Wikipedia:Reference desk/Archives/Mathematics/2012 May 14

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May 14

98 power 0=1

What is the complete procedure to solve the 98 power 0 is equal to 1? — Preceding unsigned comment added by Abachayo (talk • contribs) 14:02, 14 May 2012 (UTC)

That 98⁰ = 1 is not the result of a computation, but a definition: x⁰ = 1 for all real numbers x. Besides, your query sounds suspiciously like a request for help on a homework assignment.—PaulTanenbaum (talk) 14:06, 14 May 2012 (UTC)

98ⁿ = 1·98·98···98 (n multiplications by 98) for nonnegative integer values of n. So for n=0 you have 98ⁿ = 1. (Zero multiplications by 98). Bo Jacoby (talk) 15:28, 14 May 2012 (UTC).

Paul: x⁰ = 1 for all real numbers x except x=0, as 0⁰ is one of indeterminate forms. --CiaPan (talk) 05:16, 15 May 2012 (UTC)

0⁰=(ℝ≥0). Plasmic Physics (talk) 06:10, 15 May 2012 (UTC)

Because lim_n→−0 0ⁿ = ∞, and lim_n→+0 0ⁿ = 0. Plasmic Physics (talk) 06:47, 15 May 2012 (UTC)

Are you sure 0ⁿ makes sense for negative n...? --CiaPan (talk) 09:34, 15 May 2012 (UTC)

Yes, 0ⁿ = ∞ for all n where n<0. Plasmic Physics (talk) 12:55, 18 May 2012 (UTC)

That seems a little questionable. Sławomir Biały (talk) 13:28, 18 May 2012 (UTC)

Really? If 0⁻ⁿ = 1/0ⁿ and if 0ⁿ = 0 for n > 0, then 0⁻ⁿ = 1/0 = ∞. Plasmic Physics (talk) 14:19, 18 May 2012 (UTC)

0ⁿ = 1·0·0···0 (n multiplications by 0) for nonnegative integer values of n. So for n=0 you have 0ⁿ = 1. (Zero multiplications by 0). Bo Jacoby (talk) 07:48, 15 May 2012 (UTC).

No. Plasmic Physics (talk) 08:47, 15 May 2012 (UTC)

It's a bit more subtle than that, see Exponentiation#Zero_to_the_zero_power. -- Meni Rosenfeld (talk) 10:21, 15 May 2012 (UTC)

That's what I said. Zero to the power of zero is equal to all real positive numbers including zero; it equals the modulus of zero divided by zero. Yes, it equals one, but it also equals a thousand, and zero. I equals what ever you need it to. That is why it is classed as undefined. Plasmic Physics (talk) 10:58, 15 May 2012 (UTC)

Better to just say 0^0 is an indeterminate form as others have done. Your answer stretches the word "equals" beyond any useful meaning. Certainly "0⁰=(ℝ≥0)" is not a satisfying answer, and looks like the output from a CAS that doesn't know any better. Staecker (talk) 15:52, 15 May 2012 (UTC)

The definition 0⁰=1 is assumed in the expression for polynomials and power series f(x)=Σa_ixⁱ where f(0)=Σa_i0ⁱ=a₀ and nothing else. Bo Jacoby (talk) 13:22, 15 May 2012 (UTC).

That is correct. For the expression of polynomials and power series, it is required in that instance for the answer to be one. In other instances it is not one. Plasmic Physics (talk) 15:06, 15 May 2012 (UTC)

When the exponent is the real number 0 it is indeterminate ("equal to anything"). When the exponent is the integer 0 it is equal to 1. That's the subtlety I alluded to but I didn't want to start a heated debate. -- Meni Rosenfeld (talk) 17:44, 15 May 2012 (UTC)

Carefull, it can't equal anything - only anything equal to or greater than 0. I could simplify my expression to 0⁰ ≥ 0. Plasmic Physics (talk) 09:53, 16 May 2012 (UTC)

$\lim _{t\to 0^{+}}\left(\exp \left({\frac {-1}{t}}\right)\right)^{t\pi i}=-1$. Ok, this requires the exponent to be the complex number 0. -- Meni Rosenfeld (talk) 21:44, 16 May 2012 (UTC)

In no case does it make sense to give 0⁰ a value different from one. In no case does it make sense to distinguish between real zero and integer zero. 0=0. Bo Jacoby (talk) 23:19, 15 May 2012 (UTC).

Well, your second sentence is just wrong. Certainly much of the time, there is no particular advantage in making the distinction. Sometimes, though, there is. --Trovatore (talk) 08:54, 16 May 2012 (UTC)

Your first sentence is also wrong, read Rosenfeld's link. Plasmic Physics (talk) 09:19, 16 May 2012 (UTC)

Bo, may I be the first to say: {{citation needed}}. Sławomir Biały (talk) 12:55, 16 May 2012 (UTC)

The very long discussion on the exponentiation talk page showed no case where it made sense to give 0⁰ a value different from one, and no case where it made sense to distinguish between real zero and integer zero. Bo Jacoby (talk) 17:46, 16 May 2012 (UTC).

