What does 0^0 equal?
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84 members have voted
June 12th, 2014 at 2:00:06 AM
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Just because there are discontinuities does not mean that the function has to be undefined at those discontinuities. Who cares what the limit is?
Also, it's possible (and, indeed, common) to define exponentiation of natural numbers without developing limits or the real numbers. It's a simpler problem than that. That standard way is that x ^ y is the number of unique functions that map y -> x (or, if you prefer, sets of size y and x respectively). That is 1 when x and y are both 0 (empty sets). I consider this the very definition of exponentiation of natural numbers (not a theorem).
Also, it's possible (and, indeed, common) to define exponentiation of natural numbers without developing limits or the real numbers. It's a simpler problem than that. That standard way is that x ^ y is the number of unique functions that map y -> x (or, if you prefer, sets of size y and x respectively). That is 1 when x and y are both 0 (empty sets). I consider this the very definition of exponentiation of natural numbers (not a theorem).
June 12th, 2014 at 6:50:27 AM
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Quote: pacomartin
I'm sure when future archeologists uncover things like the image above, they'll theorize these hieroglyphs were used by the ruling class to mess with the heads fo everyone else. ;)
Donald Trump is a fucking criminal
June 12th, 2014 at 7:08:12 AM
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Quote: AxiomOfChoiceSo, how many functions map the empty set to the empty set?
I am not sure what you talking about. The expression 0^0 is shorthand for two functions where lim x->c, f(x) -> 0 and g(x)->0, and the question is what is the limiting value of f^g. An empty set is something different than zero.
June 12th, 2014 at 7:41:16 AM
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Quote: pacomartinI am not sure what you talking about. The expression 0^0 is shorthand for two functions where lim x->c, f(x) -> 0 and g(x)->0, and the question is what is the limiting value of f^g. An empty set is something different than zero.
Not really. Axioms definition is arguably the most common definition used for exponentiation and it definitely is the most common definition for a combinatorist. The same thing holds for most other things. Like factorial n! does mean n*(n-1)*...*1 but the definition given is actually just the number of ways of lining up n people in a row.
So the answer really depends on what sphere of mathematics we are talking about from an analysist point of view your definition might make the most sense but limits like that aren't really used in combinatorics.
So for 0^0 we need a function mapping a 0 element set to a zero element set hence the empty set to the empty set.
June 12th, 2014 at 10:39:09 AM
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Quote: pacomartinI am not sure what you talking about. The expression 0^0 is shorthand for two functions where lim x->c, f(x) -> 0 and g(x)->0, and the question is what is the limiting value of f^g. An empty set is something different than zero.
http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
November 24th, 2017 at 5:40:56 PM
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Sorry to wake up an old thread (I'm really not sorry) but stumbled up this YouTube video that attempts to answer the question of what is 0^0.
Direct link: https://www.youtube.com/watch?v=r0_mi8ngNnM
To save some time, the teacher shows that as x approaches 0 from the positive side, x^x approaches 1. Most of the video is spent filling in a table like this one:
Direct link: https://www.youtube.com/watch?v=r0_mi8ngNnM
To save some time, the teacher shows that as x approaches 0 from the positive side, x^x approaches 1. Most of the video is spent filling in a table like this one:
| x | x^x |
|---|---|
| 1.000000000 | 1.000000000 |
| 0.500000000 | 0.707106781 |
| 0.250000000 | 0.707106781 |
| 0.125000000 | 0.771105413 |
| 0.062500000 | 0.840896415 |
| 0.031250000 | 0.897354538 |
| 0.015625000 | 0.937083817 |
| 0.007812500 | 0.962802972 |
| 0.003906250 | 0.978572062 |
| 0.001953125 | 0.987889699 |
| 0.000976563 | 0.993253843 |
| 0.000488281 | 0.996283963 |
| 0.000244141 | 0.997971356 |
| 0.000122070 | 0.998900640 |
| 0.000061035 | 0.999407887 |
| 0.000030518 | 0.999682753 |
| 0.000015259 | 0.999830789 |
| 0.000007629 | 0.999910103 |
| 0.000003815 | 0.999952406 |
| 0.000001907 | 0.999974881 |
| 0.000000954 | 0.999986779 |
| 0.000000477 | 0.999993059 |
| 0.000000238 | 0.999996364 |
| 0.000000119 | 0.999998100 |
| 0.000000060 | 0.999999008 |
| 0.000000030 | 0.999999484 |
| 0.000000015 | 0.999999731 |
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
November 24th, 2017 at 5:57:46 PM
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I've watched that video before. There's another guy that's similar, the channel is called "red pen blue pen" and is full of good math stuff.
