Simple mathematical problem (?)

[Earthborn]

As Conway has shown, Dr. Math is wrong. You can move "iota" down from 1 to get to .999999...

(Sigh)

I think what it all comes down to is a mathematical pretending game.

It all started with negative numbers. If someone had 3 apples, everyone agreed that you could take away 3, but not 4, because 'you can't have less than nothing'.

Then along comes someone who says: "But lets pretend that you can!". Some people argued that it didn't make sense, until some Arabic guy figured out that it was a convenient way of calculating debts. But that doesn't mean that there are anti-apples around: it was just a calculating trick.

Later people started to take square roots of numbers. And everyone agreed that you can't take the square root of a negative number, because no matter whether you square negative or positive numbers, the answer is always positive.

And along comes someone who dares to say: "But let's pretend that we can!" Some people argued that it didn't make sense, until someone figured out that it was a convenient way of calculating I don't know what. But that doesn't mean that there are square roots of negative numbers: it is just a calculating trick.

For a long time people all agreed that 0.999(recurring) equals 1. That if you substract 0.999(recurring) from 1, you end up with nothing. That there is no difference.

And along comes mister Conway who says: "But let's pretend that there is a difference!". And he figures out that it is a conventient way of calculating some stuff in games or something. But that doesn't mean there really is a difference: it is just a calculating trick.

You can argue that such pretend numbers exist, but for that, you need to change the definition of 'exist'.

Dr. Math should be ashamed of himself for stifling little Emily's and little Jenny's development of their own idea of how numbers work, just so that they can fit into the school system's ideas of what is right and what is wrong. Another cog in the machine......

Yes, I think it would be a much better idea to tell these kids right away that there really are no rules in mathematics, and they can make them up as they go along. Just teach them that whenever they make a mistake they can always tell their teacher that in a completely different number system the answer is correct, whatever it is. I'm sure that is going to give them a real good insight in mathematics...

In the system most kids need to know about, and most of us use for our day to day calculations, 0.999(recurring) and 1 makes not an iota of difference.

 

[69dodge]

Originally posted by Skeptoid
A sum of rationals is itself rational. 9/10¹ + 9/10² + 9/10³ ... = 0.999... = 1.

The sum of a finite number of rationals is rational. The sum of an infinite number of rationals may be irrational. (Irrational numbers have decimal expansions too; they just go on forever. Such a decimal is the sum of an infinite number of rationals.)

I certainly agree that 0.999... = 1, but that's not a correct way to demonstrate it.

 

[69dodge]

Originally posted by Suggestologist
But they are what would be required to produce the sequence .99999999..... as the result of converting a ratio of two "integers" into decimal form. A rational number.

If you convert the ratio of 1 and 3 to decimal form, what do you get?

You're claiming that not all real numbers have decimal expansions?

Obviously, it's all a matter of definition, as with anything in math. By definition, a decimal expansion represents the real number that is approached as a limit by taking more and more digits of the expansion.

I know hardly anything about Conway's surreal numbers. I'm sure they form a consistent mathematical system too. I don't know what their usual representation is. If "0.999..." represents a number other than 1 in that system, then the question of whether 0.999... equals 1 is ambiguous without an implicit or explicit understanding of which system -- real or surreal -- we're working in. I think it's reasonable to assume that anyone asking the question has never heard of surreal numbers, and so is looking for an answer in the context of real numbers.

 

[69dodge]

Originally posted by Suggestologist
Sure, but how do you get SUMMA x=1 to infinity (9/10^x) to equal 1? It gets closer and closer .... and closer and closer, but it never actually gets there.

Well, since the value of an infinite sum is defined to be the number that the successive partial sums get closer and closer to, that's not really a problem.

 

[Suggestologist]

(Sigh)
I think what it all comes down to is a mathematical pretending game.

And along comes mister Conway who says: "But let's pretend that there is a difference!". And he figures out that it is a conventient way of calculating some stuff in games or something. But that doesn't mean there really is a difference: it is just a calculating trick.

