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Is 0^0 defined or not?

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Started by Boops January 01, 2004 05:03 PM

29 comments, last by Boops 20 years, 3 months ago

Boops

128

Author

January 01, 2004 05:03 PM

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]

superpig

1,826

January 01, 2004 05:14 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
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bigbadboo

139

January 01, 2004 06:00 PM

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]

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civguy

308

January 01, 2004 06:23 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]

temp_ie_cant_thinkof_name

100

January 01, 2004 06:26 PM

When in practice would you raise zero pies to the zeroth power?

"I study differential and integral calculus in my spare time." -- Karl Marx

Boops

128

Author

January 01, 2004 06:30 PM

Thanks for the replies

superpig: your 0^0 = 0/0 proof convinced me that it''s indeed undifined

nice explanation civguy.

Cedric

158

January 01, 2004 06:34 PM

quote:Original post by Boops
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?

Strictly speaking, and AFAIK, it''s undefined.

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.

tHiSiSbOb

463

January 01, 2004 06:47 PM

quote:Original post by Cedric
quote:Original post by Boops
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?
Strictly speaking, and AFAIK, it''s undefined.

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.

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