The Shrine

This is something that always bothered me. The most common explanation would be something either take .9(repeating) and multiply it by 10, subtract itself, and then divide by 9, or take .3(repeating) times it 3 which is equal to (1/3) * 3 = 1. Now I know since the ultimate result is true it is clearly safe to do this, but I believe Algebra is not supposed to deal with infinite numbers, so you're basically implying infinite quantites can be manipulated like a number. To me this is backwards. The only way an infinite string of 9s can be manipulated like a '1' is because by calculus we know that above number converges to 1 so we can manipulate that particular instance of infinite 9s. But then this doesn't mean you can always manipulate infinity as if it's just equal to X. Here's a case where you can't manipulate infinity.

1 + -1 + 1 + -1 + ... = 1 + -1 + 1 + -1 + ...
addition is associative, so ordering of parenthese doesn't matter
(1+-1) + (1+-1) + ... = 1 + (-1+1) + (-1+1) + ...
0 + 0 + ... = 1 + 0 + 0 + ...
0 = 1

The problem with the above proof is that the sum above is divergent and you can't just put paranetheses around that. I realize those two aren't exactly the same thing but they're both 'infinite' and are both representable as an infinite series. It seems to me the simple algebra explanation cheats in that it already knows the result converges so you can manipulate it like a number, but it's not a rigorous proof. Even if you use the geometric series, you'd still need calculus to argue that r^n = 0 if r < 1 limit n->infinity. I don't expect a rigorous background in calculus for anyone seeing this problem, but I think it should be pointed out at some point that it is because advanced mathematics exist so we know that some numbers converges to a finite number instead of divergent, which is why you can just manipulate it like a number instead of getting crazy results like the one I wrote above.

I used infinity as a number in calculus, even though I wasn't supposed to. I just wrote some rules that were based on logic and common sense, and used zero as the basis for the infinity rules. Then I said that the number line was looped. Even used the "looped number line" as a basis for a science fair thing.

Frankly, I think the whole basis for calculus is mostly ridiculous. It's all about workaround around this division by zero thing because we are too blinded to just go out and say that infinity is a number. We suddenly created zero, so why can't we just do the same thing with infinity.

BTW, here were some of the rules I had. (Using I for infinity, and X for anything except 0 or I.)

0 * I = every number at the same time (this was a clue that you needed calculus to fill in the "hole")
X * I = I
X * 0 = 0
X / 0 = I
X / I = 0
0 / 0 = 0 * I = every number
I / I = I * 0 = every number
0 + 1 = a positive number close to zero
I + 1 = a negative number close to infinity

I looked at it in terms of magnetic physics, especially for multiplicative properties. If an infinitely powerful magnet was at the left side (zero) of a perfect circular tube, and a piece of metal was at the top (positive one), then the piece of metal would get pulled to the magnet (0*1=0). if two infinitely powerful magnets were on the left-right ends of the tube (0*I) and exactly 180 degrees from each other, either the tube would shatter to pieces or the magnets would rip themselves apart (0*I=every number).

Stuff like "holes in graphs" and a TAN(x) graph bother me. There are no "holes in graphs". It's just missing information that needs to be filled in some other manner.

If you think the basis for calculus is ridiculous, then you think that the basis of all Newtonian physics is also ridiculous. He invented calculus so that he could describe objects in motion.

You don't need calculus to convince yourself that 0.9 repeating is 1. All distinct real numbers have an infinite set of numbers between them. 0.9 repeating and 1 do not have any real numbers between them. Hence, 0.9 repeating and 1 must be the same number. It's not intuitive because the concept (yes, concept, not number) of infinity is not something we're wired to have an intuition for.

If you're going to say that .9 repeating = 1 because there are no numbers between them you might as well say you defined 1 as .9 repeating. Again I think you're working backwards, that because you know .9 repeating is actually 1 that's why you can be sure there are no real numbers between them. Yes you can ask "Then what is the number between 1 and .9 repeating" and obviously I can't think of any, but just because you can't think of a counterexample doesn't preclude its existence. This isn't a proof. If you want to come up with a proof on why there can be no real number between .9 repeating and 1 it's probably just as complicated as using calculus, unless you want to say those two are defined as the same number. It'll probably look like "Suppose there's a real number between .9 repeating and 1, then this number must satisfy some property that defies the definition of a real number".

