Bryan

Wow, this article is in stark contrast to the one linked by Hal in the initial post. The aspect of this story that puzzles me the most in the amount of the bond ($5000). This amount seems too high for "making terrorist threats " and too low for someone "attempting to organize an armed takeover of the school."

Maybe we should all move to Crook County, Wyoming with him....

All this kid had to do was get the story published, and then everything would have been fine. They probably would have given him an honorary PhD for creative writing. Just ask Ward Churchill. In all seriousness, this story is truly disturbing.

A lot of people are as they ought not be

How does it not follow? Is not 0.999~ the result of the partial sum if N = infinity? I would say that the limit could be defined as an approximation of the sum, but not the actual value of them. The two values are similar but not exactly the same. Why do you accept the concept of infinity when it supports your argument and disregard it when it does not? Literally 1/infinity doesn't make sense, just take it to be the smallest conceivable number. Again, the limit is equal to one, but that doesn't make the actual value equal to 1. I disagree. 1 is the limit of that sequence, 0.999~ is the value of its sum.

What is the the largest number in the sequence of natural numbers? I was talking about the defintion of a real number. I'm saying that the sum of i = 1 to i = [insert the largest number you can think of here] of 9/(10^i) is less than one. This is where we have a disgreement. I'm talking about a literal sum, you are talking about a limit. 1/infinity is infinitely small. The final answer is always an actual number, and that actual number is never 1. You can't actually obtain an actual number doing an infinite number of interations, you can describe the final answer after an infinite number of interations is 0.999~. When in the process of calculating these interations, would the answer every jump up to 1? Your proof was valid in showing that there is always a number between two numbers. Using the basis of that proof, making x a variable, and repeating it recursivly an infinite number of times, the final x you get is the largest number less than 1. You can't actually get a final x unless you put an artificial cap on infinity. Tell me one more time what is means, I'll tell you if I agree. BTW, what is your mathematical background?

The purpose of a comedy is to ridicule the ridiculous. They are funny because they present man has he ought not be. Napoleon Dynamite is not a depressing movie because it is a comedy. I'm not even sure if a comedy could be depressing. But if the exact same characters were presented from a serious point of view, it would be horrible. If you want to see a depressing movie, go watch Mystic River.

Not if N is infinity. This the same definition I gave using more complex terminology. This is why .999~ < 1. If you put a cap on the largest number, whatever the sum of .9+.09.+.009+.0009.... iterated the "largest number" of times, the final result will be less than 1. The "something" is whatever units your are working with. The actual unit you use is arbritrary. No!!!! The final answer is dependant the cap you put on the number of iterations. No matter what the cap it is, the final answer is less than 1!. The incorrect step that there is not an infinite number of steps. I understand that it can be demonstrated that there always a number that between two numbers, in fact there are an infinite number of numbers between any two numbers. This is beginning to fall into the dead horse beating catagory. Nothing you have demonstrated shows that .999~ is equal to 1. Everything I have demonstrated shows that it is less than 1.

Declaring Miss Rand's definition to be bad depends on your definition of measurement. If you define measurement as the identification of relationships, mathematics certainly qualifies as the science of measurement. As far as complex analysis goes, it can be directly used to measure the magnitude and phases of voltages and currents in electrical systems. I have no experience in number theory but I googled it and it said "A branch of mathematics that investigates the relationships and properties of numbers." That being said, you could describe number theory as the measurement of numbers themselves. I was trying to make a distinction between calculus being used as a mathematical tool to describe reality as opposed to the mathematical tools that are used to describe calculus (i.e. limits and infinite sums).

I understand that it is valid to discuss infinity; my point of this whole thread is that you can't use the concept of infinity to set a number equal to a number that it is not equal to. They are both valid concepts, the difference being that 1 can be directly related to things in reality. Do you care to give a definition? Mine was ok, just kind of sloppy. Here are some other's from google: " any number that is not imaginary Example:"1.23156..., 5, 8/6, e, square root (3)" " A number with an integer and a fractional part. The primitive types double and float are used to represent real numbers. " " A real number is one-dimensional and can be placed somewhere on the number line. The set of real numbers includes all rational and all irrational numbers. " " Any finite or infinite decimal. Any rational or irrational number. " If you accept the concept of infinity, something can be infinitely large or infinitely small. The only limit on the smallness is that it can never be zero; it always has to be something. Let’s do this with numbers: y = (x+1)/2, where x is .99: y = .995 Now substitute .995 in for x: y = (.995+1)/2: y = .9975 Keep repeating over and over, replacing x each time with the result for y the time before. What is the “final” answer after and infinite amount of iterations? .999~ This is why .999~ is the largest number that is less than 1. It is the result of infinite iterations of the recursive function x = (x+1)/2. It is the result of the infinite sum of the sequence (.9, .99, .999, .9999, …). It is an infinitely small amount less. Edit: Sorry about the text wrapping of this post, it turns out the going from the message board to gmail to word back to the message board messes it up

I think I got officially "hooked" on Law and Order last night watching Law and Order: SVU. Every time I catch the beginning of a show I'm glued to my seat for an hour. Last night was the first night I planned on watching it. One thing I like about it is how it relates episodes to recent headlines. My only complaint, which is actually a complaint of television in general, is the subtle liberal tilt of the show. For example, one of the detective’s comments in last night’s episode about people helping Columbian drug lords launder money was, "Capitalism at it's finest!" Overall though, great show.

