0^0

No, that’s not a new smiley face of mine. So Xyon and I had a discussion today on what 0^0 (zero to the power of zero) equaled. I had always learnt that was undefined/undeterminate, but he learnt that it could equal either 0 or 1, and user gets to choose.

For me,

limit of x^0 as x approaches 0 = 1
limit of 0^x as x approaches 0 = 0

These 2 lines never intersect and therefore the limit of x^y as x and y approaches 0 = UNDEFINED (as I recalled in my calculus book).

However, we were looking a number sequences and they defined that 0^0 = 1, which started this whole debate in the 1st place.

We punched it into Google and got: 0^0 = 1.
We then tried to punch it into Maple and got the same answer.

Xyon then brought up a good point on how exponents are defined. Although this may not be the correct definition, it’s what has persisted it appears: x^y = 1 * (x multipled y times), therefore 1 * (any number multiplied 0 times) will always = 1.

However, from Math Forum:

According to some Calculus textbooks, 0^0 is an “indeterminate form.” What mathematicians mean by “indeterminate form” is that in some cases we think about it as having one value, and in other cases we think about it as having another.

The following is a list of reasons why 0^0 should be 1.

Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from the right.

From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x , if the binomial theorem is to be valid when x=0 , y=0 , and/or x=-y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.

Published by Addison-Wesley, 2nd printing Dec, 1988.

As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0) ; but Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) –> 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^g(x) –> 1 .


Random Crap:

Jeopardy – What’s a Ho(e) (from MS newsgroup) – A rather old video, but is always funny and should give you a few chuckles. Alex Trebeck asks, “This term for a long-handled gardening tool can also mean an immoral pleasure seeker” and Ken Jennings replies with, “What’s a ho(e).” It took Alex a moment before realizing the pun. Looking at the scores, Ken Jennings was dominating.

India Traffic (from MS newsgroup) – video proof that traffic lights are not needed!

Is Your Mechanic Cheating? (mirror) (from MS newsgroup) – Joel Grover reports on his investigation into Jiffy Lube shots in the Southland. That’s why you should never get your car tuned at a Jiffy Lube, especially if you’re in southern California. I can’t believe how organized this cheating was! It seems to be directly ordered from the district manager.

PS3 controller spelling mistake (from MsticAzn) – Sony mistakenly spells ‘Select’ ‘Serect’ on their new PS3 controller. It’s common for Japanese to mix up Ls and Rs since they make the same sound in Japanese.

Could coffee protect your liver against alcohol?Drinking coffee may shield the liver from the worst ravages of alcohol, a study of more than 125,000 people suggests. The risk of developing alcoholic cirrhosis of the liver dropped with each cup of coffee they drank per day. “Consuming coffee seems to have some protective benefits against alcoholic cirrhosis, and the more coffee a person consumes the less risk they seem to have of being hospitalised or dying of alcoholic cirrhosis,” says Arthur Klatsky at Kaiser Permanente Medical Care Programme in Oakland, California, US, who led the study. Haha, there’s no more need to worry about your liver since you can fix it by drinking coffee. I always find it interesting people are always trying to find good health benefits from food that are generally bad for health.

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