What is 0^0 equal to?
By Murray Bourne, 22 Jan 2009
In the introduction to Laws of Integral Exponents, I mention the debate about the value of 00. Is it zero or is it one?
Why is it a problem? Look at the following 2 patterns:
Multiply 0 as many times as you like, you get 0.
0^3 = 0
0^2 = 0
0^1 = 0
0^0 = 0
But then again, any number raised to the power 0 is 1:
3^0 = 1
2^0 = 1
1^0 = 1
0^0 = 1
That's why there is dispute about the value of 0^0.
On the page linked to before, I wrote:
It is most commonly regarded as having value 1
An interesting conundrum. Sometimes our wonderful, (normally) consistent system of math breaks down.
See the 42 Comments below.
22 Jan 2009 at 9:22 pm [Comment permalink]
Ah....
x^0=1 because of the rule x^s divided by x^t = x^(s-t)
If s is larger than t, we have more x in the numerator, so with canceling all of the common x's we are left with (s-t) in the numerator.
For example: x^5 divided by x^3 we have (x*x*x*x*x) divided by (x*x*x) so we can cancel three of the x's in the numerator and three in the denominator (because x divided by x = 1)
leaving two x's remaining, and therefore x^2
But if we have the same number of x's in numerator and denominator then we can cancel all of them - or divide all of them leaving 1*1*1*1... which is clearly 1.
It is not convention that makes this true. It is what follows from the basic laws of algebra.
x^0= 1 is not a convention, it is a consequence.
22 Jan 2009 at 9:28 pm [Comment permalink]
Thanks, Susan
But if x = 0, we cannot divide as you are suggesting, because dividing by 0 (or any power of 0) is undefined.
23 Jan 2009 at 6:55 am [Comment permalink]
Exactly!
0^0 is not a conundrum it is an impossibility!
Because dividing by 0 doesn't make sense!
When I teach dividing by 0 as impossible, I say to my students -
Suppose there are 20 students in class and I brought in 40 donuts how many would each student get? They say easy - 2 because 40/20=2
So I say on a day like today, when I bring 0 donuts to divide up, how many do the 20 students get, and logically the answer is none because 0/20=0
Ah but if I show up for class and none of my students show up, then what do I do? I can't dive my 20 donuts up between 0 students! It's impossible. Plus - if none of them show up, they're in big trouble!
It's those little math stories that help what seems mystical and tricky just seem common sense!
24 Jan 2009 at 3:49 am [Comment permalink]
Nice one Susan, I like the way you interpret it. I am not Math teacher myself but such math questions always intrigue me. A quick question to both you and Zac. Are there any significantly known patterns in Prime numbers. I worked with a Prof for some time in college on that for a little time, but never had time to see it through..
25 Jan 2009 at 9:11 am [Comment permalink]
I personally don't think that stories and common sense are a way to decide something in math. Complex numbers don't make sense or fit any story but they are valid nonetheless.
A different way to approach this problem is using L'Hopital's rule: lim x->0 of x^x = y
take the natural log of both sides:
lim x->0 x ln x = ln y
lim x->0 lnx/(x^-1) = lim x->0 (1/x)/(-x^-2) = lim x->0 -x = 0 = ln y
y = 1 = lim x->0 x^x
25 Jan 2009 at 1:43 pm [Comment permalink]
hi Eman
the way u interpret the result is great
but i am a little confused about the step(2nd last)
lim x->0 lnx/(x^-1) = lim x->0 (1/x)/(-x^-2) = lim x->0 -x = 0 = ln y
can u kindly simply brief it again..............
thanks a lot.
25 Jan 2009 at 3:23 pm [Comment permalink]
Hi Zac,
Thanks for posting this intriguing conundrum. It is interesting to consider why most mathematicians use 0^0=1 as well as whether or not the possibility of alternative choices represents a breakdown of consistency in mathematics.
The argument that 0^3=0,0^2=0,0^1=0 (etc) imply 0^0=0 does, at first, seem just as convincing as the other one, but actually it is a bit "one-sided", since one could just as well say 0^(-3) is undefined (or "infinite" if you like to put it that way) and ditto for 0^(-2), 0^(-1), etc, so why shouldn't the same be true for 0^(-0) (which is of course the same as 0^0)?
On the other hand, x^0=1 is true not just for positive x but also negative ones (and in fact for all nonzero complex numbers). So to extend the definition by defining 0^0=1 has the advantage of creating a continuous extension of x^0 to all real (and complex) numbers.
So x^0=1 can be usefully extended in a continuous way to all numbers but 0^x=0 cannot. This is why mathematicians generally adopt the 0^0=1 definition. But of course it remains just a definition - as is each step in the process of extension from the more primitive cases of b^p where b and p are both positive integers and where the power is defined by repeated multiplication.
In fact, it often happens that a concept or operation defined in a restricted context has more than one reasonable extension to a broader context and so we can't use both without giving them different names. Often people are careless and use the same name for the extended operation and it is usually not a problem because everyone makes the same choice. But if different people make different choices and both fail to change the name then this does lead to an inconsistency. I guess whether or not this is a breakdown of the consistency of mathematics depends on whether by "mathematics" we mean the set of statements made by mathematicians or the fundamental reality that those statements are intended to describe. Perhaps we should cconsider the existence of alternative interpretations for 0^0 as an inconsistency of mathematicians than of mathematics.
cheers,
Alan
P.S. With regard to the wording of the "repeated multiplication" definition, wouldn't "multiply b by itself p times" give us b^(p+1)? (or perhaps b^(2^p) if you interpret it another way?).
Personally I prefer the clumsier-sounding (but clearer) "multiply together p copies of b".
25 Jan 2009 at 4:48 pm [Comment permalink]
[...] This post at squareCircleZ (a very nice enrichment and support website for students and teachers of mathematics) raises the conundrum of how to define 0^0 if all positive x give x^0=1 and 0^x=0. It is indeed a useful exercise for students to consider why most mathematicians use 0^0=1 as well as whether or not the possibility of alternative choices represents a breakdown of consistency in mathematics. [...]
25 Jan 2009 at 4:59 pm [Comment permalink]
Saagar: Here are some articles on the spacing of prime numbers:
http://mathforum.org/library/drmath/view/68490.html
http://mathworld.wolfram.com/PrimeNumberTheorem.html
25 Jan 2009 at 5:49 pm [Comment permalink]
I'm sorry, Dave. I'm afraid I can't do that.
Something messed up. :-(