In response to Sine's point, clearly Calculus exists so that you can actually do stuff with infinity, like x^2/x lim x->infinity is equal to infinity, because infinity^2/infinity = infinity. But sometimes you also get dangerous stuff like a divergent infinite series, such as the 1+ (-1) + 1 + (-1) + ... example so you need Calculus to tell you which infinity you can't mess around with like a number.

Kupek wrote:If you think the basis for calculus is ridiculous, then you think that the basis of all Newtonian physics is also ridiculous. He invented calculus so that he could describe objects in motion.

Hey, I'm not saying you don't NEED calculus, but the reason why it's needed is ridiculous. And using it for planetary motion fits: there's a lot of infinite variables buried in planetary motion. However, Calculus takes all of the logical-based arguments we learned for 12 years, and throws it all out the window. Calculus is needed, but the concepts around it are dancing around the obvious without actually saying it.

Kupek wrote:You don't need calculus to convince yourself that 0.9 repeating is 1. All distinct real numbers have an infinite set of numbers between them. 0.9 repeating and 1 do not have any real numbers between them. Hence, 0.9 repeating and 1 must be the same number. It's not intuitive because the concept (yes, concept, not number) of infinity is not something we're wired to have an intuition for.

Errr, just because 0.9 repeating and 1 do not have any real numbers between them doesn't mean they are the same number. That argument is flawed. The numbers 1 and 2 don't have any integer numbers between them, but they aren't the same number. And actually, the number in-between 0.9r and 1 is 0.0r1. That's what you get when you subtract 0.9r from 1.

The issue is that 0.9 repeating was never the answer in the first place. It's just the best thing we could come up with in our limited number system. 3/3 = 1. It will always equal one. 1/3 + 1/3 + 1/3 = 1 and 2/3 = 2/3 != 0.6 repeating. No, it doesn't equal that. Once we convert it to decimal, we are making the number inaccurate. The best way to represent it is in fractions.

I think by the definition for real numbers (not integers) if there are no real numbers between two real numbers then they're by definition the same real number. However how do you know there are no real numbers between 0.9 repeating and 1? Have you written down every one of them to check nothing is in between them? Sure you can say clearly there are no real numbers between them but then you can say clearly in the Monty Hall problem your chance of getting the car is the same whether you switch or not and that's wrong.

The only reason you know there are no real numbers between those two entites is because you already know .9 repeating is actually 1, but you can't use that fact to prove the same thing. If you're making an argument based on real numbers then you've to start assuming there are some real numbers between .9 repeating and 1 and then show that it is not possible.

The problem I have isn't the answer to these things but that explanation often boil down to 'trust me, this is right'. Monty Hall has the same problem as a lot of time you end up having to tell people 'trust me the odds of switching is 2/3' which is why a lot of people still never get it. Explaining this stuff is actually pretty hard and I don't like to see people handwave over it.

That there must be a number between two distinct real numbers is a fact of the real numbers; there's no proof behind it. It's a property, a given. The real numbers can be thought of as existing on a line. If there are no points on a line between two points, then they must be the same point.

Sine, you're confusing the real numbers and the integers. Pi is a real number, but pi is not an integer. It's not even a rational number. Integers are whole numbers. Rational numbers are numbers that can be expressed as the division of integers (such as 3/4 or 9/8) - note that all integers are rational numbers since you can simply express them as x/1. Real numbers are all non-imaginary numbers (no i part) - which includes the rationals and integers, of course. If a number is a real number but not a rational number, then we say it is an irrational number. Pi, for example, is an irrational real number.

Perhaps this is enough to demonstrate that you don't know as much about numbers as you thought you did. And if that's true, then perhaps your intuition will lead you astray.

Also, Newtonian mechanics can be used to express planetary motion, but it is also used for things like throwing a ball. In other words, it's the most basic physics that we do. Newton invented calculus (although we use Leibniz's notation) so that he could express "an object's velocity is its rate of change in distance with respect to time" mathematically. In calculus, we would say that v = dx/dt, where v is the velocity, x is distance, and t is time. It means that velocity is the first derivative of distance with respect to time. Limits come up because we want instantaneous values. That is, we want to consider what happens when the change in time (dt) approaches zero. Notice that while we're talking about limits, we're not talking about infinity.

How do you know there is no real number between .9 repeating and 1? Because you can't think of any? That's not a valid proof. You might as well say I defined .9 repeating as 1. If you're not required to show a proof on why there can be no number between those two I can easily say there must be a number between those two without showing a proof for that either, because it's 'clearly true'.

