What does 0^0 equal?
Poll
 | 13 votes (15.47%) | ||
| | 31 votes (36.9%) | ||
| | 4 votes (4.76%) | ||
| | 24 votes (28.57%) | ||
| | 2 votes (2.38%) | ||
| | 2 votes (2.38%) | ||
| | 8 votes (9.52%) |
84 members have voted
September 16th, 2013 at 6:41:29 PM
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The title pretty much says it all.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 16th, 2013 at 6:47:23 PM
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Quote: WizardThe title pretty much says it all.
I will not cheat and use google, I will guess it is undefined. Just as you cannot take a pile of 28 objects and divide them into equal piles of 0 with none left over I say you cannot take 0 of a number and get an answer.
But then again it tool me 3 years to really get algebra, and even then I basically got a "C."
All animals are equal, but some are more equal than others
September 16th, 2013 at 6:58:36 PM
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RATS! Really?
If the House lost every hand, they wouldn't deal the game.
September 16th, 2013 at 7:02:19 PM
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DUHHIIIIIIIII HEARD THAT!
September 16th, 2013 at 7:04:56 PM
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I say "undefined" because:
(a) 0x = 0 for all x <> 0, so the limit as x approaches 0 of 0x = 0.
(b) for all real numbers a, xax = e(ax ln x).
As x approaches 0, 1/x approaches positive infinity, as does -ln x.
Using L'Hopital's rule, the limit of x * (-ln x) = the limit of (-ln x) / (1/x) = the limit of (-1/x) / (-1/x2) = the limit of x.
Therefore, the limit of ax ln x as x approaches 0 = a * the limit of (x ln x) as x approaches 0 = a * the limit of x as x approaches 0 = 0.
The limit of eax ln x as x approaches 0 = e0 = 1.
(a) 0x = 0 for all x <> 0, so the limit as x approaches 0 of 0x = 0.
(b) for all real numbers a, xax = e(ax ln x).
As x approaches 0, 1/x approaches positive infinity, as does -ln x.
Using L'Hopital's rule, the limit of x * (-ln x) = the limit of (-ln x) / (1/x) = the limit of (-1/x) / (-1/x2) = the limit of x.
Therefore, the limit of ax ln x as x approaches 0 = a * the limit of (x ln x) as x approaches 0 = a * the limit of x as x approaches 0 = 0.
The limit of eax ln x as x approaches 0 = e0 = 1.
September 16th, 2013 at 7:08:09 PM
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DUHHIIIIIIIII HEARD THAT!
September 16th, 2013 at 7:20:33 PM
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I came up with -4.28 And I think I can prove it.
Shed not for her
the bitter tear
Nor give the heart
to vain regret
Tis but the casket
that lies here,
The gem that filled it
Sparkles yet
September 16th, 2013 at 7:32:11 PM
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I found my calculator and came up with:
OR
OR
"It's not called gambling if the math is on your side."
September 16th, 2013 at 7:54:02 PM
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I think this question tests my memory more than my math skills, neither of which are very solid these days.
First, zero raised to a non-zero integer power equals zero: 0x=0.
Second, a non-zero integer raised to the zero power equals one: x0=1.
If the non-zero integer x in each of these statements is replaced by zero, the statements conflict on the answer, which is the point of the question.
I think the answer is established by a conventional definition, and I think I remember that it is defined as 00=1; i.e., the second statement holds by convention. (But my memory has failed me many times before.)
First, zero raised to a non-zero integer power equals zero: 0x=0.
Second, a non-zero integer raised to the zero power equals one: x0=1.
If the non-zero integer x in each of these statements is replaced by zero, the statements conflict on the answer, which is the point of the question.
I think the answer is established by a conventional definition, and I think I remember that it is defined as 00=1; i.e., the second statement holds by convention. (But my memory has failed me many times before.)
September 16th, 2013 at 7:59:08 PM
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DUHHIIIIIIIII HEARD THAT!
September 16th, 2013 at 8:02:02 PM
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#NUM!
September 16th, 2013 at 8:33:14 PM
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It has been suggested that I should state this publicly. Here are my virtual pushups.
I got 0 by a perfectly logical process. Wrong.
I got 0 by a perfectly logical process. Wrong.
If the House lost every hand, they wouldn't deal the game.
September 16th, 2013 at 8:38:29 PM
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Let's example what x^x approaches as x gets close to 0. We'll start with x=8 and keep dividing by 2.
I submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
| x | x^x |
|---|---|
| 8.00000000 | 16777216.00000000 |
| 4.00000000 | 256.00000000 |
| 2.00000000 | 4.00000000 |
| 1.00000000 | 1.00000000 |
| 0.50000000 | 0.70710678 |
| 0.25000000 | 0.70710678 |
| 0.12500000 | 0.77110541 |
| 0.06250000 | 0.84089642 |
| 0.03125000 | 0.89735454 |
| 0.01562500 | 0.93708382 |
| 0.00781250 | 0.96280297 |
| 0.00390625 | 0.97857206 |
| 0.00195313 | 0.98788970 |
| 0.00097656 | 0.99325384 |
| 0.00048828 | 0.99628396 |
| 0.00024414 | 0.99797136 |
| 0.00012207 | 0.99890064 |
| 0.00006104 | 0.99940789 |
| 0.00003052 | 0.99968275 |
| 0.00001526 | 0.99983079 |
| 0.00000763 | 0.99991010 |
| 0.00000381 | 0.99995241 |
| 0.00000191 | 0.99997488 |
| 0.00000095 | 0.99998678 |
| 0.00000048 | 0.99999306 |
| 0.00000024 | 0.99999636 |
| 0.00000012 | 0.99999810 |
| 0.00000006 | 0.99999901 |
| 0.00000003 | 0.99999948 |
| 0.00000001 | 0.99999973 |
I submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 16th, 2013 at 8:42:44 PM
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Click. That is the definitive explanation for me. Brain matter wrinkling audibly. Thank you.
If the House lost every hand, they wouldn't deal the game.
September 16th, 2013 at 8:45:42 PM
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Quote: beachbumbabsIt has been suggested that I should state this publicly. Here are my virtual pushups.
I liked my Pepsi One answer, easier to understand.
"It's not called gambling if the math is on your side."
September 16th, 2013 at 8:49:29 PM
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I said zero, but can also see the argument for Error.
Here's some random thoughts...
Why can't you calculate the square root of a negative number?
Why will most calculators (including the app on my iPhone) allow cube roots of negatives?
What are the rules governing that?
Is zero a real number?
For x^y where both are positive, the result is a curve that runs from 1 to x when y is a fraction, and from x to infinity when y>1. When y is 0, that curve finaly terminates at 1.
But that's for real numbers.
At least that's the way I interpret it.
Here's some random thoughts...
Why can't you calculate the square root of a negative number?
Why will most calculators (including the app on my iPhone) allow cube roots of negatives?
What are the rules governing that?
Is zero a real number?
For x^y where both are positive, the result is a curve that runs from 1 to x when y is a fraction, and from x to infinity when y>1. When y is 0, that curve finaly terminates at 1.
But that's for real numbers.
At least that's the way I interpret it.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
September 16th, 2013 at 8:52:19 PM
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So 0^0 is one? Is this like the what is 0.999999... repeating thing again? (Not to bring up that again).Quote: Wizard
I submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
"Dice, verily, are armed with goads and driving-hooks, deceiving and tormenting, causing grievous woe." -Rig Veda 10.34.4
September 16th, 2013 at 9:06:51 PM
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Quote: WizardI submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
Agreed. However, the fact that limx-->af(x)=b does not imply that f(a)=b. That requires that you either assume or prove that f(x) is continuous at x=a. Can you prove that xx is a continuous function at x=0? If you take it as continuous just by assumption, then you have just assumed the answer to your original question.
I think that the "conventional definition" I mentioned earlier that 00=1 was selected to provide continuity at x=0, while the function is discontinuous if the other 0x=0 relationship is accepted. Of course, this "conventional definition" makes 0x a discontinuous function at x=0.
September 16th, 2013 at 9:16:51 PM
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Quote: teddysSo 0^0 is one? Is this like the what is 0.999999... repeating thing again? (Not to bring up that again).
No. 0^0 is undefined, if for no other reason than because 0^x approaches 0 as x approaches 0. If that's not good enough for you, x^(-a/ln |csc x - cot x|) approaches e^a, so it can be made to equal anything. Although the ultimate reason is that it's undefined simply by convention - if a function that were equal to x^x at every point, but not defined at the Gaussian integers, were useful for some purpose, the fact that this function has a limit at the Gaussian integers wouldn't change the fact that it would be undefined.
The trick to poker is learning not to beat yourself up for your mistakes too much, and certainly not too little, but just the right amount.
September 16th, 2013 at 10:09:54 PM
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Quote: JB#NUM!
So, what is #NUM factorial?
Quote: DocCan you prove that xx is a continuous function at x=0?
Can you prove it isn't? Without such proof, 1 seems as legitimate an answer as "undefined."
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 16th, 2013 at 10:16:14 PM
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If I remember inverses ... x^a =y so x=y^(1/a)
If the answer is 1, then the answer to 1^(1/0) should be 0.
For 1/0, as you approach zero as a denominator under 1, its value approaches infinity.
BUT can you say for certainty that 1/0 is infinity...because you never get there.
So, I vote "undefined" . Because even though you are approaching an answer of 1 in this example ... you never get there.
