever) always equals infinity.

On the other hand, there is a theoretical number bigger than infinity, and IIRC it's called "aleph naught" (can't make the special characters show up, but it's written as the first letter in the Hebrew alphabet followed by a subscript zero.)

[Happy Monkey]

It's probably something to do with shifting one sigma over a term before comparing/adding them, but I don't remember my sigma math very well.

[tw]

Originally Posted by Clodfobble

It is by definition possible. Infinity is not a number, infinity plus one (or multiplied by two, or whatever) always equals infinity.

That is not exactly true.

∞ + 1 > ∞

Other strange things occur. For example a function divided by t does not become infinity as t approaches zero. It becomes an impulse of one. I don't remember exact details - this was many decades ago. But ∞ + 1 also is not same as ∞. ∞ + 1 is approximately ∞ which is good enough for calculations involving reality. But that is an approximation not valid for rigorous proofs or this algebraic solutions.

Of course, we can change an assumption. Same is accomplished in Euclidean geometry where two parallel lines never meet. We simply change some underlying principles (to create a different type of geometry) so that two parallel lines do meet at ∞. Suddenly the rules of that geometry change because we are using a completely new geometry (forgot the name of that geometry).

But we are using the domain of standard algebra.

So how does that S = -1 come about? Something in the equation before 2S + 1 = S is wrong because 2∞ + 1 = ∞ is wrong. There is apparently some restriction in algebraic rules used that I just don’t see. I just don't recognize the mistake - an overlooked restriction.

[dar512]

Originally Posted by tw

That is not exactly true.

∞ + 1 > ∞

That's not the way I learned it all those years ago in HS. I learned ∞ + 1 = ∞. These guys agree with me.

[tw]

Originally Posted by dar512

That's not the way I learned it all those years ago in HS. I learned ∞ + 1 = ∞. These guys agree with me.

I don't know what you read. But when I read their citation, your concept of ∞ does not agree with what "These guys" say.

Some quotes from that website are

If we want to say that infinity times infinity is bigger than infinity, then we have to show how the set with infinity-times-infinity members (the rational numbers) cannot be put into a one-to-one correspondence with the set that has an infinite number of members (the counting numbers). ...

Cantor's work revealed that there are hierarchies of ever-larger infinities.

and

These things are so obvious they seem silly. However, if we want to know the size of an unknown quantity, but the counting task is tricky, we can try to put the unknown quantity in one-to-one correspondence with some known quantity. This is the strategy that Georg Cantor used to compare different sizes of infinity.

Did you understand their story of Hotel Infinity?

We have two sets. Set A = {1, 2, 4, 8, 16, 32 ...}. Set B = {2, 4, 8, 16, 32, 64 ...}. Two examples of infinity. But to be equal, then 1 = 2; 2 = 4; 4 = 8; etc. Clearly they are not equal. IOW we have two different sizes of infinity.

But again, some defining condition in the original problem 1) is violated and 2) causes 2∞ + 1 = ∞ . I just don't see the algebraic mistake because I do not see the violated restriction.

Yes, ∞ + 1 = ∞. But they are not the same size ∞. Shall we talk about Schrodinger's Cat? It's a weird, weird, weird world. Fortunately, when it makes no sense, we can go out back and urinate on the bible. Then things change.

[tw]

Originally Posted by dar512

These guys

Appreciate from your post "Cantor Sets". I have struggled without success to make something useful from the concept (that I was never comfortable with). But Cantor Sets are fundamental to some concepts thta appear to have future, great, and useful purpose. I just cannot say how or why. But the concepts may underlay a whole new and useful concept in science and math. Or maybe it just still remains too much of a mystery to me - in which case I am only babbling uselessly.

Meanwhile it still does not explain Shocker's 'cool math trick'.

[Clodfobble]

Originally Posted by tw

We have two sets. Set A = {1, 2, 4, 8, 16, 32 ...}. Set B = {2, 4, 8, 16, 32, 64 ...}. Two examples of infinity. But to be equal, then 1 = 2; 2 = 4; 4 = 8; etc. Clearly they are not equal. IOW we have two different sizes of infinity.

No, to be equal in size, there must be a one-to-one correlation between terms, 1 --> 2, 2 --> 4, 4 --> 8, etc. The infinte sets you listed are equal in size. Doesn't matter that their terms are different, for every next term in set A there will be exactly on next term in set B.

An example of a larger infinity would be a "set of sets," i.e, {Set A, Set B, Set C...} where each set is also infinite. However, and here's where it gets wacky, this set can be the same size as a normal infinte set if you count it the right way.

Think of it as a grid like the first picture below. If you start counting down one column, you'll be going downward for infinity and never get to column B. BUT, if you count back and forth along the diagonals as shown in the second picture, you can reach infinity in both the horizontal and vertical directions. Thus the set of all infinite sets does have a one-to-one correlation with an infinite set as long as you count it this particular way.

This is the same as the part of the Hotel Infinity story where an infinite nunbedr of buses arrive each with an infinite number of people, assuming the website is using the same allegory my professor used.

[lumberjim]

Originally Posted by tw

- in which case I am only babbling uselessly.

ah hah!