By definition of what it is, 0.9 repeating has an infinite number of 9s. There is no digit greater than 9, and there is "no room" for anything between that infinite sequence and 1.

If you want more formalism, then here you go: http://news.ycombinator.com/item?id=692648

Consider:

Proof:
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus:
x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

Kupek wrote:Also, Newtonian mechanics can be used to express planetary motion, but it is also used for things like throwing a ball. In other words, it's the most basic physics that we do. Newton invented calculus (although we use Leibniz's notation) so that he could express "an object's velocity is its rate of change in distance with respect to time" mathematically. In calculus, we would say that v = dx/dt, where v is the velocity, x is distance, and t is time. It means that velocity is the first derivative of distance with respect to time. Limits come up because we want instantaneous values. That is, we want to consider what happens when the change in time (dt) approaches zero. Notice that while we're talking about limits, we're not talking about infinity.

When you're talking about division over zero, you're talking about infinity. Period. The only reason for the "limits" or "when a number approaches zero" is because we don't have an expression for x/0. We would never say "when a number approaches one" because we can already solve that algebraically.

Kupek wrote:Sine, you're confusing the real numbers and the integers. Pi is a real number, but pi is not an integer. It's not even a rational number. Integers are whole numbers. Rational numbers are numbers that can be expressed as the division of integers (such as 3/4 or 9/8) - note that all integers are rational numbers since you can simply express them as x/1. Real numbers are all non-imaginary numbers (no i part) - which includes the rationals and integers, of course. If a number is a real number but not a rational number, then we say it is an irrational number. Pi, for example, is an irrational real number.

Perhaps this is enough to demonstrate that you don't know as much about numbers as you thought you did. And if that's true, then perhaps your intuition will lead you astray.

Uhh, no, I'm not confused. I know all this. This is eighth grade math. Don't insult my intelligence by declaring that I don't know the definition of a irrational real number, or that I'm somehow confused between real numbers and integers.

I was pointing out that your explanation of "Hence, 0.9 repeating and 1 must be the same number" is flawed, and I used an integer based system as an example. You still haven't said anything about 0.0r1, either, which also disproves your theory.

My point is that infinite decimal representations (like pi or 2/3 or 1/7) cannot be accurate expressed as decimals. They become inaccurate because of the limits of the decimal system itself. Yes, it's important to understand the numbers behind those fractions or numbers, but also, it must be understood that the decimal system is not perfect and can break.

Fractions aren't perfect, either. Irrational numbers are a good example of this. The most accurate way to illustrate pi is "pi". That representation has all of the digits down to the most infinite point. Once you try to actually represent that on even a fractional system, the number is flawed. It is no longer "pi", but a close approximation to it.

Mathematics is perfect. The systems we use to prove or illustrate mathematics is flawed.

Take Euler's Identity: e^(i*pi) + 1 = 0. This is the most brilliant formula in mathematics, yet it's the mathematical equivalent of shooting a bullet into a black hole and somehow getting a response. Imaginary numbers is this black hole in our field of view. It is the antimatter of our mathematical system. Yet, somehow, we magically created another identity for i, besides just the sqrt(-1), using real numbers and pi. Even more baffling is the identity of pi and e using imaginary numbers.

i = (base e log of -1) / pi
pi = (base e log of -1) / i
e = the (i*pi) root of -1

Then you get into the bizarre physics of zero. Nevermind infinity. We don't even need to use my theoretical mathematics. Let's use a number that has been in mathematics for centuries:

E = e^(i*pi) + 1
0 = E / X (where X is any number besides 0 or inf)

You get that? I just defined zero in an identity formula using a variable. I could have just used 0/1, but of course, 0/2 is the same thing, as is any 0/X except X=0 or inf. I could have even used 0*X (and included zero into X). But, where does that get us? Well, for one, it means that you can ram all kinds of things into this formula and it's still "accurate". For example: 0 = ((E / X) * Y) ^ Z. (X = any number besides 0 or inf, Y = any besides inf, Z = any positive number besides inf)

Two, I could take the E formula completely out and get:

0 * (1/E) = (E/X) * (1/E)
0 = 1/X -> 0 * 2X = (1/X) * 2X
0 = X

0 / E = (E*Y) / E
0 = Y

E root of (0) = E root of (E^Z)
0 = Z

Do you get that? I just mathematically proved that zero is every number. I just broke algebra.

Mathematics is perfect. The systems we use to prove or illustrate mathematics is flawed.