No, I don't see the particular number .999~, directly referring to anything in reality. Infinity only exists as an abstract concept; therefore infinite series only exist as concepts. Let's be clear about mathematics. In ITOE, Ayn Rand aptly defines mathematics as "the science of measurement". Calculus isn't made possible by infinite series and limits, it is made possible by reality. It was developed as a manner of describing and measuring things in reality. Infinite sums and limits are used to describe calculus concepts, not the other way around. Mathematically speaking, a real number is a non-complex number, so yes, it’s a real number. It is an abstract number the sense that it only exists as a concept. I never said "infinitesimal" so I can't help you there. I demonstrated how .999~ is the largest number less than 1 with function you provided [f(x) = (1+x)/2 ], again this is just an abstract concept, perhaps I wasn't clear. And yes it is a sequence of rational numbers, specifically the sequence 0.9 + 0.09 + 0.009 + ... I'm not using the definitions interchangeably, its just that .999~ fits all these definitions. The limit of the infinite sum 0.9 + 0.09 + 0.009 + ... does indeed equal 1. But this does not mean that the actual number .999~ is equal to 1. Just because the limit of something equals 1 does not mean that the something is 1. 0.999~ can be approximated by 1 because its pretty damn close, but its still not exactly 1. Nothing is exactly 1 but 1.

I'm in the minority here, but I still hold that .999~ is not 1. Abstractly speaking it is the largest number that is less than 1. The difference between .999~ and one is inifinitely small, but there is a difference. For a number to exist in reality, it needs to be represented as a measurement of something. Since inifinity does not actually exist, inifinite decimal numbers don't exist. In the example of 1 and .999~, lets pretend you have something with a mass of 1 unit. You remove 1 single electron from it. Now the mass is 1 unit - the mass of one electron. Its a measureable amount and a finite number. You can't take an infinitely small amount away, because ininity doesn't exist. A number can't really approach a number. When I said the .999~ approaches 1, I meant that the infinite sum 0.9 + 0.09 + 0.009 + ... approaches 1. It never actually equals one because the sum never ends, but everytime you carry out another iteration of it, you get a number that is closer to 1. This is why I said that .999~ doesn't actually exist in reality. The function you gave above is a perfect example of what .999~ is. Lets say that you performed (1+x)/2 over and over starting with x = .99 and then replaced x with the result and performed the operation again. If you did this an inifinite number of times, your "final" answer would be .999~. Obviously pi and the sqrt(2) exist, They just can't be 100% accurately represented using decimal numbers. That has nothing to do with their nature, it has to do with the nature of the things they describe.

Is this not the case for any brain teaser?

This is an Objectivist message board, its purpose is to exchange Objectivist ideas. There is absolutely nothing wrong with coming here, asking questions and trying to gain knowledge. That's why I'm here As long as your intentions are honest, most everyone here is more than happy to help you learn.

(3-3)(3+3) does equal 3(3-3), they both equal zero. There is nothing invalid about that line. He did apply the initial condition that a = b. This being the case, the only way that a+b = b is if they are both 0. That is the line that is the "trick" in the "proof", not any other one.

((-x)(-x))^1/2 is equal to positive or negative x (the absolute value of x). x = ((-x)^1/2)((-x)^1/2) is only equal to negative x, not positive x. While it is true that the absolute value of x does equal -x, this does not mean that x = -x.

If a = b, the only way that a + b = b is if they are equal to 0. There is no other number that can be added to itself and get the same number. So at this point you know that b = 0 and only the final "magical" step that appears to make 2 = 1 is invalid. This step is valid if a and b are 0.

Your mistake is the that you say: 3 * (1/3) = 0.99999999999999999 (repeating) 3 multiplied by 1/3 is 1. Not anything other than 1! 1 divided by 3 is not a number that can be accurately represented by a decimal. You can put as many 3s at the end of .333 until you drop dead, you still won't get exactly 1/3. By the same method you can put 9s at the end of .999 until you drop dead, you will never get exactly 1.

That formula is just a way to approximate inifinite sums. The infinite sum does approximately equal 1, but its is actually slightly smaller. In the abstract sense, .999~ is the largest number that is less than 1. The square root of negative one is a mathematical tool that does represent things in reality. In fact, I used complex numbers this morning to represent the power in a transformer in my electrical machinary lab.

This is a quote from the other thread, I took upon myself to combine them because this thread contains the actual topic. The infinite series (0.9 + 0.09 + 0.009 + ...) is just a big long list of 9s if you actually calculate the sum of the series. .999~ is theoretically the largest number less than one, but it doesn't actually exist in reality. Let's pretend you have 1 cup of coffee. You take the smallest sip of it that you possibly can. You now have less coffee in the cup than you had before. We'll say that you now have .999~ cups of coffee. But in reality, no matter how small of a sip you took, you still removed a measurable amount of coffee from the cup. You can't take an infinitely small amount of coffee out of the cup, which is why .999~ doesn't actually exist.

There is no number between them, .999~ is the largest number less than 1. 0.999~ doesn't actually exist in reality.

No matter how many 9s you put at the end of the "infinite" decimal if you multiply it by 9 you will get a number less than 9. .999~ < 1, therefore 9*.999~ < 9.

The argument in the other thread is based on an algebraic error, not calculus. You can never say that .999999~ = 1, it approaches 1 but never quite equals it.

1/3 doesn't really = .3333 repeating. 1/3 is an irrational number, meaning that it can't be accurately represented as a decimal. Anytime you represent 1/3 as a decimal it is only an approximation, no matter how many 3s you put on the end of it Edit: I made a mistake, 1/3 is not an irrational number. An irrational number is a number that cannot be represented as an exact ratio of two integers. 1/3 is just a ratio that cannot be represented as a terminal decimal.