Search

Navigation

User login

Upcoming events

No upcoming events available

There are 553 members of the Warwick Mathematics Society, of which 0 are new today!
We're 110% of the way toward our target of 500 members.
You can join up on the UWSU website.

Who's new

Ask a hard question, get an easy answer

Posted: Submitted by cj on 23 May 2007 - 9:18pm.

Joined: 2006-10-10
Posts: 524

This is an issue very personal to me. I like to talk about mathematics, however I find a lot of my discussions with people not taking university mathematics fall into a depressingly similar routine of trying to correct misconceptions. I'm talking about things like division by zero, or why 0.999..... = 1.

I know we have some truly gifted expositors of mathematics here, and what I am after is some thought-out, simplistic answers to some of the most commonly debated questions in amateur mathematics. I can then hone my arguments a bit better next time they come up in conversation. I don't mind getting links to pages which explain them well, but I'm also interested to see just how clear we can get our explanations :) So, some questions:

1. Why is division by zero not allowed? In particular, explain why it can't just be interpreted as "sharing n apples to 0 people" i.e. n/0 = 0

2. Why isn't infinity a number?

3. Why is 0.999... = 1?

That's enough for now ^^

cosmin

Posted: 23 May 2007 - 11:56pm

Joined: 2006-11-02
Posts: 1291

1) Well, we have $0x = 0y$ for all $x$ and $y$ . Thus, allowing division by 0 would allow you to conclude that $x = y$ , which we don't really want obviously (unless we're working in the ring with one element :D). This generalizes to zero divisors in arbitrary rings for example.

2) Because it wouldn't be compatible with the operations for example (and in fact with the whole ordered field structure of the reals): for instance, would we have $\infty + 1 = \infty$ and thus substracting $\infty,$ $1 = 0$ ?

3) A nice way to see this is: $\sum_{n=1}^\infty \frac{9}{10^n} = \frac{9}{10}\cdot\frac{1}{1-10^{-1}} = 1$ . The cool thing about this is that it generalizes to any base $b$ : if $c = b-1,$ $\overline{0.ccc...} = 1$ since it's just $\sum_{n=1}^\infty \frac{b-1}{b^n} = \frac{b-1}{b} \cdot \frac{1}{1-b^{-1}} = \frac{b-1}{b-1} = 1$ . If you don't want to use infinite series, you have some slightly sloppier alternatives involving multiplication of decimals. For example, note that $1 = 3 \cdot \frac{1}{3} = 3 \cdot 0.333... = 0.999...$ or say $10 \cdot 0.999... - 9 = 0.999...$ so $0.999...$ is a solution of $10 x - 9 = x,$ i.e. $x = 1$ .

Posted: 24 May 2007 - 2:57am

Joined: 2006-10-10
Posts: 524

One really cool reason I heard for why infinity can't be a number is, what would its decimal representation be? All numbers need a decimal rep. Seems nice and explain-y

darthsteven

Posted: 24 May 2007 - 9:02am

Joined: 2006-08-31
Posts: 699

2. Surely Infinity is a concept rather than a number, and thus doesn't fit well on the real number line.

------------
Regards
Steven Jones
Website Editor
The Warwick Mathematics Society

Newtonswig

Posted: 27 May 2007 - 11:28pm

Joined: 2006-10-01
Posts: 439

Indeed, though on the riemann sphere it's positively at home...

Or the one point compactification of the real line...

Or non-standard analysis.

Yay infinity!!!

darthsteven

Posted: 2 June 2007 - 11:14am

Joined: 2006-08-31
Posts: 699

If these sorts of questions really interest you, do some reading about the p-adic numbers, and non-Archimedian fields in general. You can do some really cool things, and it makes the 'standard' analysis and algebra look fairly mundane.

------------
Regards
Steven Jones
Website Editor
The Warwick Mathematics Society

tom bainbbridge

Posted: 7 January 2009 - 12:32am

Joined: 2009-01-07
Posts: 1