The problem with attempting to make a statement like 0*infinity = ? is that the result changes depending on what is 0 and what is infinity. For example x * (1/x) is 1, but x^2 * (1/x) is infinity even though they're both infinity * zero. The problem is that you need L'hopital's rule to correctly analyze all the cases of 0 and infinity interact so it's a bad idea to teach it in basic math. Not all infinities are created equal. For example the set of real numbers is far more *infinite* than the set of integers because for every 2 integer you can find infinte number of real numbers between them but not vice versa. And then here you get into number theory which is pretty complicated (I never quite understood it either).

As an aside I saw someone said that the fact that you have elegant formulas like Euler's equation is probably an accident. There's no reason why mathematic is supposed to be elegant or simple based on what we know of the universe. Most of the time it's just a mess that's tough to sort out.

Well, 0*I is like two black holes crashing into each other. It's one of those "indeterminate forms", where you have to use some other method to figure it out. Another one is 0^0. I can't believe people used to believe that 0^0 = 1. One rule says that 0^X = 0, and another rule says that X^0 = 1. So, obviously, either one of those rules is flawed, or 0^0 = an IF.

Funny, most of these paradoxes involve zero or infinity. I guess that's what happens when you include black holes in mathematics.

I was pointing out that your explanation of "Hence, 0.9 repeating and 1 must be the same number" is flawed, and I used an integer based system as an example.

I explained the various kinds of numbers because the above statement betrays a lack of understanding. What is true for integers is not necessarily true for real numbers. That you did not realize this indicated to me that you don't understand what those terms actually mean.

Limits come up in Newtonian mechanics because there is no "rate of change" in an instant. We're not trying to avoid dividing by zero, we're trying to determine an instantaneous rate. Recall that the word rate implies the passage of time. Algebra does not let us express this. Calculus does. If we were to throw out calculus as it is now, we would not be able to do basic physics - things as basic as if I throw a ball, where will it be in two seconds?

Here's what it comes down to: you are wrong. The world-wide community of mathematicians, scientists and engineers disagree with you. You can either accept that you're wrong and actually learn about how things work, or you can continue spinning your wheels.

Nerds.

Imakeholesinu wrote:Nerds.

True dat.

Imakeholesinu wrote:Nerds.

Isn't that the pot calling the kettle black? :-)

Touche Zeus.

Kupek wrote:Limits come up in Newtonian mechanics because there is no "rate of change" in an instant. We're not trying to avoid dividing by zero, we're trying to determine an instantaneous rate.

Well, that is exactly division by zero. Instant is zero, and when you're trying to compute a rate of change in that instance, you'll run into division by zero. The most basic time derivative is dx / dt. dt = instant = 0. Division by zero.

That is the whole point of calculus. I'm not saying that it's not useful in day-to-day applications, but it is a workaround to a problem that has yet to be solved. It is not a solution in and of itself.

Kupek wrote:Here's what it comes down to: you are wrong. The world-wide community of mathematicians, scientists and engineers disagree with you. You can either accept that you're wrong and actually learn about how things work, or you can continue spinning your wheels.

I understand that I'm in a position of thinking I'm right and the world is wrong. However, I'm am merely applying logical thinking to a mathematical problem. Not everything in mathematics has a proof, and things have been invented out of thin air just because it made logical sense.

Zero is a perfect example. There is no proof of zero. It just made logical sense that mathematics needed a concept of nothingness, so the number was invented. Many of the properties around zero are mere logical arguments without any basis in mathematics.

And it's not the first time that my logical arguments win out. I've seen a few English rules changed to my way of thinking because people like me thought the old rules didn't make logical sense. Eventually, logic will win out over rules that dance around a gap implying the obvious. (Blah, blah, blah, Sagan/Bozo quote.)

Practically, .(9) and 1 are identical. Mathematics doesn't ever discriminate here and you can prove their equality (again, mathematically) in a variety of ways. Similarly, rational numbers like 1/3 and 1/7 do in fact equal the nonterminating decimals our base 10 system needs to use in order to represent them, and that shouldn't be too difficult to wrap one's head around. Conceptually however, the debate gets more interesting.

When I was younger, I was convinced if you kept dividing a number in 2, you'd never reach 0. How could you? I've since opened my mind to this a little more and like to think that I now understand the idea of infinity a little better. It was years however between the time I was told that values needn't be finite or even static to work with them mathematically and the time I began to truly understand it.

An instantaneous rate of change is much more subtle than "divide by zero."

http://math.ucr.edu/home/baez/crackpot.html