Peano Postulates - Page 5

xilman · 2005-08-26, 17:07

Quote:

Originally Posted by mfgoode

Can anyone kindly explain to me how do we arrive at 0^0 =0 ?
when a^0 always = 1 Surely by continuity it should also be eqaul to 1 when a = 0?
2) By convention 0! =1 and 1! is also equal to 1. so 0 = 1 A paradox!

Mally

Both of these are jokes, right?

1) 0^a = 0 for all a>0.

As Bob has said, 0^0 is usually left undefined, but can be defined as either 0 or 1, depending which limit makes more sense in any given context. Almost always lim a->0 (a^0) is more useful in practice than lim a->0+ (0^a), not least because 0^a is undefined for a<0.

2) seems to be a variation on the frequently produced "paradoxes" based on the equality (-1)*(-1) = 1 * 1. A neat one, that I've not seen before.

Paul

trilliwig · 2005-08-26, 17:11

Quote:

Originally Posted by ET_

Sounds like Winston Churchill's quote about his "twin nation"...

Something like "They are not different from us... apart from language"

"Two nations divided by a common language". :D

mfgoode · 2005-08-28, 15:05

It was a great pleasure indeed receiving replies from both of you Bob and
Paul whom I consider to be the stalwarts of m/forum.
I got my information strangely enough from ‘The Penguin Dictionary of Curious and Interesting Numbers by David Wells. This book is a must for anyone interested in meaningful numbers from -1 and i to Graham’s Number
I have mentioned t his English author before, who has the rare distinction of having been a Cambridge scholar in maths and failing his degree. But he has various other feathers in his cap and each of his books on maths is a masterpiece. I strongly recommend his books.
I advanced his argument as a^0 = 1, so why not 0^0 ?
He says “Not so, 0^a is always zero so by the same argument from continuity 0^0 =0. This is much the same as Paul’s answer.
Regards 0! I came across this beautiful derivation but I cannot recall where and by whom.
If you go down the factorial line from n! you get
n! =n*(n-1)
Similarly 4! =4*(3!) right down the line coming to
2! = 2* (1!)
So 1! = 1! * 0 !.
Thus we take 0! as 1 also to make max. sense and max. use.
The symbol (!) is an operation and not an integer. Hence though the operation yields the same value the integers 0 and 1 cannot be equated.
For more information on the origin and properties of Zero, Garo gave an interesting website earlier on and I give the link to it.
http://www-gap.dcs.st-and.ac.uk/%7Eh...pics/Zero.html
Brahmagupta speaks of 0^2 and sq. rt. 0 also = 0
Another excellent book on zero is ‘The Nothing that IS’---
A Natural History of Zero’ by Robert Kaplan
Oxford Univ. Press. ISBN 0-19=514237-3 (pbk)
Mally.

2005-08-28, 15:05	#47
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²·3³·19 Posts	Peano Postulates It was a great pleasure indeed receiving replies from both of you Bob and Paul whom I consider to be the stalwarts of m/forum. I got my information strangely enough from ‘The Penguin Dictionary of Curious and Interesting Numbers by David Wells. This book is a must for anyone interested in meaningful numbers from -1 and i to Graham’s Number I have mentioned t his English author before, who has the rare distinction of having been a Cambridge scholar in maths and failing his degree. But he has various other feathers in his cap and each of his books on maths is a masterpiece. I strongly recommend his books. I advanced his argument as a^0 = 1, so why not 0^0 ? He says “Not so, 0^a is always zero so by the same argument from continuity 0^0 =0. This is much the same as Paul’s answer. Regards 0! I came across this beautiful derivation but I cannot recall where and by whom. If you go down the factorial line from n! you get n! =n(n-1) Similarly 4! =4(3!) right down the line coming to 2! = 2* (1!) So 1! = 1! * 0 !. Thus we take 0! as 1 also to make max. sense and max. use. The symbol (!) is an operation and not an integer. Hence though the operation yields the same value the integers 0 and 1 cannot be equated. For more information on the origin and properties of Zero, Garo gave an interesting website earlier on and I give the link to it. http://www-gap.dcs.st-and.ac.uk/%7Eh...pics/Zero.html Brahmagupta speaks of 0^2 and sq. rt. 0 also = 0 Another excellent book on zero is ‘The Nothing that IS’--- A Natural History of Zero’ by Robert Kaplan Oxford Univ. Press. ISBN 0-19=514237-3 (pbk) Mally.