That's just false. It did indeed show cases where it made sense to distinguish between real zero and integer zero. --Trovatore (talk) 19:15, 16 May 2012 (UTC)

Bo, this seems to be your private opinion, and while you're certainly entitled to it, it is not a widely accepted mathematical truth. Indeed, that discussion page and article indicate very good reasons that 0^0 should not be regarded as the same as 1 in all circumstances. Sławomir Biały (talk) 23:14, 16 May 2012 (UTC)

Of course I am not entitled to a private opinion. It is up to Trovatore and Sławomir to prove that 0≠0 and 0⁰≠1. Bo Jacoby (talk) 05:15, 17 May 2012 (UTC).

Does my limit function not prove anything besides that I can use mathematical notation? Plasmic Physics (talk) 05:41, 17 May 2012 (UTC)

I direct your attention please to the article Exponentiation, the discussion page Talk:Exponentiation, as well as to the discussion that has already taken place here. Sławomir Biały (talk) 14:35, 17 May 2012 (UTC)

The power x^y can be defined for real values of x and nonnegative integer values of y by x^y = 1·x·x···x (y multiplications by x). The power x^y can be defined for positive real values of x and real values of y by x^y = e^{y log(x)} (where the exponential function e^x is defined by the power series e^x = Σ_n xⁿ/n! and the logarithm is defined by the equation x = e^log(x) ). These two definitions give the same result in the intersection of their domains, that is for positive real values of x and nonnegative integer values of y. For y>0 the limit for x→0⁺ is lim_x x^y = 0^y = 0. For x>0 the limit for y→0⁺ is lim_y x^y = x⁰ = 1. No redefining or undefining of 0⁰ = 1 can make these limits equal. Bo Jacoby (talk) 13:03, 17 May 2012 (UTC).

This argument undercuts itself. Being equal on the common domain does not make these the same function. Indeed, the very fact that these functions have different domains makes them different functions. x=0 is in the domain of one but not the other. It's just that simple. Sławomir Biały (talk) 14:45, 17 May 2012 (UTC)

Being equal on the common domain allows the functions to be merged into a single function, x^y , which is uniquely defined on the union of the two domains. Bo Jacoby (talk) 14:57, 17 May 2012 (UTC).

And that's a third function, not the same as the other two. Sławomir Biały (talk) 16:26, 17 May 2012 (UTC)

Yes, and? Bo Jacoby (talk) 16:45, 17 May 2012 (UTC).

When people say that 0^0 is not defined, they mean the second function, not the first or third. Sławomir Biały (talk) 17:29, 17 May 2012 (UTC)

No. Trovatore et al. insist that the expression 0⁰ equals one when the exponent is integer zero, but is undefined when the exponent is real zero. This means that 0≠0, which makes no sense. Bo Jacoby (talk) 18:31, 17 May 2012 (UTC).

There isn't a common domain. The set of integers and the set of reals does not intersect, you have to map integers into the reals. Dmcq (talk) 17:48, 18 May 2012 (UTC)

Real number says: "The real numbers include all the rational numbers". Trovatore and Dmcq do not accept that. They believe that 0≠0. It can't be helped. Bo Jacoby (talk) 20:02, 18 May 2012 (UTC).

Accessibility is important in an article like that. Saying the integers are isomorphic to a subset of the reals when you consider the standard operations of addition, multiplication and the less than sign would not be accessible language in the lead. And you're perfectly well entitled to define 0⁰ as 1 for the reals if you like. It could be done generally but it isn't. The exponentiation article explains the usual conventions and the reasons and we should normally give that answer on this reference desk. Dmcq (talk) 22:08, 18 May 2012 (UTC)

I was not suggesting to say that "the integers are isomorphic to a subset of the reals when you consider the standard operations of addition, multiplication and the less than sign" in the lead of the exponentiation article. The reader knows that 0=0. Claiming that 0≠0 is not accessibility. It is insanity. Bo Jacoby (talk) 11:27, 19 May 2012 (UTC).

An argument with a fine ancestry as in Saccheri about Euclids 5th axiom "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines" Dmcq (talk) 17:39, 19 May 2012 (UTC)

Dmcq really seems to believe that 0≠0, and he also believes that this is mainstream mathematics. Bo Jacoby (talk) 19:19, 19 May 2012 (UTC).

98ⁿ = |98/98|·98₁·98₂···98_n.

0ⁿ = |0/0|·0₁·0₂···0_n.

0⁰ = |0/0|. Plasmic Physics (talk) 06:38, 19 May 2012 (UTC)

In fact, a real zero is not the same as 32bit signed integer zero. You have to cast one of them to the other datatype in order to compare them... --77.125.208.4 (talk) 12:29, 20 May 2012 (UTC)

After having cast one of them to the other datatype, what is the result of the comparison 0=0? True or false? Bo Jacoby (talk) 13:46, 20 May 2012 (UTC).