Bit of maths

[Lazlo]

Originally Posted by Sixsquid

Nothing is finite. Nothing works perfectly in our universe because we are limited by our own perceptions.
Consider, how many days in a year? So what to we do ? We add em together to make a a full number to fit in our realm of understanding and bung in February once every 4 years

... eeeeerrrrmmmm ... not quite

It's a leap year every 4 years, except every 100 years except every 1000 years. Which is why Feb 29 2000 was such a big deal for us thicko programmers

[Letz]

Originally Posted by Lazlo

... eeeeerrrrmmmm ... not quite

It's a leap year every 4 years, except every 100 years except every 1000 years. Which is why Feb 29 2000 was such a big deal for us thicko programmers

Yeah, but you know what I meant

[Lazlo]

Originally Posted by Sixsquid

Yeah, but you know what I meant

Yes I did, sorry.

I think the universe works just fine, it's just we humans that have a problem fitting ourselves within it. For instance, there's no particular reason why we should even bother with leap years. We could just stick to 365 days and agree to let the seasons come a quarter of a day later every year.

[uwila]

0.9 recurring is not the same as 1. Our way of writing it down as a series of 9s is simply an approximation to the value. The exact value can be written as

1 - e

where 'e' is an arbitrarily small value. The mistake in the maths in the OP comes by assuming that multiplying by 10 gives 9.9r whereas it really gives
10 -10e We can make 'e' as small as we like, but it is never 0.

One ninth is 0.1r so one might think that nine ninths are 0.9r. However, 0.1r is simply an approximation to 1/9 and not the exact value, so 0.9r is only an approximation to 9/9.

Here's another one to ponder:
Infinity is the biggest number - everyone knows that...wrong, there are different sizes of infinity. There is countable and uncountable

Think of the 'natural numbers', which is the number series 1,2,3,4.....infinity. So we can always think of one more number by adding 1 to the last one.

Now think of all the fractions between 0 and 1. As it happens, there are many more fractions between 0 and 1 than whole numbers.

Firstly, there is 1/2, 1/3, 1/4, 1/5 ...which is a never ending series because we can always make a new fraction. We also have 2/3, 2/5, 2/7, 2/9...another infinitely large series of new fractions. You can see that we could invent lots more sequences of fractions all between 0 and 1 so there are even more fractions than whole numbers. In fact the number of rational fractions (those which are n/m where n and m are integers) between 0 and 1 is uncountable.

Confused? I should hope so, I have been for the last 30 years since university...

[JAG]

Originally Posted by uwila

0.9 recurring is not the same as 1. Our way of writing it down as a series of 9s is simply an approximation to the value. The exact value can be written as

1 - e

where 'e' is an arbitrarily small value. The mistake in the maths in the OP comes by assuming that multiplying by 10 gives 9.9r whereas it really gives
10 -10e We can make 'e' as small as we like, but it is never 0.

One ninth is 0.1r so one might think that nine ninths are 0.9r. However, 0.1r is simply an approximation to 1/9 and not the exact value, so 0.9r is only an approximation to 9/9.

Here's another one to ponder:
Infinity is the biggest number - everyone knows that...wrong, there are different sizes of infinity. There is countable and uncountable

Think of the 'natural numbers', which is the number series 1,2,3,4.....infinity. So we can always think of one more number by adding 1 to the last one.

Now think of all the fractions between 0 and 1. As it happens, there are many more fractions between 0 and 1 than whole numbers.

Firstly, there is 1/2, 1/3, 1/4, 1/5 ...which is a never ending series because we can always make a new fraction. We also have 2/3, 2/5, 2/7, 2/9...another infinitely large series of new fractions. You can see that we could invent lots more sequences of fractions all between 0 and 1 so there are even more fractions than whole numbers. In fact the number of rational fractions (those which are n/m where n and m are integers) between 0 and 1 is uncountable.

Confused? I should hope so, I have been for the last 30 years since university...

Fantastic, seriously, fantastic!

[captainnemo]

The Greeks recognised the concept of paradox which is often depicted in the race between a hare and a tortoise over an infinite distance where it seems the tortoise would always be that fraction ahead where starting before the hare.

Another paradox, my favourite is the Cretan or Liar's Paradox..

Epimenides of Crete states that all Cretans are liars. True or false?

Maths is supposed to be precise, right or wrong answers but there is still room for deviation. Pi, the ratio of circumference to diameter has never been finally quantified (3.142 does for most of us but it sort of goes on a bit more) Then there is the uncomfortable step change for sub-atomic particle physics. Einstein went to the grave wrestling with the problem of a unifying theory (which probably gave Newton a wry satisfaction)

Actually, all I really know about maths is that if I'd failed the O level first time then I'd never have got through a retake.....calculators on phones etc have made my life so much better.