All that the numbers are, are ways of doing "calculating tricks". As I've pointed out before. You can count items, and you can measure distances with the "real" numbers. But those are two fundamentally different uses for numbers.

How much sense does it make to divide a distance by a distance? The answer is that it depends on what you are using the numbers for.

You can argue that such pretend numbers exist, but for that, you need to change the definition of 'exist'.

No, you just have to expand the definition of "number". Just as we did when the "number" ZERO was added, and the negative "numbers", and the rationals, etc.

Yes, I think it would be a much better idea to tell these kids right away that there really are no rules in mathematics, and they can make them up as they go along.

Not that there are no rules. But that there are different opinions depending on the perspective one takes. Oh, but let's turn them into mindless zombies instead, who believe everything their teachers tell them, that's much better -- yeah -- and let's teach them that numbers reflect the objective eternal truth of reality.

In the system most kids need to know about, and most of us use for our day to day calculations, 0.999(recurring) and 1 makes not an iota of difference.

I'm sorry, but there is a difference. The difference is the difference between dogmatic pedagogy and teaching kids to think and explore things for themselves.

 

[Zep]

OK, let's work it this way with a simple mind experiment.

Consider we have a perfect string that is exactly 1 metre long. An operation on this string that we do is to cut off and discard 90% of the current length each time. Perform the operation once - we are left with 10cm of string. Do it again, 1cm. Again, 1mm. Repeat this operation ad infinitum. Question: At what point do we have "no string"? Answer: Never. There is always "some" string left over if we remove only 90% of the remainder at each operation.

This is the quivalent of refuting the mathematical notion that:

Suppose 0.9~ = 1, then 1 - 0.9~ = 0
But since we can demonstrate that 1 - 0.9~ <> 0 then it follows that 0.9~ <> 1.

Again, think Cantor Sets!

 

[Suggestologist]

I think it's reasonable to assume that anyone asking the question has never heard of surreal numbers, and so is looking for an answer in the context of real numbers.

How do they know what system they're looking for, until they find the answer that satisfies them?

 

[69dodge]

Originally posted by Suggestologist
My preferred definition of a rational number might go something like: can be UNIQUELY represented as the ratio of two integers.

What does that mean?

I guess you agree that 2/3 is a rational number. Also, that 4/6 is a rational number. But they're the same rational number. What do you mean by "uniquely"?

 

[slimshady2357]

Your argument is circular. If you assume that .9... is rational then it can be written as a fraction. If you don't assume that, then it can't.

Of course 0.9999... is rational! You can't get an irrational number by adding finite amounts of rational numbers.

1/3 = 0.3333... exactly and 1/3 + 1/3 + 1/3 = 0.999.... exactly.

And trivially, since 1/3 + 1/3 + 1/3 = 1 also, 1 = 0.999.... exactly

Please, anyone who thinks otherwise, read the other thread first. If I remember correctly xouper gives some excellent explanations.

Adam

 

[Suggestologist]

What does that mean?

I guess you agree that 2/3 is a rational number. Also, that 4/6 is a rational number. But they're the same rational number. What do you mean by "uniquely"?

I mean that it's unique on the decimal side. 4/6 always produces the same decimal pattern. It doesn't sometimes produce .7777777... and other times produce .6666666...

 

[Suggestologist]

Of course 0.9999... is rational! You can't get an irrational number by adding finite amounts of rational numbers.

1/3 = 0.3333... exactly and 1/3 + 1/3 + 1/3 = 0.999.... exactly.

And trivially, since 1/3 + 1/3 + 1/3 = 1 also, 1 = 0.999.... exactly

Please, anyone who thinks otherwise, read the other thread first. If I remember correctly xouper gives some excellent explanations.

Adam

1/3 produces the decimal pattern: .3333333..... ; that does not mean that they are EQUAL. Depending on one's perspective, they may or may not be equal.

 

[slimshady2357]

Well, it does depend on your perspective. If you believe the definition of rational number that you have described, then obviously .9.... is rational. But that's still a circular argument from definition. My preferred definition of a rational number might go something like: can be UNIQUELY represented as the ratio of two integers. But, If .9.... = 1, then why doesn't 1 divided by 1 produce .9..... ?

