0.99999... = 1

[ZoraxDoom]

Are you out of your mind? Any recurring or terminating decimal is rational.

Is it? Whoops. Haven't done much of that.
Anyways, I'm sure there is a law on this written somewhere.

 

[tassietiger]

I don't believe or like that concept of Infinity then. I understand what you are saying, but I don't understand how people will accept that 0.999...infinity is equal to 1. I don't like it. It's just a conceptual view that if you get closer and closer to something, you will eventually get there, but it is like dividing a number by 2, diving the result by 2, divide that by 2, and so on till infinity. You get closer and closer to 0, and after a while you can take it as 0 for practical use, but it should never, truely, be equal to 0.

If you want to consider 0.999 reocurring equal to 1 for practical reasons, then fine, but it should really never, ever be exactly equal to 1. Maybe it is in the concept of inifity, but not to me.

It isn't just for practical reasons. Mathematically, they are equal.

Infinity is a hard concept to agree with. It's only really used to figure out what sequences are tending towards, and has very little practical use. But until you agree with the infinity concept you probably won't be able to understand why 0.999... = 1.

There is no law written on this stuff. To be accurate, you would leave 0.333... as 1/3, and just multiplied that by 3 to get 3/3, which is 1. If you do go on to do sequences and series in maths, this will all make so much more sense.

That dividing example you gave is a perfect example, by the way. 1/infinity actually is equal to 0, and that is one of the main ways infinity is used.

 

[Neo95]

Please correct me if I am wrong........

0.999999..... has infinite no of significant figures .We take it as 1 for cutting down of complicated calculations.The actual no is taken for very accurate measurements for more the number of significant figures the more accurate is the measurement.

And Abby and ABBh have also proved that 0.99999....=1.So why the hell are we going from the topic of 0.9999... to infinity.
hypocrites

 

[ZoraxDoom]

So basically, we've summed up:

Practically: You can assume 0.999... = 1
Normally: 0.9999... is not = 1
But under the concept of infinity, 0.999... =1

Alrigh, I guess that is that?

BTW: When is the last thread where you (Tassie) and I were doing an on-topic dicussion, and everyone else were making jokes around us? Has PC gone upside down? :help

 

[Neo95]

BTW: When is the last thread where you and I were doing an on-topic dicussion, and everyone else were making jokes around us? Has PC gone upside down? :help

Who You and I or you and Tassie?

 

[Harrypotter_fan]

0.999... is 1, not taken to be 1.

 

[Skater]

0.99999... = 0.99999...

If it = 1, then it would be 1.

 

[Harrypotter_fan]

Every number can be represented as decimal and fraction. So 0.999... = 1/1, like 0.444... = 4/9.

 

[MattW]

It isn't just for practical reasons. Mathematically, they are equal.

The problem seems to come in applying normal logic to mathematical logic. While it makes full mathematical sense for them to equal each other, it is hard to logically say that two non identical things are equal.

Something that has always stuck out at me as being illogical is the fact that something to the power of zero is one. It seems to me that these sort of things come more from mathematical convenience rather than normal logic.

When I first came across the x^0 thing I didn't like the idea of it being one. To attempt my logic:
7^3 gives you three sevens, (7)(7)(7) or 343. 7^1 gives you just the one seven, (7) which equals 7.

Therefore if you have no 7s, as would be the case continuing that pattern with 7^0, where does the 1 come from?

I googled to try to find the sort of proof that you get with the thing this thread is about, and basically it rests on the argument that because:
n^x ? n^y = n^(x + y)

n^0 has to equal one so that n^x ? n^0 fits in with the formula.
(via http://www.homeschoolmath.net/teaching/zero-exponent-proof.php)

Which goes back to the idea that some of these concepts only exist because it made it easier to do something else. So I'd argue that practical reasons causes a proof to be made, it is just that sometimes that proof has potential flaws depending on your interpretation.

That would be why nearly everything in maths is a theorem and all the 'laws' have a bunch of exceptions.

Though on the other hand, I won't pretend to be as good at maths as many others in this thread, but I think there is a psychological barrier to accepting quite a lot of the proofs in maths.

On a side note I also think you can have a negative 0 (I've actually written an answer on a test as -0) and that a number divided by zero is infinity, so my thoughts on the complexities of maths may not be all that valid.

 

[Kshitiz_Indian]

The problem seems to come in applying normal logic to mathematical logic. While it makes full mathematical sense for them to equal each other, it is hard to logically say that two non identical things are equal.

Something that has always stuck out at me as being illogical is the fact that something to the power of zero is one. It seems to me that these sort of things come more from mathematical convenience rather than normal logic.