November 25th, 2017 at 1:11:05 AM
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trick Question: it's Mr Magoo's spectacles.Quote: WizardWHAT DOES 0^0 EQUAL?.
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Proverbs 18:2
A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.
November 25th, 2017 at 5:28:39 AM
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I just watched a video yesterday that showed the answer to power(sqrt(i), sqrt(i))
November 25th, 2017 at 7:01:32 AM
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Oddly. the limit approaching 0 can be shown to be both 0 or 1 depending on whether only the exponent approaches zero or both the exponent and the number approach zero.
lim 0^x x->0+ =0
lim x^x x->0+ =1
Take your pick.
(Apologies if anyone else has posted this.)
lim 0^x x->0+ =0
lim x^x x->0+ =1
Take your pick.
(Apologies if anyone else has posted this.)
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 10:28:54 AM
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Quote: QFITOddly. the limit approaching 0 can be shown to be both 0 or 1 depending on whether only the exponent approaches zero or both the exponent and the number approach zero.
lim 0^x x->0+ =0
lim x^x x->0+ =1
Take your pick.
(Apologies if anyone else has posted this.)
It has been said that 0^x=0 and x^0=1. Clearly at least one must be wrong for a value of x=0.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
November 25th, 2017 at 11:02:12 AM
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Quote: WizardIt has been said that 0^x=0 and x^0=1. Clearly at least one must be wrong for a value of x=0.
When I was in high school, I developed a transfinite system for solving limits which resolved this.
0^2= a number infinitely smaller than 0.
0^1=0
0^0=1
0^-1=∞
0^-2=∞^2
So, there would be no contradiction as 0^x was not = 0 in my system.
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 11:58:46 AM
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As in -∞?Quote: QFIT... = a number infinitely smaller than 0.
November 25th, 2017 at 12:06:30 PM
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No, -∞ has an infinitely large magnitude. In my system, 0 was an infinitely small positive number. 0^2 was smaller yet,
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 1:56:58 PM
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By how much do 0 and 0^2 differ?
November 25th, 2017 at 2:00:34 PM
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Infinitely. 0^2 is 1/∞^2
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 2:10:59 PM
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Quote: QFITInfinitely. 0^2 is 1/∞^2
Do you mean infinitely small?
November 25th, 2017 at 2:12:10 PM
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Yes. Infinitely smaller than 0.
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 2:34:20 PM
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I meant the difference between them is infinitely small, isn't it or shouldn't it be? If I'm understanding you correctly.
November 25th, 2017 at 2:37:58 PM
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If you subtract the larger from the smaller, you will get an infinitely small number. If you divide the smaller into the larger, you will get infinity.
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 25th, 2017 at 5:32:30 PM
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Quote: QFIT... an infinitely small number.
In terms of magnitude, I think your description provides an adequate definition of what I consider zero. I thus have to conclude that your 0 and 0^2 are both equal to what I believe is zero (as are mine.) As I suspected, it seems we are not likely to agree on definitions of basic terms, so a discussion probably won't satisfy anyone.
November 25th, 2017 at 5:41:02 PM
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The well-known geometric series is
∑ x^n = 1/ (1-x) where the summation is over n = 0 to ∞ and x is any number between -1 and +1.
If we set x = 0 we have ∑ 0^n = 1 (again, summation is over n = 0 to ∞)
The only way this identity can be true is if 0^0 = 1, because all other terms of ∑ 0^n will be equal to 0. Thus, 0^0 = 1.
This is probably mathematically equivalent to what the Wizard posted, but I found a lazier way of doing it!
∑ x^n = 1/ (1-x) where the summation is over n = 0 to ∞ and x is any number between -1 and +1.
If we set x = 0 we have ∑ 0^n = 1 (again, summation is over n = 0 to ∞)
The only way this identity can be true is if 0^0 = 1, because all other terms of ∑ 0^n will be equal to 0. Thus, 0^0 = 1.
This is probably mathematically equivalent to what the Wizard posted, but I found a lazier way of doing it!
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
November 25th, 2017 at 5:52:56 PM
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Quote: DocIn terms of magnitude, I think your description provides an adequate definition of what I consider zero. I thus have to conclude that your 0 and 0^2 are both equal to what I believe is zero (as are mine.) As I suspected, it seems we are not likely to agree on definitions of basic terms, so a discussion probably won't satisfy anyone.
Yep. I defined zero as a positive number for convenience -- not the typical definition (albeit used as an indication of direction in limit theory). This allowed me to use it, and infinity, in simple algebraic "work" in solving limit theory problems. As a result, I was able to complete a limit theory test in 11th grade in less than a minute. And my teacher had to give me an A as all my answers were correct, even though I didn't show the proofs in the required manner. But, I developed it when I was 16 just for fun and because I'm lazy (and to get over on my math teacher).