I may think about it some more while having a beer & working on the formula
for the area of a circle with radius two. ..a pepperoni circle
Pi=2*2 a
If the answer is 1, then the answer to 1^(1/0) should be 0.
For 1/0, as you approach zero as a denominator under 1, its value approaches infinity.
BUT can you say for certainty that 1/0 is infinity...because you never get there.
So, I vote "undefined" . Because even though you are approaching an answer of 1 in this example ... you never get there.
I may think about it some more while having a beer & working on the formula
for the area of a circle with radius two. ..a pepperoni circle
Pi=2*2 a
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 16th, 2013 at 10:19:23 PM
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Quote: teddysSo 0^0 is one? Is this like the what is 0.999999... repeating thing again? (Not to bring up that again).
Quote: NewToCrapsBecause even though you are approaching an answer of 1 in this example ... you never get there.
Oh no, not that topic again.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 16th, 2013 at 10:26:30 PM
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Is this like asking what was here before the BIG BANG ?
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 16th, 2013 at 10:41:39 PM
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Quote: WizardOh no, not that topic again.
There's a huge difference between the two. That "...," or whatever denotes the repetition, denotes a limit at infinity, whereas 0^0 denotes an attempt to evaluate a binary function at invalid parameters. I guess if you've never seen the concept of a limit before that conversation, you might be reminded of it every time someone mentions a limit, but that's the fault of education.
The trick to poker is learning not to beat yourself up for your mistakes too much, and certainly not too little, but just the right amount.
September 17th, 2013 at 3:27:18 AM
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Quote: WizardLet's example what x^x approaches as x gets close to 0. We'll start with x=8 and keep dividing by 2.
x x^x 8.00000000 16777216.00000000 4.00000000 256.00000000 2.00000000 4.00000000 1.00000000 1.00000000 0.50000000 0.70710678 0.25000000 0.70710678 0.12500000 0.77110541 0.06250000 0.84089642 0.03125000 0.89735454 0.01562500 0.93708382 0.00781250 0.96280297 0.00390625 0.97857206 0.00195313 0.98788970 0.00097656 0.99325384 0.00048828 0.99628396 0.00024414 0.99797136 0.00012207 0.99890064 0.00006104 0.99940789 0.00003052 0.99968275 0.00001526 0.99983079 0.00000763 0.99991010 0.00000381 0.99995241 0.00000191 0.99997488 0.00000095 0.99998678 0.00000048 0.99999306 0.00000024 0.99999636 0.00000012 0.99999810 0.00000006 0.99999901 0.00000003 0.99999948 0.00000001 0.99999973
I submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
Is it possible to see this on a graph as well as the chart? Might help some of us out.
All animals are equal, but some are more equal than others
September 17th, 2013 at 4:00:06 AM
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The pertinent operator here is the power function.
If you were to take 3^2 you would get 9 because you have 3 units in one dimension and 3 units in a second dimension then just count up the units 3+3+3. That's easy to understand but it gets complicated when you start using non-integer values for the power function. For example, the square root of 9 can be written as 9^.5 so what does that mean? 9 half a time? You can't lay out 9 as a half of a dimension (although that's exactly what's happening) because that's not intuitive. Half dimensions don't happen in real life.
Since the pertinent operator is the power function, the answer has to be what it is for all other numbers and that's 1.
If you were to take 3^2 you would get 9 because you have 3 units in one dimension and 3 units in a second dimension then just count up the units 3+3+3. That's easy to understand but it gets complicated when you start using non-integer values for the power function. For example, the square root of 9 can be written as 9^.5 so what does that mean? 9 half a time? You can't lay out 9 as a half of a dimension (although that's exactly what's happening) because that's not intuitive. Half dimensions don't happen in real life.
Since the pertinent operator is the power function, the answer has to be what it is for all other numbers and that's 1.
Someday, joor goin' to see the name of Googie Gomez in lights and joor goin' to say to joorself, "Was that her?" and then joor goin' to answer to joorself, "That was her!" But you know somethin' mister? I was always her yuss nobody knows it! - Googie Gomez
September 17th, 2013 at 5:02:51 AM
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Quote: WizardQuote: DocCan you prove that xx is a continuous function at x=0?
Can you prove it isn't? Without such proof, 1 seems as legitimate an answer as "undefined."
The difficult part is that there have been several relevant functions mentioned:
f(x)=xx
f(x)=0x
f(x)=x0
and others.
If you do the same exercise, start with x=8 and at each step reduce x by 50%, you can (usually) find a limit that f(x) approaches as x approaches zero. Unfortunately, not all of the functions approach the same limit, as not all of them are continuous at zero. By declaring the value of 00, you establish which are continuous and which are not. It's still a matter of answering the original question by assuming the answer.
September 17th, 2013 at 5:28:25 AM
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Quote: WizardOh no, not that topic again.
