The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief that 0.999… should have a last 9.

with negative factors and bits it becomes very useful. At A-Level you get taught to ignore the factors of i, but that's not the correct thign in more advanced stuff. It's all very boring for most people.

So like the square root of -9 is 3i. If you are after information better than I can give it, it's a good idea to go to a university maths library and ask for Mary Boas. It's like the university maths/physicist/engineers bible!

than zero, negative numbers, fractional and irrational numbers. All were invented to express some kind of abstract concept that wasn't previously thought to be a 'proper number'.

With just the positive natural numbers (which at one point was all we thought of as a number), you can't represent the solution to this equation:

1 + x = x

So people invented the notion of zero, and made a symbol for it.

Then, they wanted to solve equations along the lines of:

1 + x = 0

and so invented the idea of a negative number. This must have seemed pretty damn strange and intangible at first.

Then of course people want to talk about ratios, and represent solutions to equations like

2x = 1

So they invent the concept of a rational number / fraction.

Then Pythagoras famously proves that the length of the hypoteneuse of a certain triangle - the square root of 2 - is not rational. This leads to greeks getting their knickers in a philosophical twist about how is it even possible to have a number that isn't a ratio of whole numbers?

But they get over it, and invent the notion of a square root and irrational numbers.

Finally, people (more physicists and algebraists this time, but still) want to represent the solution to an equation like

x^2 + 1 = 0

It turns out one can construct an expanded number system (a field, to be precise) which extends the reals and makes these things possible to represent. They happen to be very useful.

It's got to the point now in some of the fields where they're waiting for the mathematicians to come up with the mathematical tools necessary to carry the research further.

Though I always preferred the abstraction of mathematics rather than the application of physics. It was a pretty stupid choice by me really. I actually never finished my degree and I'm considering going back to do maths as part of my career development.

I believe it was further distributions, like Chi squared tests, things like that. It was 5 years ago for me now.
I did every maths module, though I didn't take exams in the final 3.

are two different ways of writing the same number.

It's not the case that every decimal sequence represents a unique Real number. This sometimes confuses people a little. When the decimal representation ends in recurring 9's, the representation isn't actually unique.

If you're interested in understanding why this is the case, it may help to get a first-year analysis textbook and read some of the basic proofs about limits and the real numbers.

It'll also help to see how the real numbers are commonly defined/formalized. There are various ways but Dedekind cuts and equivalence classes of Cauchy sequences are probably the most common.

If you've done a bit of calculus though - especially if you've used calculus for something useful like physics - you may be able to just 'see' why 0.9999... and 1 must be the same real number. The intuition of what a real number is, kinda captures the fact that the difference between 0.9999... and 1 is irrelevant

is that pontificating about these things has absolutely no point unless you're working from a precise defintition of what a real number constitutes, and what an expression like '0.999...' represents in formal terms.

Mathematicians do have these definitions, and they lead to it's being equal to one.

the limit of the sequence is equal to one, and the limit of the sequence is what an expression like '0.999...' is taken to represent over the real numbers.

Again, the term 'limit' here has a very precise definition - as I say it's pointless arguing these things unless you have precise definitions.

I see. You can tell I bunked out of maths and failed physics due to my lack of maths.

I'm still not comfortable with the idea though.

Say for example you removed a microscopic chip from the standard kilogramme, surely it wouldn't be THE kilogramme anymore. It'd be close enough for most purposes though.

There is a notion of a limit of a sequence (0.9, 0.99, 0.999, 0.9999, ...). This idea of a limit has a very precise definition, and in this case is (provably) equal to one.

numbers are ALWAYS theoretical. if you take a number to refer to a weight or other specific type of quantity, then the quantity is always going to be infinitesimally less or more than the number by which you are referring to that quantity.

this being the case, why is the practical / theoretical distinction valid?

## i read about this only yesterday

http://en.wikipedia.org/wiki/0.999...

still...can't quite...understand

</slow>

## yeah I saw it yesterday too.

I can follow it just about. It still seems slightly silly though.

