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

[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)