Is 0^0 defined or not?
My HP48GX calculator sais it's 1
Maple sais it's 1.
But at school I learned it's undefined, and so sais my TI40x calculator.
However, since Maple is made by mathematicians (I hope so ), who like defenitions and should shudder to program something that's mathematically incorrect, why did they say 0^0 = 1?
Trying some things myself, I found that
lim_x-->0 x^x = 1,
as well as
lim_x-->0 -x^-x
lim_x-->0 -x^x
lim_x-->0 x^-x
lim_x-->0 x^0
But the only limitthat becomes 0 instead of 1 is:
lim_x-->0 0^x
So is that the only one limit that sais the opposite of all the other ones, and is that why they say it's undefined, but because all others say it's 1, they defined it as 1 in Maple?
Anyway, personally I'd like if 0^0 was 1, it would make life so much easier
EDIT: Hmm I just found this: http://home.att.net/~numericana/answer/algebra.htm, look at the second topic, some guy there is claiming that 0^0=1. Is he right, or is he going to be flamed by any mathematician that sees him?
[edited by - Boops on January 1, 2004 6:13:26 PM]
x^a / x^b = x^(a-b)
0^n / 0^n = 0^(n-n) = 0^0
0^n = 0
.: 0^0 = 0/0 = undefined.
HOWEVER
a / a = 1
let a = 0^n
.: 0^0 = 0^n/0^n = a/a = 1.
I dunno, I suspect that the programs in question might be being lazy... checking for ^0 first and automatically returning 1, without checking the 0 that you''re raising.
Richard "Superpig" Fine
- saving pigs from untimely fates, and when he''s not doing that, runs The Binary Refinery.
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0^n / 0^n = 0^(n-n) = 0^0
0^n = 0
.: 0^0 = 0/0 = undefined.
HOWEVER
a / a = 1
let a = 0^n
.: 0^0 = 0^n/0^n = a/a = 1.
I dunno, I suspect that the programs in question might be being lazy... checking for ^0 first and automatically returning 1, without checking the 0 that you''re raising.
Richard "Superpig" Fine
- saving pigs from untimely fates, and when he''s not doing that, runs The Binary Refinery.
Enginuity1 | Enginuity2 | Enginuity3 | Enginuity4 | Enginuity5
ry. .ibu cy. .y''ybu. .abu ry. dy. "sy. .ubu py. .ebu ry. py. .ibu gy." fy. .ibu ny. .ebu
"Don''t document your code; code your documentation." -me
"I''m not a complete idiot; parts of me are missing." -}}i{{
but it doesnt make sense in practice
0 pies raised 0 times = 1 pie?
[edited by - bigbadboo on January 1, 2004 7:02:20 PM]
0 pies raised 0 times = 1 pie?
[edited by - bigbadboo on January 1, 2004 7:02:20 PM]
0^0 can be defined to be 1. It doesn't violate any other rules or lead to contradiction, so it's a definition such as the axiom a+(b+c) = (a+b)+c. It could've been defined to be 0, or even 42. But 0^0=1 is actually a pretty useful definition that takes some special cases away from certain formulas (e.g. Taylor series). On the other hand, there are some formulas where e.g. 0^0=0 would've been a better definition and such definition would've removed those special cases, but those formulas are in the minority.
[edited by - civguy on January 1, 2004 7:24:53 PM]
[edited by - civguy on January 1, 2004 7:24:53 PM]
Thanks for the replies
superpig: your 0^0 = 0/0 proof convinced me that it''s indeed undifined
nice explanation civguy.
superpig: your 0^0 = 0/0 proof convinced me that it''s indeed undifined
nice explanation civguy.
quote:Original post by BoopsStrictly speaking, and AFAIK, it''s undefined.
So is that the only one limit that sais the opposite of all the other ones, and is that why they say it''s undefined, but because all others say it''s 1, they defined it as 1 in Maple?
Another related example is sin(x)/x. It''s undefined at x=0. But since this function is used so often, and its limit is 1, mathematicians have called it sinc(x), which is sin(x)/x if x!=0 and 1 if x = 0.
quote:Original post by Cedricquote:Original post by BoopsStrictly speaking, and AFAIK, it''s undefined.
So is that the only one limit that sais the opposite of all the other ones, and is that why they say it''s undefined, but because all others say it''s 1, they defined it as 1 in Maple?
Another related example is sin(x)/x. It''s undefined at x=0. But since this function is used so often, and its limit is 1, mathematicians have called it sinc(x), which is sin(x)/x if x!=0 and 1 if x = 0.
I may be incorrect, but I remember my dad (Ph.D. in Mathematics) telling me that it is 1.
My TI83+ says it''s ERR:DOMAIN.
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quote:telling me that it is 1.
... if x->0
[edited by - DerekSaw on January 1, 2004 8:37:18 PM]
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