## It only works if there are infinite 9s at the end

which is obviously impossible in a practical situation, but theoretically...

## yeah or

10/3=3.3333333333333....

3.33333333333333333......x3=9.9999999999......

9.99999999999999....=10

## but there are

infinite 9's

infinite

that means infinity.

## fool

The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief that 0.999… should have a last 9.

## There are number systems in which infintessimals exist

and they're pretty cool / interesting. The hyperreals are one.

Not in the standard real numbers though.

## I love maths.

As it tends to infinity, it becomes indistinguisable from one. Infinity is like one of the most useful things ever. Its in my top 3.

1. Imaginary Numbers

2. Zero

3. Infinity

That's a good list there.

## my personal favourite

is the square root of negative one. does that count as an imaginary number?

## Yes.

It is i

It is also the most useful number ever when doing physics.

## How do you use it?

Also: Roots of negative numbers

WTF

## exactly

## Well when solving equations

with negative factors and bits it becomes very useful. At A-Level you get taught to ignore the factors of i, but that's not the correct thign in more advanced stuff. It's all very boring for most people.

So like the square root of -9 is 3i. If you are after information better than I can give it, it's a good idea to go to a university maths library and ask for Mary Boas. It's like the university maths/physicist/engineers bible!

## sqrt(-1) is no less weird, when you think about it,

than zero, negative numbers, fractional and irrational numbers. All were invented to express some kind of abstract concept that wasn't previously thought to be a 'proper number'.

With just the positive natural numbers (which at one point was all we thought of as a number), you can't represent the solution to this equation:

1 + x = x

So people invented the notion of zero, and made a symbol for it.

Then, they wanted to solve equations along the lines of:

1 + x = 0

and so invented the idea of a negative number. This must have seemed pretty damn strange and intangible at first.

Then of course people want to talk about ratios, and represent solutions to equations like

2x = 1

So they invent the concept of a rational number / fraction.

Then Pythagoras famously proves that the length of the hypoteneuse of a certain triangle - the square root of 2 - is not rational. This leads to greeks getting their knickers in a philosophical twist about how is it even possible to have a number that isn't a ratio of whole numbers?

But they get over it, and invent the notion of a square root and irrational numbers.

Finally, people (more physicists and algebraists this time, but still) want to represent the solution to an equation like

x^2 + 1 = 0

It turns out one can construct an expanded number system (a field, to be precise) which extends the reals and makes these things possible to represent. They happen to be very useful.

## oops

by 1 + x = x, I meant 1 + x = 1.

:)

## I like you matt.

You share my nerdiness? You a physics graduate?

## Oh wait! I read your link at the bottom of the page!

Ah ha! I wish I had done maths instead of Theoretical Physics.

## why thankyou

Maths is fun.

That page is rather out of date now! I should update it I guess.

Theoretical physics is pretty hardcore though. Some very interesting maths has arisen from it.

## Very much so.

It's got to the point now in some of the fields where they're waiting for the mathematicians to come up with the mathematical tools necessary to carry the research further.

Though I always preferred the abstraction of mathematics rather than the application of physics. It was a pretty stupid choice by me really. I actually never finished my degree and I'm considering going back to do maths as part of my career development.

## my sister showed me her

year 9 maths homework the other day and I couldn't do it. Something to do with simultaneous equations. Maths is silly.

## imaginary numbers are da bomb

I once saw a whole hour long programme on the nature of the number 1.

## Maths

is cool. What were everyone's favourite Maths A-Level modules? Mine was M2!

## s2

## S2?

Statistics is Maths for girls!

## aka useful maths

## P6

Also S3 was good!

## Wow

we're getting into Further Maths territory here! FP2 was my favourite of the further modules. I didn't even do S3. What did it entail?

## s2 was in my further maths

but not s3

## If I remember correctly

I believe it was further distributions, like Chi squared tests, things like that. It was 5 years ago for me now.

