Re: .999~ = 1
 

I'm glad to see so many people discussing mathematics and proofs. When doing so, please don't refer to your "trusty TI-89" calculator, or even the latest PC or Apple supercomputer. All computers run in binary and have problems handling the easy math problems that a 4th grader can understand (can a computer really express exactly that 1/3 + 1/3 + 1/3 equals 1?). Besides the inherent flaws with computers using binary, you also have to remember that someone had to program your calculator or computer and they are very capable of making mistakes just like you and I. Programming algorithms and software for math relies heavily on a ridgid set of rules (can you divide by zero, or take the ln of a negative?).

Mathematical proofs are carefully constructed arguments based on either known (and accepted truths) like axioms and definitions, and also theorems which have been previously proven. All of the above "proofs" or mysterious equations have a flaw in them somewhere which is not readily apparent to the reader. Unless you know the rules, you do not know what can or cannot be done. This is like the FIRST competition, is it not? If you don't know an obscure rule that other teams have read about in a team update or the manual, you are at a distinct disadvantage.

Over the years, math has evolved and our eyes have been opened to things our ancestors never thought possible. Let me cite a few examples.

Negative numbers. When caveman first started counting their rocks (1,2,3...), they didn't have a need for negative numbers. Did they exist? Sure, but it wasn't until people started borrowing sheep from their neighbor and they "owed 2" did the concept of being in the hole with a negative 2 sheep did this make any sense.

Irrational numbers. Pythagoras nearly made himself go crazy trying to find a fraction that would satisfy the hypotenuse of a right triangle with the two legs equal to 1. The ancient greeks thought all numbers could be represented by fractions. Of course, we know that the hypotenuse is √2, which is not rational. The introduction of rational numbers changed our way of viewing numbers and the associated rules. Transcendental numbers rocked our world in a similar way.

Non-integer exponents. A fifth grader can understand that x²=x*x and that x³=x*x*x. If you asked them, what is x^6, they would say "multiply 6 x's together". If you asked them, what is x^(-2) or what is x^(2.154), they would say "you can't do that!". A Calculus student wouldn't even flinch.

How about 0/0? Is this 1 or 0? Isn't there a rule that says anything divided by itself is equal to 1? But isn't there also a rule that says 0 times anything is 0, and since 0 is in the numerator, this must be 0. Hmmm. Go find L'Hopital and ask him what to do.

Imaginary numbers? No need to say anything. People still don't understand them.

Quaternions? These are extensions of Imaginary numbers. Think i²=-1 is hard to understand. Wait until you find out that j²=-1 and k²=-1 but that i≠j≠k.

Again, once you know the rules, your eyes become opened. If you don't understand the rules of the game, someone that does is sure to either pull the wool over your eyes or beat you at the game.

Sorry for the long post, but its been awhile since I've had the opporunity to talk about math with such a willing audience.

Let me end with my favorite quote of all time:
"God made the integers; all the rest is the work of man." - Leopold Kronecker

Dave...

Re: .999~ = 1
 

These things are pretty accurate when it comes to knowing about certain things. The TI-83's actually know when a fraction is non termaniting like 1/3. That is one of the very first things that I thought was odd about it. Though some of the algorithims are inaccurate like most if not all of the calculus functions. They are always off by a little bit.

Re: .999~ = 1
 

One would believe that .999 is not equal to one.


In order to make .999 equal to 1, you first have to declare that your tolerance for the project is +/- .0005. This would allow you to round any number greater than .9995 up to 1.

You would have to look at what type of calculation you are doing. If it was calculating the length of a peice of angle iron, I think your tolerances could be within +/- .005 or even +/- .05. If you were calculating the molar mass of a particle of angle iron though, you might want to have a tolerance of +/- 5.0 x 10^-29 (just to use as an example.)

Finally, remember that if you design something in CAD that the CNC machine or Machinist that makes the part will have some degree of error (hence the +/-). So if you send a part that is measured to be .999 with a tolerance of +/- any thing, you will most likely get a part that measures 1.

Just my 25 cent worth:)

Gabe Goldman
Prez and Founder of VCU Robotics Club

Re: .999~ = 1
 

This is exactly why the proof that I posted with e^(i*pi) doesn't work.

Re: .999~ = 1
 

e^(pi*i) IS actually -1. The problem is somewhere else...

Re: .999~ = 1
 

yes it is actually -1 but the problem is when i compare e^(3*i*pi) to e^(i*pi) cause even though they are both -1 they are not equal

Re: .999~ = 1
 

cut it down for space reasons, but no reason to apologize, in fact it is wholeheartedly welcomed by me, and the fact that you mentioned mathamatical terms i can't understand at this time intruiges me. now if only i could do better in pre calculus....

Re: .999~ = 1
 

thats easy, you cant divide by zero..we did that one in my algebra 4 class

Re: .999~ = 1
 

a problem here - S(n) is not true for all n. The proof of S(n+1) involves at least 3 distinct persons, and thus is not applicable for S(2). Thus, the entire sequence breaks down, as the S(n+1) proof therefore does not work for S(3), et cetera, ad infinitum.

Re: .999~ = 1
 

Right, but the TI-89 isn't solving a problem like ln[e^(3*i*pi}] by first multiplying 3, i, and pi together, then raising e to that result, then taking the ln of that. That kind of calculation would introduce error due to storing numbers in binary. Instead it applies known mathematical rules to produce the correct answer with little to no actual computation. For example, if you type "sin(x + 90)" (in degrees) where x is some unknown variable and press enter, the TI-89 will display "cos(x)". Obviously it knows the rules. It either instantly recognized the +90 as a shift to cos, or it expanded sin(a + b) to sin(a)cos(b) + sin(b)cos(a) and simplified.

That's basically what a human does when they prove something. They go step by step, and each step must be backed by a known and proven mathematical rule. If a calculator was programmed to know all the rules, why couldn't it produce a similar proof? How would checking an answer on the calculator be any different than applying those rules yourself? From my experience, I'm pretty sure the TI-89 is programmed to know all the rules. If you've never used one, it is quite an impressive calculator.

Obviously there is an error in the proof, since we all know that 3 != 1. I was simply using my TI-89 to figure out the location of the error, not the specific rule that was broken.

Re: .999~ = 1
 

also, a note on the 3 = 1 proof -
given a value X,
e^(X*i) = i*sin(X) + cos(X) (eulers law or something like that... forget the exact name)

thus, we are working on a unit circle, and in radians. As far as reference angles go, 3*pi does actually equal pi. However, this is only in the reference angle form. Essentially, when computing e^(X*i), the operation is one-way. You cannot definitively find X given the value of e^(X*i) - you can only find one or more possible X's.

Re: .999~ = 1
 

Exactly what I was saying earlier. This is related to sines and cosines.

sin(0) = sin(2*pi) does not imply 0=2*pi. It implies 0 rad = 2*pi rad. Just as pi rad = 3*pi rad.

Re: .999~ = 1
 

not quite - 0 rad does not equal 2pi rad - they are different angles. However, the key lies in the fact that sine and cosine are not definitively reversible operations.

Re: .999~ = 1
 

Hmmm... What are those acos and asin buttons on my calculator for? Sine and cosine are "reversible" operations since there are inverse functions.

I think that what you really mean is that the sine and cosine functions are not one-to-one mappings.

Re: .999~ = 1
 

hence definitively reversible. also known as not being one-to-one, as you said. Essentially, the inverse function is not actually a function. it's all good :)