Is 1 = 0.9999......

[Alphathree33]

No, engineering applications of calculus are all about approximations. After all, none of us needs infinitely many decimal places of accuracy.

The math of calculus is exactly correct. The mathematicians of the past thousand years weren't kinda right, they really were.

e = mc^2 isn't a good example here. For the purpose of my previous post let's remember that it's E for energy and not e for the base of a natural logarithm.

The speed of light, c, is not defined exactly and so yes I agree with you for all calculations done with that equation for measured values (mass, speed of light), your answer will be approximate.

Again, properties of the real number system are not approximate. They are axioms. These things work by definition, not by proof or by punching a lot of numbers on your calculator and scratching your head.

 

[Haircut]

Slightly off topic, but the speed of light is defined exactly, or rather the definition of 1 metre is derived from the speed of light so it is exact by default.

Other than that your last two posts have pretty much summed up the situation here. 0.9999... is 1 because math has defined it that way.

 

[Alphathree33]

Originally posted by: Haircut
Slightly off topic, but the speed of light is defined exactly, or rather the definition of 1 metre is derived from the speed of light so it is exact by default. Other than that your last two posts have pretty much summed up the situation here. 0.9999... is 1 because math has defined it that way.

I thought one meter was defined by some abstract french definition of so many millionths of the circumference of the earth or some such thing?

Anyway let's not get people off track .

 

[Haircut]

Originally posted by: Alphathree33

Originally posted by: Haircut
Slightly off topic, but the speed of light is defined exactly, or rather the definition of 1 metre is derived from the speed of light so it is exact by default. Other than that your last two posts have pretty much summed up the situation here. 0.9999... is 1 because math has defined it that way.

I thought one meter was defined by some abstract french definition of so many millionths of the circumference of the earth or some such thing?

Anyway let's not get people off track .

Link
I think we need to get people off track as this thread isn't getting anywhere on the 0.999...=1 discussion

 

[SilentRunning]

My point about e=mc^2 is that that is not an actual equation, It is an approximation of the actual equation.

And you seem to be missing the point

1/3 = 0.3333........ + 10^(-n) as n approaches infinity


to say that as n approaches infinity 10^(-n) = zero is wrong, it approaches zero.

Nothing you can say will change that fact. So 0.33333...... is an approximation of 1/3

 

[Alphathree33]

Originally posted by: Haircut

Originally posted by: Alphathree33

Originally posted by: Haircut Slightly off topic, but the speed of light is defined exactly, or rather the definition of 1 metre is derived from the speed of light so it is exact by default. Other than that your last two posts have pretty much summed up the situation here. 0.9999... is 1 because math has defined it that way.

I thought one meter was defined by some abstract french definition of so many millionths of the circumference of the earth or some such thing? Anyway let's not get people off track .

Link I think we need to get people off track as this thread isn't getting anywhere on the 0.999...=1 discussion

Saw the link. Hmmm. I think the french originally defined it the way I said and then modern scientists took the meter and messed with it a bit because that integer they chose for the denominator of the number in the definition (299 etc) made the meter still almost the same as it was before, but now they could define light exactly with it, and they couldn't before

 

[Alphathree33]

Originally posted by: SilentRunning
My point about e=mc^2 is that that is not an actual equation, It is an approximation of the actual equation. And you seem to be missing the point 1/3 = 0.3333........ + 10^-n as n approaches infinity to say that as n approaches infinity 10^-n = zero is wrong, it approaches zero. Nothing you can say will change that fact. So 0.33333...... is an approximation of 1/3

I agree with you that as n approxaches infinite, that function will approach zero. As I said numerous times in my other posts... unless n "equals" infinite, the function does not "equal" its limit. Saying "as n approaches some number, the function equals some number" implies that there is some really big, but finite value for n that you can find where the function actually does hit its limit. Again, that flies in the face of everything I just said I agree with. For finite values of n, the function will not reach its limit. (Again, let's assume no oscillation here.)

[ On a side note, if a function oscillates then it does hit its limit infinitely many times while on the way to approaching its limit, lol ]

So, I agree with your assumption and you're right I'm not going to change it.

For all finite times that you write 3, 0.3333... is an approximation of 1/3. But by definition, putting three dots at the end of the number means that you don't have a finite number of 3s, you have infinitely many.

Assume for a second that 0.3 repeating was only an approximation for 1/3. Rational numbers can be represented in the form m/n where m and n are integers. Therefore 1/3 must be a rational number since 1 and 3 are integers. Rational numbers can also be represented identically accurately in decimal form. If 0.3 repeating is only an approximation for 1/3, then it must in fact be equal to some other number. Now, 1/3, as a rational number, is lacking an identically accurate decimal representation. If it is not 0.3 repeating, what is it?