I did every maths module, though I didn't take exams in the final 3.

## P3

It made everyone very angry, which was hilarious.

## thats not cool

## and that is why

art>science :P

## this is quite a silly thing to say.

## ..

That is WAY cool.

## ..

No, you're silly.

## 0.99999... and 1

are two different ways of writing the same number.

It's not the case that every decimal sequence represents a unique Real number. This sometimes confuses people a little. When the decimal representation ends in recurring 9's, the representation isn't actually unique.

If you're interested in understanding why this is the case, it may help to get a first-year analysis textbook and read some of the basic proofs about limits and the real numbers.

It'll also help to see how the real numbers are commonly defined/formalized. There are various ways but Dedekind cuts and equivalence classes of Cauchy sequences are probably the most common.

If you've done a bit of calculus though - especially if you've used calculus for something useful like physics - you may be able to just 'see' why 0.9999... and 1 must be the same real number. The intuition of what a real number is, kinda captures the fact that the difference between 0.9999... and 1 is irrelevant

## Another way of putting it:

0.9999... is basically just an abbreviation for:

"The limit, as n tends to infinity, of the sum from i=1 to n of 9*10^(-i)"

This, of course, evaluates to one.

:-)

## In this expression, the phrase

"tends to infinity" has a very precise defined meaning, which you'll learn if you study a bit of analysis.

'Infinity' is not some mystical magic number in these expressions.

## matt is right !

## even if there are infinite 9s

then it can only be 'as close to one as possible', but can never be 'one' as thats an independant integer, surely.

It may be that the difference is so small that it's 'effectively' equal to 'one', but it can never 'be' one.:-S

*goes to hide inferior maths corner*

## no, it's equal to 1

your justification would be correct if there were not infinite 9's, but there are, it's the same as 1

## The problem here

is that pontificating about these things has absolutely no point unless you're working from a precise defintition of what a real number constitutes, and what an expression like '0.999...' represents in formal terms.

Mathematicians do have these definitions, and they lead to it's being equal to one.

## but even if it does carry on to infinity

it's still not one. It'll be very, very, very, very, very, close. But not one, at any point.

## The sequence itself never reaches one, but

the limit of the sequence is equal to one, and the limit of the sequence is what an expression like '0.999...' is taken to represent over the real numbers.

Again, the term 'limit' here has a very precise definition - as I say it's pointless arguing these things unless you have precise definitions.

## hmm.

I know what you mean. How can you place a limit on infinity though?

## 'Limit' is used as a technical term

not in the vague colloquial sense of the word.

If you'd like to see the definition:

http://en.wikipedia.org/wiki/Limit_of_a_sequence

## a ha

I see. You can tell I bunked out of maths and failed physics due to my lack of maths.

I'm still not comfortable with the idea though.

Say for example you removed a microscopic chip from the standard kilogramme, surely it wouldn't be THE kilogramme anymore. It'd be close enough for most purposes though.

## No matter how microscopic

the chip would still have a finite, non-zero size. A size that's a real number which could be written down.

From a mathematicians point of view, this makes all the difference.

The notion of something infinitessimal is not a positive real number and can't ever form the difference between two real numbers.

## There's no such thing

as an infinitessimal real number.

There is a notion of a limit of a sequence (0.9, 0.99, 0.999, 0.9999, ...). This idea of a limit has a very precise definition, and in this case is (provably) equal to one.

## Oh maths

## I think i get it now

If theres infinite 9's, the "difference" between 1 and 0.99... must be infinitly small, i.e not there?

Is that right?

## God damn!

Here I was ignoring my maths degree and DiS makes me feel bad that there's worksheets just waiting for me upstairs.

## we're doing this now.

Silly.

## thing is

numbers are ALWAYS theoretical. if you take a number to refer to a weight or other specific type of quantity, then the quantity is always going to be infinitesimally less or more than the number by which you are referring to that quantity.

this being the case, why is the practical / theoretical distinction valid?

be gentle. me no know maths.