this morning

[ajva]

I cycled to the tube station; it saved me 10 mins.

Stef *will* be proud. I wonder if I can get him to study some algebra in return. If he's on a mission to get me ultra-fit, then surely I can be on a mission to cure him of his maths-dunceness?

*ponders*

Comments

[jhg.livejournal.com]

Where did you get a bike from?

Where did you leave it at Leytonstone?

And how come it didn't get nicked?

Oh, and 0.9 recurring does not equal 1!!


J

bikes and algebra

[ajva.livejournal.com]

1) Stef gave me one he'd fixed up from old tin cans, yoghurt pots and washing-up bottles etc.

2) D-locked into the bike racks outside.

3) Well, I've not gone home yet, so maybe it has done.

4) Oh for fuck's sake.

Anne. xxx

P.S. x=0.9(rec)
so 10x=9.9(rec) [shifting it all up a column, as one does when multiplying by 10]
10x-x=9x
is the same as
9.9(rec)-0.9(rec)=9
therefore 9x=9
therefore x=1

Re: bikes and algebra

[wechsler.livejournal.com]

I have the nagging feeling that this is much the same as the proof that x=x+1, based on the "fact" that (infinity)=(infinity)+1.

But then I already said I was crap at maths. Took me two beers to work out the "goat on a game show" one.

defence...

[ajva.livejournal.com]

*sigh*

I knew I shouldn't have started this.

I am indeed talking about "the ordinary real numbers you learn in school". Sorry if I didn't make that clear.

But look, imagine infinity as a package, right? One single package of infinity? You can't change what's in the package, but you can add or subtract the package to anything else.

So you can't say infinity=infinity+1 because you can't add 1 to infinity. That's like slipping a brick into a locked box. But you could put the brick on top of the box. They're still two separate things, though.

You can multiply 0.9(rec) by 10 and that's what it means to shift things one place to the left in this number system. By definition. What you are left with after the decimal point is exactly the same as what you started with, so you can take both away as they are the same thing - the locked box. It neatly gets rid of the problem of mixing up finite and infinite numbers, since you don't have to.

bleurgh maths

Re: defence...

[wechsler.livejournal.com]

Yeah, point. There's no point me sitting here thinking "sums to infinity converge but never meet" when they *do*.

Take it as a mark of just how bored I am today that my mind even tried to make something of that. If people don't answer some emails today I really and going to have to go and club them.

Re: defence...

[ajva.livejournal.com]

Och I know what you mean. Fret not. I think you're just thinking too much, to be honest. Isn't that probably a good thing? ;o)

Re: defence...

[wechsler.livejournal.com]

No. It makes my brain hurt ;)

Re: defence...

[ciphergoth.livejournal.com]

To add to what Anne's said:

You can assign the symbols in "infinity = infinity + 1" meanings such that the whole expression is meaningful and true[1], but then the "+" symbol becomes extremely ill-behaved. In particular, it no longer has a sister called "-" that does what you expect, so you can't just "subtract infinity from both sides" and expect it to work. Infinity is a tricky bugger like that.

But when Anne proves that 0.9(rec) = 1, she's using all the normal meanings of the symbols, so they're extremely well behaved - when we refer to the real numbers as a "field", it's another way of saying that all these symbols are incredibly well behaved! And so her proof is sound.

[1] in more than one way - I know of two! One of which is so weird that 1 + infinity = infinity but infinity + 1 > infinity!

interesting shit

[ajva.livejournal.com]

This is, indeed, where loads of interesting shit starts happening. I have only a vague idea about it, but am determined to read more once I have done with categorising wallpaper patterns and colouring in dodecahedra etc.

locked box

[ajva.livejournal.com]

No. Bad analogy. It's much more like adding a drop to the ocean. The ocean's still the ocean, and the drop has disappeared.

Excuse me but I had to say that. :o)

Re: bikes and algebra

[babysimon]

Suprised no-one else took you up on the gameshowgoat thing. That's something most people get with great difficulty - I had to write a program to simulate doing it 1000000 times when I was 16 to be convinced.

Re: bikes and algebra

[jhg.livejournal.com]

Bollocks. You're quite right.

In fact, now I see it again, we went through this proof at school, and I was impressed at the time. Ah well.

Well, I hope your bike does survive the day - but having one that looks shit is the best tactic for it, so it should be all right.

I would worry for the survival of any bike at Walthamstow Central (or Blackhorse Road), however.

Remember to check the tires before you jump onto it!

I am very tempted to bring my own bike up from home, but I really don't have the space to keep it!

I'll ask Soph, and if I'm feeling really cheeky, I'll ask Stef to help me fix it up...


J

Re: bikes and algebra

[ajva.livejournal.com]

Oh yes, I'd ask Stef for help to fix up your bike if I were you. He's far too polite to refuse and besides I think he still feels guilty for stealing your girlfriend. ;o)

[ciphergoth.livejournal.com]

I believe there are number systems where you can have numbers infinitesimally smaller than 1, but they are incredibly weird and full of head fuck. If you're thinking about the ordinary real numbers you learn in school, 0.9999 recurring is exactly equal to one; it's just a different representation of the same number.

aaagh!

[thekumquat.livejournal.com]

This is where maths and I part company...
sticking with our standard real numbers, surely 0.9999etc can't be equal to 1 but is equal to 1 - (1/infinity)? Isn't that the point???

I ought to point out that I got an effortless A in GCSE maths and an effortful U in A-level maths. It was the idea of "feeling" your way to a solution for an equation that got me - logic I was fine with.

Which meant I never got to grips with things like inverse sines and cosines etc. After learning for years that the inverse of anything is 1/anything, to eventually suss that the cause of my mistakes was that suddenly an inverse sine was *not* 1/sin, did my head in.

As for complex numbers... "We try solving this equation, but it's difficult, so we'll multply everything by the square root of minus one, as you do, and turn it back again later when we do the washing up, and hope it comes out in the wash."

And people call the subject logical???

Re: aaagh!

[ciphergoth.livejournal.com]

The thing you're looking for when you say "1/infinity" is an infinitesimal. As I said before, there *are* number systems that include infinitesimals, but they have many counterintuitive properties. There are no infinitesimals among the real numbers - for every real number x > 0, there is a perfectly ordinary real number n such that 1/n < x. So either two numbers are exactly the same, or they are some distance apart, with a bit of the number line separating them.

It's clear that there's no distance between 0.9(rec) and 1 - they're the same to within 1/1000 or 1/1000000, or any fraction you care to name. So they're identical.

Or to put it another way, there's something mesmerising about the decimal expansion of 0.999999... that makes people think it must be different from 1. If it were, what would the decimal expansion of 1 - 0.999999... be?

The complex numbers are actually the best behaved numbers of all...

Re: aaagh!

[ajva.livejournal.com]

Might I just step in at this point and say...yes, Hessie! You're absolutely right!

You say 0.9(rec)=1-(1/infinity)

And indeed, it is so.

But what do you get if you divide anything at all by 0? Infinity.
Therefore, what do you get if you divide anything at all by infinity?
0

so your original equation becomes

0.9(rec)=1-(1/infinity)=1-0=1

So there. You are completely and utterly correct. :o)