 

[SilentRunning]

Assume for a second that 0.3 repeating was only an approximation for 1/3. Rational numbers can be represented in the form m/n where m and n are integers. Therefore 1/3 must be a rational number since 1 and 3 are integers. Rational numbers can also be represented identically accurately in decimal form. If 0.3 repeating is only an approximation for 1/3, then it must in fact be equal to some other number. Now, 1/3, as a rational number, is lacking an identically accurate decimal representation. If it is not 0.3 repeating, what is it?



"Rational numbers are simply numbers that can be written as fractions or ratios. "

No where does it state that rational numbers must have a accurate decimal representation. Computers and calculators sure have confused people.

 

[Alphathree33]

Originally posted by: SilentRunning
Assume for a second that 0.3 repeating was only an approximation for 1/3. Rational numbers can be represented in the form m/n where m and n are integers. Therefore 1/3 must be a rational number since 1 and 3 are integers. Rational numbers can also be represented identically accurately in decimal form. If 0.3 repeating is only an approximation for 1/3, then it must in fact be equal to some other number. Now, 1/3, as a rational number, is lacking an identically accurate decimal representation. If it is not 0.3 repeating, what is it? "Rational numbers are simply numbers that can be written as fractions or ratios. " No where does it state that rational numbers must have a accurate decimal representation. Computers and calculators sure have confused people.

What is this "it" to which we are referring?

This is also an axiom of the number system... that rational numbers have identically accurate decimal representations. That's what separates them from irrational numbers which can only be approximated with decimal representation. Essentially you're trying to tell me that 0.3 repeating is not a rational number because you don't think it's equal to 1/3, and therefore it has no fractional representation and therefore it is not rational. This is a contradiction since all real numbers are rational unless, in decimal form, they have a non-terminating, non-repeating string of digits.

 

[Alphathree33]

... so 0.3 repeating is rational
... so it must be identically equal to some number m/n where m, n are integers
... and if that number is not 1/3, I ask again what number it IS.

 

[SilentRunning]

Originally posted by: Alphathree33

Originally posted by: SilentRunning
Assume for a second that 0.3 repeating was only an approximation for 1/3. Rational numbers can be represented in the form m/n where m and n are integers. Therefore 1/3 must be a rational number since 1 and 3 are integers. Rational numbers can also be represented identically accurately in decimal form. If 0.3 repeating is only an approximation for 1/3, then it must in fact be equal to some other number. Now, 1/3, as a rational number, is lacking an identically accurate decimal representation. If it is not 0.3 repeating, what is it? "Rational numbers are simply numbers that can be written as fractions or ratios. " No where does it state that rational numbers must have a accurate decimal representation. Computers and calculators sure have confused people.

What is this "it" to which we are referring?

This is also an axiom of the number system... that rational numbers have identically accurate decimal representations. That's what separates them from irrational numbers which can only be approximated with decimal representation. Essentially you're trying to tell me that 0.3 repeating is not a rational number because you don't think it's equal to 1/3, and therefore it has no fractional representation and therefore it is not rational. This is a contradiction since all real numbers are rational unless, in decimal form, they have a non-terminating, <STRONG>non-repeating</STRONG> string of digits.

But the mere fact that it has a non-terminating string of digits means that the decimal representation is an approximation.

Because it is non terminating that means there must always be a remainder.

 

[Alphathree33]

Originally posted by: SilentRunning

Originally posted by: Alphathree33

Originally posted by: SilentRunning Assume for a second that 0.3 repeating was only an approximation for 1/3. Rational numbers can be represented in the form m/n where m and n are integers. Therefore 1/3 must be a rational number since 1 and 3 are integers. Rational numbers can also be represented identically accurately in decimal form. If 0.3 repeating is only an approximation for 1/3, then it must in fact be equal to some other number. Now, 1/3, as a rational number, is lacking an identically accurate decimal representation. If it is not 0.3 repeating, what is it? "Rational numbers are simply numbers that can be written as fractions or ratios. " No where does it state that rational numbers must have a accurate decimal representation. Computers and calculators sure have confused people.

What is this "it" to which we are referring? This is also an axiom of the number system... that rational numbers have identically accurate decimal representations. That's what separates them from irrational numbers which can only be approximated with decimal representation. Essentially you're trying to tell me that 0.3 repeating is not a rational number because you don't think it's equal to 1/3, and therefore it has no fractional representation and therefore it is not rational. This is a contradiction since all real numbers are rational unless, in decimal form, they have a non-terminating, non-repeating string of digits.

But the mere fact that it has a non-terminating string of digits means that the decimal representation is an approximation.

It means that writing out a finite number of those digits would be an approximation, yes.