[Lazlo]

So.... using an arbitrarily small e we get:


a = 1 - e
10a = 10 - 10e
10a - a = (10 - 10e) - (1 - e)
9a = 10 - 10e - 1 + e
9a = 9 - 9e
a = 1 - e

Hurrah! Sounds reasonable to me.

Originally Posted by uwila

Here's another one to ponder:
Infinity is the biggest number - everyone knows that...wrong, there are different sizes of infinity. There is countable and uncountable

Think of the 'natural numbers', which is the number series 1,2,3,4.....infinity. So we can always think of one more number by adding 1 to the last one.

Now think of all the fractions between 0 and 1. As it happens, there are many more fractions between 0 and 1 than whole numbers.

Firstly, there is 1/2, 1/3, 1/4, 1/5 ...which is a never ending series because we can always make a new fraction. We also have 2/3, 2/5, 2/7, 2/9...another infinitely large series of new fractions. You can see that we could invent lots more sequences of fractions all between 0 and 1 so there are even more fractions than whole numbers. In fact the number of rational fractions (those which are n/m where n and m are integers) between 0 and 1 is uncountable.

And then of course there are irrational numbers (like pi) that fall between the rational numbers. And then on top of that there are imaginary numbers!

There's an analogy I read in one of Simon Singh's books which is to do with hotel bookings.

Imagine a hotel with an infinite number of rooms.

An infinite number of guests turn up. No problem: there is room for them all. The hotel is now fully booked.

Another guest turns up. No problem: we just get all the existing guests to have a look at their room number and move to that room number plus 1. There is now one room free.

Another infinite number of guests turn up. No problem again: we just get all the existing guests to have a look at their room number and move to that room number multiplied by 2. There is now an infinite number of rooms free, space for everybody.

And so on.

[dry suit diver]

Originally Posted by captainnemo

The Greeks recognised the concept of paradox which is often depicted in the race between a hare and a tortoise over an infinite distance where it seems the tortoise would always be that fraction ahead where starting before the hare.

Another paradox, my favourite is the Cretan or Liar's Paradox..

Epimenides of Crete states that all Cretans are liars. True or false?
Maths is supposed to be precise, right or wrong answers but there is still room for deviation. Pi, the ratio of circumference to diameter has never been finally quantified (3.142 does for most of us but it sort of goes on a bit more) Then there is the uncomfortable step change for sub-atomic particle physics. Einstein went to the grave wrestling with the problem of a unifying theory (which probably gave Newton a wry satisfaction)

Actually, all I really know about maths is that if I'd failed the O level first time then I'd never have got through a retake.....calculators on phones etc have made my life so much better.

thats as bad as the directive we received the other day

signature stamps are not to be used on correspondence, any documents with a stamp will not be deemed to be signed and are to be disregarded

whats at the bottom of the memo


a stamped signature. this one has had the policy dept busy for 2 days trying to give me an answer

[uwila]

Originally Posted by Lazlo

Imagine a hotel with an infinite number of rooms.

An infinite number of guests turn up. No problem: there is room for them all. The hotel is now fully booked.

Another guest turns up. No problem: we just get all the existing guests to have a look at their room number and move to that room number plus 1. There is now one room free.

Another infinite number of guests turn up. No problem again: we just get all the existing guests to have a look at their room number and move to that room number multiplied by 2. There is now an infinite number of rooms free, space for everybody.

And so on.

But now imagine you have a guest list where every guest was sent an invitation upon which a rational fraction between 0 and 1 was written (see my earlier post). Assume that you've used all the rational fractions available.*

The hotel has rooms numbered 1, 2, 3, 4, 5, ... to infinity.

Does the number of guests exceed the number of rooms available? Can the hotel find rooms for everyone?

*This won't work during any postal strike, 'cos most people won't actually receive the invitations

[Lazlo]

Originally Posted by uwila

But now imagine you have a guest list where every guest was sent an invitation upon which a rational fraction between 0 and 1 was written (see my earlier post). Assume that you've used all the rational fractions available.*

The hotel has rooms numbered 1, 2, 3, 4, 5, ... to infinity.

Does the number of guests exceed the number of rooms available? Can the hotel find rooms for everyone?

*This won't work during any postal strike, 'cos most people won't actually receive the invitations

Well, one way of looking at it is:

No. There are more rational numbers between 0 and 1 than there are positive integers, therefore there is not enough rooms.

But another way of looking at it would be:

Yes: Take every invitation note, look at the rational number n/m and convert that rational number to a positive integer by combining the two numbers into one: e.g. 1/3 becomes 13, 26/9216 becomes 269216 and so on. Now everyone has a unique room number, and everyone is accommodated.