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 26th, 2017 at 2:48:08 PM
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Quote: QFITWhen I was in high school, I developed a transfinite system for solving limits which resolved this.
0^2= a number infinitely smaller than 0.
0^1=0
0^0=1
0^-1=∞
0^-2=∞^2
So, there would be no contradiction as 0^x was not = 0 in my system.
In your system, what was 1 - 0.9999.... (repeating forever)?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
November 26th, 2017 at 2:55:14 PM
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Zero., which I considered a positive number.
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
November 26th, 2017 at 3:04:18 PM
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Quote: QFITZero., which I considered a positive number.
What would you call the probability of drawing pi if picking a random number from the uniform distribution from 0 to 10?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
November 26th, 2017 at 3:14:09 PM
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10x0. Just kidding. This was over a half century ago. I wouldn't remember what I had for dinner last night if it weren't for the turkey in the 'fridge.:)
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
October 24th, 2018 at 7:52:07 PM
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Quote: QFITZero., which I considered a positive number.
Why do you consider it a positive number?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 24th, 2018 at 10:09:35 PM
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Or maybe zero isn’t even a number. So formulas and descriptions don’t always apply.
It’s all about making that GTA
October 25th, 2018 at 3:53:11 AM
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For the purposes of quickly solving limit problems, I define 0 as approaching zero. 2x0 approaches 0 twice as quickly. 0^2 approaches infinitely more quickly, etc.
"It is impossible to begin to learn that which one thinks one already knows." -Epictetus
October 25th, 2018 at 4:39:31 AM
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Interesting idea which means 0^1 approaches zero (at say 5mph) then 2x0 approaches at 10mph. 0^2 approaches very fast. 0^-1 goes the other way towards infinity. Thus 0^0 never moves - so where it starts who knows! Of course with train strikes, power problems near Paddington etc. hardly any trains arrive on time!Quote: QFITFor the purposes of quickly solving limit problems, I define 0 as approaching zero. 2x0 approaches 0 twice as quickly. 0^2 approaches infinitely more quickly, etc.
October 25th, 2018 at 6:57:03 AM
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Quote: charliepatrickOf course with train strikes, power problems near Paddington etc. hardly any trains arrive on time!
I would love to see the lot of those London subway workers fired. I still haven't forgiven them for striking while I was in town. Nobody has been able to explain to me why you Londoners put up with it. You should take a lesson from how Reagan handled the striking air traffic controllers.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 26th, 2018 at 2:31:41 AM
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Quote: WizardI would love to see the lot of those London subway workers fired. I still haven't forgiven them for striking while I was in town. Nobody has been able to explain to me why you Londoners put up with it. You should take a lesson from how Reagan handled the striking air traffic controllers.
I do not think any pol in our lifetimes will have the guts to do that. Reagan did the right thing and succeeded. Everyone thought he was nuts for trying, but it showed he meant business. A local mayor will be too afraid of losing re-election.
All animals are equal, but some are more equal than others
October 27th, 2018 at 5:42:24 PM
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Quote: AZDuffmanI do not think any pol in our lifetimes will have the guts to do that. Reagan did the right thing and succeeded. Everyone thought he was nuts for trying, but it showed he meant business. A local mayor will be too afraid of losing re-election.
I take it you mean politician. Didn't Thatcher fire the striking garbage workers?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
October 27th, 2018 at 6:28:30 PM
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Quote: WizardI take it you mean politician. Didn't Thatcher fire the striking garbage workers?
Yes politician. Can’t say I know if she did. The controllers was a huge thing as they thought they were irreplaceable at least in the short to middle term.
Ironically who it really scared to death was the USSR. When the Cold War ended it came out they knew he meant business.
All animals are equal, but some are more equal than others
July 1st, 2020 at 6:20:33 AM
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This thread was mentioned elsewhere, so looked it up. This guy explains why "undefined" is the correct answer to this problem. He has fantastic videos of all the university level math that I've completely forgotten.
https://www.youtube.com/watch?v=12Nae7qYxs4
https://www.youtube.com/watch?v=12Nae7qYxs4
July 1st, 2020 at 11:30:05 AM
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Quote: QFITWhen I was in high school, I developed a transfinite system for solving limits which resolved this.
I dropped into this reincarnated thread near the response above, which is what drew my attention more than the fascinating mathematics..
High school? Really? What grade? Did that help getting girls, which was what I tried to develop, with no success, when I was in high school.