When I first came across the x^0 thing I didn't like the idea of it being one. To attempt my logic:
7^3 gives you three sevens, (7)(7)(7) or 343. 7^1 gives you just the one seven, (7) which equals 7.

Therefore if you have no 7s, as would be the case continuing that pattern with 7^0, where does the 1 come from?

I googled to try to find the sort of proof that you get with the thing this thread is about, and basically it rests on the argument that because:
n^x ? n^y = n^(x + y)

n^0 has to equal one so that n^x ? n^0 fits in with the formula.
(via http://www.homeschoolmath.net/teaching/zero-exponent-proof.php)

Which goes back to the idea that some of these concepts only exist because it made it easier to do something else. So I'd argue that practical reasons causes a proof to be made, it is just that sometimes that proof has potential flaws depending on your interpretation.

That would be why nearly everything in maths is a theorem and all the 'laws' have a bunch of exceptions.

Though on the other hand, I won't pretend to be as good at maths as many others in this thread, but I think there is a psychological barrier to accepting quite a lot of the proofs in maths.

On a side note I also think you can have a negative 0 (I've actually written an answer on a test as -0) and that a number divided by zero is infinity, so my thoughts on the complexities of maths may not be all that valid.

I can solve your x^0 problem.

See, in mathematics, we multiply any value by 1, it doesn't make a difference does it?

So instead of writing 7^3 as 7*7*7, I could write it as 1*7*7*7, doesn't make a difference does it? So what we did was, we multiplied 1 with 7 three times.

Now lets take 7^0. We follow the same logic, we multiply 1 with 7 zero times, or in other words, we just don't multiply 7 with 1 at all, so that leaves us only with 1. I guess you caught my drift, and if you didn't, tell me and I'll explain in a simpler manner!

 

[svijay]

Three points.

1. Two unequal real numbers must have a positive difference.

2. Two unequal real numbers must have a real number in between.

3. Every real number has a unique representation as an INFINITE decimal. Terminating representations are a luxury that very few real numbers can afford - such representations are unofficial and purely for convenience.

This must be the slow class.

Terminating representations are like nicknames, people. They are not official. Except for zero, there is no other real number whose official decimal representation you can ever write down. If you look at the birth certificate of 24/5 you'll find 4.799999... People write 4.8 for convenience and out of kindness to the typesetter. Like we say Pele instead of Edson Arantes do Nascimento when we refer to the great man.

I know this may fly in the face of what you may have been taught, but it's true. Some rational numbers have two representations, one of which is much shorter than the other. But they are as equal as 42 and XLII.

If you agree with points 1 and 2 in my previous post and still think 0.999... does not equal 1, you are welcome to compute the difference or construct a real number in between.

 

[MattW]

I can solve your x^0 problem.

I was going to put something in to my post about an implied ?1 in power related equations, but when you learn powers you learn it as multiplying x by itself n number of times and it is only when you need the x^0 to equal one that you would ever bring it in.

Also, by that logic, 0^0 should equal 1, but instead you get a math error (on a calculator).

 

[Kshitiz_Indian]

I was going to put something in to my post about an implied ?1 in power related equations, but when you learn powers you learn it as multiplying x by itself n number of times and it is only when you need the x^0 to equal one that you would ever bring it in.

Also, by that logic, 0^0 should equal 1, but instead you get a math error (on a calculator).

Well I just tried in the windows calculator on the scientific calculator part, and using the x^y function in it, I used 0 ^ 0 and got 1.

 

[tassietiger]

So basically, we've summed up:

Practically: You can assume 0.999... = 1
Normally: 0.9999... is not = 1
But under the concept of infinity, 0.999... =1

Alrigh, I guess that is that?

BTW: When is the last thread where you (Tassie) and I were doing an on-topic dicussion, and everyone else were making jokes around us? Has PC gone upside down? :help

Nah, this is the only thing I'm agreeing with anyone on and I'm willing to argue it all day.

0.999... = 1

In any situation. Also I don't get what you mean by off-topic discussions. I've never been involved in one, I don't know whether or not the case is the same with you.

 

[AbBh]

I'm surprised that this is still going on as its not a hard concept to grasp.
Its not an approximation or an assumption that 0.999... is one. If you say its not then prove that a number exists between 0.999... and 1. Subtract 0.999... from 1 and the result will be an infinite sequence of 0's.
For example
1 - 0.99 = 0.01

But there is no end to 0.999... so the result of subtracting it from 1 will be 0.

However, I'm not convinced that numbers like 0.999... should exist as it just doesn't seem right. If there was a 'complete' numeric result for divisions like 1/3, 1/9 etc then none of this would be an issue. Theoretically a number has infinite number of digits after the decimal point but no real world application is going to really need that.