Question

What is 0^0 ?
is it 1 or 0?
Please, explain

Answer 1

1

Answer 2

Explain, please.

Answer 3

(anything)^0 =1, you count exponent
but 0^(anything) =0, you count the base
Why do you choose the first one but the second one?

Answer 4

by definition a^0 = 1 for any a... including 0

Answer 5

\[\lim_{x \rightarrow 0}e^{\ln(x^x)}=e^{\lim_{x \rightarrow 0}x^x}=e^{\lim_{x \rightarrow 0} \frac{\ln(x)}{\frac{1}{x}}}=e^{\lim_{x \rightarrow 0}\frac{\frac{1}{x}}{-\frac{1}{x^2}}} \\ =e^{\lim_{x \rightarrow 0}\frac{x}{-1}}=e^{0}=1\]
but we aren't talking about limits
i always under 0^0 and 0/0 is indeterminate form
it cannot be determined at 0 or 1

Answer 6

it cannot be determined as 0 or 1

Answer 7

enter 0^0 in calculator and you get 1. ... it is defined to be 1 to keep it consistent with all other a^0

Answer 8

\[0^0=0^{n-n}=\frac{0^n}{0^n}=(\frac{0}{0})^n\]

Answer 9

I watch this video, at 17:00, the prof use 0^0 =1, I wonder why, That's why I make question here http://www.youtube.com/watch?v=KJuSx1EXdd8

Answer 10

i always tell my students a^0=1 as long as a doesn't equal 0

Answer 11

\[\large (1-1)^n = \sum\limits_{k=0}^n \binom{n}{k}(-1)^{n-k}\]

plugin n = 0

Answer 12

it looks like that video is talking about limits

Answer 13

So, 2 profs, you and the prof in the video tap, teach the students (I am one of them) in 2 different ways? @myininaya .
You guys confuse students . hahah

Answer 14

*tape

Answer 15

i also don't see where you are talking about in the video where he says 0^0 is 1

Answer 16

At 15:00, he is talking about geometric sequence with r^n , at 17:00 , he expands the sequence as 1,0,0,0...

Answer 17

when r =0

Answer 18

a^3 = a*a*a*1
a^2 = a*a*1
a^1 = a*1
a^0 = 1

this work for a=0 as well.

@myininaya we know 0*3 = 0 even though 0/0 is not 3 ... same way we can define 0^0=1 even though we can't do 0^n/0^n

Answer 19

it doesn't matter what he called 0^0 the sequence numbers still get closer to 0

Answer 20

i will never believe 0^0 is 1

Answer 21

BUT IT IS :'(

Answer 22

\[0^0=0^{n-n}=\frac{0^n}{0^n}=(\frac{0}{0})^n \]
how do you argue this then @PaxPolaris ?

Answer 23

eh no 0^0 is problematic so mathematicians tend to avoid it

Answer 24

but i guess we could play with that arguement
0^5=0
but 0^5=0^{5-0}=0^5/0^0

Answer 25

but still 0^0 is an indeterminate form
this means it cannot be determined

Answer 26

https://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtml

Answer 27

what ever you do you will different results therefore we don't know what does that mean

Answer 28

\[0^0=0^{n-n} \ne\frac{0^n}{0^n}=(\frac{0}{0})^n\]

Answer 29

http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

Answer 30

i know i already found a countexample for that arugment
but there are other arguments against 0^0 being 1

Answer 31

If you want to believe in binomial theorem, define it as 1. You could define it as 0 and simply exuse binomial theorem and million other things your defintion contradicts with... below is not affected by whatever you choose
\[\large \lim\limits_{x,y \to 0} x^y = DNE\]

Answer 32

Conclusion: Depend on what the current professor says, we define whether 0^0 =1 or 0^0 =0 LOL

Answer 33

\(\large \dfrac{0}{0}\) is undefinable based on my understanding

we can define 0^0, 1^infty etc based on context..

Answer 34

Thanks you all. I really don't know this is a controversy problem in Math

Answer 35

if we believe that 0^0=1 a lot of things will get complicated lol
so 0^0 is problematic @myininaya gave already some counterexamples

Answer 36

well the counterexample i gave i was actually able to give a counterexample to that counterexample
saying that logic probably wasn't the best route to choose
--
bottom story is 0^0 cannot be determined (it is why they call it an indeterminate form)

Answer 37

if 0^0 was a number you would never have needed to use l'hosptal rule for that form

Answer 38

Exactly! limit is a different story

Answer 39

basically in number theory and sequences/series you can assume 0^0 as 1. but in multivariable calculus or when you consider limits of two different functions, you need to consider it as indeterminate. be aware of whats going on, then you will be fine..

Answer 40

much ado about nothing (about 0)

Answer 41

And I do agree with @ganeshie8 on the part you can define 0^0 as 1 for certain purposes.