Ahhh, you're using some other definition of rational.

Btw, the one you give at the end would ceratinly exclude a lot of rationals..... possiblly all of them

1/3 = 2/6 = 3/9
1/2 = 2/4 = 3/6

etc....

Adam

 

[slimshady2357]

OK, let's work it this way with a simple mind experiment.

Consider we have a perfect string that is exactly 1 metre long. An operation on this string that we do is to cut off and discard 90% of the current length each time. Perform the operation once - we are left with 10cm of string. Do it again, 1cm. Again, 1mm. Repeat this operation ad infinitum. Question: At what point do we have "no string"? Answer: Never. There is always "some" string left over if we remove only 90% of the remainder at each operation.

That is not correct, you will have no string after an infinite number of cuts.

I think the problem you're having is that you're thinking of 0.9999... as a process. It's not a process. 0.999... is a representation for an infinite number of 9's already being there. It is complete, so to speak.

Adam

 

[69dodge]

Originally posted by Suggestologist
I mean that it's unique on the decimal side. 4/6 always produces the same decimal pattern. It doesn't sometimes produce .7777777... and other times produce .6666666...

That's just because the standard long division algorithm yields a terminating decimal, in cases where there is one. A slightly different algorithm would yield the equivalent decimal that ends in an infinite string of 9s. How do you know the standard algorithm is the only correct one?

 

[Suggestologist]

Ahhh, you're using some other definition of rational.

Yes. Exactly. It depends on the definition used.

Btw, the one you give at the end would ceratinly exclude a lot of rationals..... possiblly all of them

1/3 = 2/6 = 3/9
1/2 = 2/4 = 3/6

etc....

Adam

No. Read my just prior post.

 

[slimshady2357]

1/3 produces the decimal pattern: .3333333..... ; that does not mean that they are EQUAL. Depending on one's perspective, they may or may not be equal.

I'm not sure if you're talking from the perspective of 'surreal' numbers, as I've never heard of them, but I have no idea what you mean by 'produces the decimal pattern'.

Do you mean the actual act of dividing? Sort of like writing it out?

Adam

 

[Suggestologist]

That's just because the standard long division algorithm yields a terminating decimal, in cases where there is one. A slightly different algorithm would yield the equivalent decimal that ends in an infinite string of 9s. How do you know the standard algorithm is the only correct one?

It may not be the only correct one.

The underlying problem is that people think of mathematics as static. But in fact, it's a process, and you have to think about what you are doing with that process in order to get the results you want.

A simple example is the square root. Taking a square root can imply that a negative number or imaginary number is the correct result. But negatives and imaginaries may not make sense in the context of the mathematical proposition you are trying to solve. So you have to reject them, not because they are mathematical errors, but because they don't make semantic sense for the real world problem.

 

[Suggestologist]

I'm not sure if you're talking from the perspective of 'surreal' numbers, as I've never heard of them, but I have no idea what you mean by 'produces the decimal pattern'.

Do you mean the actual act of dividing? Sort of like writing it out?

Adam

Yes. When you convert between rational numbers and decimal numbers, you sometimes lose information. Just like when you translate a word between languages, the translated word often means something close to the original word, but it doesn't quite mean exactly the same thing. It may have slightly (or very) different connotations, or associations.

 

[69dodge]

Originally posted by Suggestologist
A simple example is the square root. Taking a square root can imply that a negative number or imaginary number is the correct result. But negatives and imaginaries may not make sense in the context of the mathematical proposition you are trying to solve. So you have to reject them, not because they are mathematical errors, but because they don't make semantic sense for the real world problem.

Sure, no argument there.

 

[Earthborn]

I'm sorry, but there is a difference. The difference is the difference between dogmatic pedagogy and teaching kids to think and explore things for themselves.

Or, differently formulated: allowing kids to think and explore things within a set of rules they might actually understand, or confusing them even more than they are with things that make them think (even more than they already do) 'When am I ever going to need this?'