Math Problem

[adrian jaye]

more to the point are you likly to use it in the real world ?

Originally posted by Souk
The mean, median, unique mode, and range of a collection of eight counting numbers are each 8. What is the largest number that can be in this collection?

[cashflyer]

Originally posted by adrian jaye
more to the point are you likly to use it in the real world ?

We used it HERE. Are you trying to imply that PPOT is not "the real world"?????

[alf]

[klaucke]

Originally posted by T_Samner
The second and third pictures show that an infinite series is defined as the limit of its sequence of partial sums as n tends to infinity. The first picture shows how the limit of a sequence is defined. The fact that epsilon is greater than 0 shows that the sequence never actually equals the limit S - it can just be made arbitrarily close by choosing a large enough N. So again, we are talking about a limit. The sum of the infinite series in your calculation approaches 1 but is not actually equal to 1. And 0.99999... with an infinite number of 9’s is still not equal to 1. Only 1 is equal to 1.

While what you posted is somewhat true in general, that is not the case here. There is an explicit formula for the sum of an infinite geometric series as I posted above. You do not need to take any limits w/ n,N, and epsilon in order to derive the explicit form. Like I said, check around, it's been known for hundred of years. Just because a sequence never reaches it's limit, does not mean the series of sums defined by it doesn't when taken ad infinitum.

Wikipedia has a proof for the geometric series here:
http://en.wikipedia.org/wiki/Geometric_series

And here is an article on .999... = 1:

[T_Samner]

I looked at your links and researched it a bit more and you are, in fact, correct. Had you used the algebraic argument I would have accepted the fact more quickly.


Let x = 0.999... Then 10x = 9.999...

10x = 9.999...
- x = 0.999...
------------------
9x = 9

And therefore x is also equal to 1.

[Flatbutt1]

klaucke, the algebraic and fractional proofs are more amenable to my level of math, thanks

[T_Samner]

Even though you were correct that 0.999... is equal to one, after reading the discussion on Wikipedia some more I think the method of proof that you used here (with an infinite geometric series) only proves the case for the limit like I said.

The problem I have with the proof using infinite geometric series like you used is illustrated in the second picture below. It doesn't say that r raised to the nth power equals 0 as n tends to infinity it just says that r raised to the nth power approaches 0. That means that the sum of the series in Example 1 (and in your proof) only approaches a / (1 - r).


.


Some more information from my real analysis book (see 16.3 f) seems to indicate that x to the power of n only equals 0 when x is equal to 0 and only approaches 0, as n tends to infinity, when the absolute value of x is less than one.





----

The information on Wikipedia is also a bit confusing:

...the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.[5]

A sequence (x0, x1, x2, ...) has a limit x if the distance |x- xn| becomes arbitrarily small as n increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:

[K.B.]

How about this rule breaker:

Start out with
X = 1

subtract 1 from each side of the eqn
X - 1 = 0

divide each side by (X-1)

X - 1		0
______	=	______
X - 1		X - 1

so we get

1 = 0

[klaucke]

To divide by 0 (x-1 in this case) is nonsensical.

[K.B.]

Ok but X is really 1 - .9999........

[T_Samner]

If anyone wants an easy to use math software program, I recommend Derive 6.1 from Texas Instruments. You can even download a fully functional 30 day trial version and if you decide to buy it, it's only about $80. I have Mathematica 5.0 but Derive does 90% of what I need and it's a lot easier to use because the input is less finicky.

[snowman]

You all forget that because I thats a capital I, say so, 0.99999 exactly equals ONE. Anyone who disagrees is wrong, why , because I say so. Not only that mothe nature is also likely to agree as she is a bitccch and cannot make up her mind. Is it one, is it 0.999999, is it 1.000001. Thats reality and also why engineers make many more bucks than mathmeticians.

[klaucke]

Originally posted by snowman
Thats reality and also why engineers make many more bucks than mathmeticians.

Those that aren't outsourced to India or China.

[T_Samner]

I like math because things can be proven and you can know that they're true. Just like with the 0.999... question. Initially I didn't believe it but it can be proven to be true so then there's no more arguing about it. Gotta love that.

[Taz's Master]

So how much do you have to reduce 1 by before no longer equals itself?

[MotoSook]

Originally posted by adrian jaye
more to the point are you likly to use it in the real world ?

I often gave my math teachers a hard time when in high school, and I've asked the same thing. Now I don't see such excercises as a waste of time, lacking "real world" use.

The problem was given to a group of Jr. High School students, and I'm sure they are all questioning its merit. I acutally think the question is a great mind excercise. Without the use of Excel or other means unavailable to students in Jr. HS, they'll have to deductively reason their way toward the right answer. Not all of them will understand it, but if the teacher explains it to the class adequately after their work has been turned in, you may hear the light bulbs going on above their heads.

The problem(s) in math or science are meerly the tools used in teaching problem solving and reasoning skills that make great thinkers. Unfortunately (as I'm seeing with the stuff our kids bring home), some teachers (and administrators) don't seem to understand this. Our school district is a great school district (and we pay top tax dollars to live here), yet the programs they have instituted for math makes me wonder if they are teaching good reasoning skills. I worry that some of the teachers don't understand the goal of teaching math to grade school and Jr. HS students.

[T_Samner]

Originally posted by Taz's Master
So how much do you have to reduce 1 by before no longer equals itself?

Any reduction from 1, no matter how small, means that it's no longer equal to 1. It turns out that 0.9 repeating is just another decimal expression for 1.

[snowman]

Why is it that when you get to partial differential equations, i.e. "real math for the real world", you have to resort to guessing, and approximations to solve problems?

Why is it that a zillion people can solve math problems, yet only a very few can set up the problem to begin with? The setup is the most important part in the "real" world. A friend of mine, who was in calculus class in HS, could not figure out how to calculate the length of guy wires for an antenna, until a triangle was mentioned. I have found that this is not uncommon, for even highly educated people. Its those 8th grade word problems that usually give them the most trouble.