0.3... is just a symbol that says "there are infinitely many threes here and we can't write them all".

We could define a number system in which 1 was 0.3... and then it would be much easier to write, but other numbers would become more difficult. The same is true for rationally defined lengths. A right triangle of sides 1 and 1 will always have an irrational hypoteneuse. If you define that hypoteneuse to be rational, the other two sides will be irrational in length in that number system.

Anyway getting back on track, you're saying that all non-terminating strings of digits are approximations? ... approximations of what? Again, if 0.3 repeating is an approximation for 1/3, then what is it actually equal to? Or are you trying to introduce some fuzzy definition of 0.3 repeating such that it is an approximation for everything an in fact equal to nothing? Are you a Descartes fan? Because if you were, I don't think you'd exist anymore.

At least agree with me on the fact that all numbers are exactly equal to SOMETHING. I mean, if we can't agree on that, we might as well let the arts department define our number system for us.

 

[SilentRunning]

At least agree with me on the fact that all numbers are exactly equal to SOMETHING. I mean, if we can't agree on that, we might as well let the arts department define our number system for us.

Well, all terminating decimals are exactly equal to something.

 

[Alphathree33]

Originally posted by: SilentRunning
At least agree with me on the fact that all numbers are exactly equal to SOMETHING. I mean, if we can't agree on that, we might as well let the arts department define our number system for us. Well, all terminating decimals are exactly equal to something.

No... agree with me that all rational numbers (even numbers that repeat forever) ARE EQUAL TO SOME INTEGER RATIO. Rational numbers are defined this way!!!

This is fundamental math we're in disagreement on here, not even calculus anymore.

 

[Haircut]

Originally posted by: SilentRunning
At least agree with me on the fact that all numbers are exactly equal to SOMETHING. I mean, if we can't agree on that, we might as well let the arts department define our number system for us.

Well, all terminating decimals are exactly equal to something.

OK, if you disagree that 0.33333... = 1/3 can you please point out where exactly you disagree with the geometric series formula here
If we let r = 0.1 and multiply the whole sum by 3 we get that 0.3333... = 1/3.

 

[SilentRunning]

I am rusty on the math as it has been more than 25 years ago, but what do you think the three lines after the S represent, they do not represent equal to.

 

[Kerouactivist]

no

 

[Alphathree33]

Originally posted by: SilentRunning
I am rusty on the math as it has been more than 25 years ago, but what do you think the three lines after the S represent, they do not represent equal to.

You're right, they represent "identically equal to" meaning "another way of saying the same thing"

 

[Alphathree33]

Originally posted by: bthorny
no

Oh jesus christ read some of the frickin thread before you post.

 

[Kerouactivist]

Originally posted by: Alphathree33

Originally posted by: bthorny
no

Oh jesus christ read some of the frickin thread before you post.

I did I just decided that it might be best to end the discussion with the truth

 

[Haircut]

Originally posted by: Alphathree33

Originally posted by: SilentRunning
I am rusty on the math as it has been more than 25 years ago, but what do you think the three lines after the S represent, they do not represent equal to.

You're right, they represent "identically equal to" meaning "another way of saying the same thing"

Thanks, I was just about to post the same thing.

It appears that some people here have some issues with the concept of infinity. I remember similar things in the HT thread.

 

[Alphathree33]

Originally posted by: Haircut

Originally posted by: Alphathree33

Originally posted by: SilentRunning I am rusty on the math as it has been more than 25 years ago, but what do you think the three lines after the S represent, they do not represent equal to.

You're right, they represent "identically equal to" meaning "another way of saying the same thing"

Thanks, I was just about to post the same thing. It appears that some people here have some issues with the concept of infinity. I remember similar things in the HT thread.

Well the greeks had trouble with infinite too, but thank goodness somebody cracked it otherwise we wouldn't have integration. You know that's kind of important for circuits.

 

[SilentRunning]

OK, part of the proof


For -1<r<1 , the sum converges as n approaches infinity ,in which case

Sure seems to be stating an approximation to me. Converge, does not equate to equal to.

They are stating that r^(n+1) approaches zero as n approaches infinity

That is all for now, must do something constructive.

 

[Haircut]

Originally posted by: SilentRunning
OK, part of the proof


For -1<r<1 , the sum converges as n approaches infinity ,in which case

Sure seems to be stating an approximation to me. Converge, does not equate to equal to.

They are stating that r^(n+1) approaches zero as n approaches infinity

That is all for now, must do something constructive.

True, for all real n it does not equal 0. The point is that infinity is not on the (standard) real line.

For a sum where that sum approaches a value as n tends to infinity the infinite sum is equal to that value.

 

[MadRat]

Hence, objective values cannot be used to define subjective ones.