So the number 0.999 etc. with a googolplexian of 9's is not the same as 1?
I will start inputting it into my computer to see if it is true.
I'll report back when I am done.
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 17th, 2013 at 5:35:59 AM
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Quote: NewToCrapsIs this like asking what was here before the BIG BANG ?
Or what came first, the chicken OF the egg?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 17th, 2013 at 5:48:32 AM
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Quote: WizardOr what came first, the chicken OF the egg?
Did water vapor condense to liquid water or did liquid water evaporate to water vapor first, back at that BIG BANG?
PS - still inputting 9's.
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 17th, 2013 at 5:55:40 AM
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Quote: DJTeddyBearHere's some random thoughts...
Why can't you calculate the square root of a negative number?
Why will most calculators (including the app on my iPhone) allow cube roots of negatives?
What are the rules governing that?
Is zero a real number?
To get the square root of x, you have to find what number multiplied by itself will give you x. A negative number multiplied by itself will never give a negative number. (You can calculate the square root of a negative number using imaginary numbers, where the number i is defined as the square root of -1. The square root of -9 is 3i. But that's an entirely different subject.)
Likewise for cube root of x, you have to find what number multiplied by itself a total of three times will give you x. A negative number multiplied by a negative number yields a positive, multiplied by the third negative gives a negative result.
0 is a real number, as real numbers are defined as all rational and irrational numbers. 0 is a rational number; it is not the absence of a value.
September 17th, 2013 at 6:48:42 AM
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0^0 ? Maybe I'm nuts here, but, 0^0 is nothing multiplied to the power of nothing, right? My guess is, the answer is, nothing.
Wudged, 0 is the definition of an absence of value. Seriously, look it up. Rational or irrational, that's the definition of 0.
Wudged, 0 is the definition of an absence of value. Seriously, look it up. Rational or irrational, that's the definition of 0.
September 17th, 2013 at 6:54:23 AM
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Quote: WizardI submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
I really don't know math, but I understand some things. For instance 1/0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 is valid, while 1/0 is not. therefore 0.0000000000001 <> 0
Donald Trump is a fucking criminal
September 17th, 2013 at 6:57:46 AM
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Quote: NewToCrapsSo the number 0.999 etc. with a googolplexian of 9's is not the same as 1?
I will start inputting it into my computer to see if it is true.
I'll report back when I am done.
Still putting in 9's !!
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 17th, 2013 at 6:59:46 AM
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On the other hand, there's the Greek story about Pyhtagoras and the mathematicians. Shorn of details, Pythagoras decides to ask Athena, Goddess of Wisdom, to determine the thorny problem. But he does this by offering to sacrifice 0^0th of a lamb to Her. Naturally Athena instructs him to slaughter exactly one lamb.
Of course, there is a glaring flaw in this story (I mean, at least one).
I had the day off yesterday, so for all practical purposes I'm operating on Monday mood, BTW
Of course, there is a glaring flaw in this story (I mean, at least one).
I had the day off yesterday, so for all practical purposes I'm operating on Monday mood, BTW
Donald Trump is a fucking criminal
September 17th, 2013 at 7:25:05 AM
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Quote: WizardLet's example what x^x approaches as x gets close to 0. We'll start with x=8 and keep dividing by 2.
x x^x 8.00000000 16777216.00000000 4.00000000 256.00000000 2.00000000 4.00000000 1.00000000 1.00000000 0.50000000 0.70710678 0.25000000 0.70710678 0.12500000 0.77110541 0.06250000 0.84089642 0.03125000 0.89735454 0.01562500 0.93708382 0.00781250 0.96280297 0.00390625 0.97857206 0.00195313 0.98788970 0.00097656 0.99325384 0.00048828 0.99628396 0.00024414 0.99797136 0.00012207 0.99890064 0.00006104 0.99940789 0.00003052 0.99968275 0.00001526 0.99983079 0.00000763 0.99991010 0.00000381 0.99995241 0.00000191 0.99997488 0.00000095 0.99998678 0.00000048 0.99999306 0.00000024 0.99999636 0.00000012 0.99999810 0.00000006 0.99999901 0.00000003 0.99999948 0.00000001 0.99999973
I submit for the consideration of the forum that as x approaches 0, x^x approaches 1.
I like this. I picked 0, but it was just a guess because there was nothing on the line and I don't care about being wrong. I like that you thought about it and applied some established reasoning. I understand the argument that the function may not be continuous, but I still like this.
A falling knife has no handle.
September 17th, 2013 at 7:43:16 AM
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Me too, Mosca.
Nareed, can we thank Windows 8 for the fact that your post is the only one I've ever seen beyond the margin of the thread?
Nareed, can we thank Windows 8 for the fact that your post is the only one I've ever seen beyond the margin of the thread?
If the House lost every hand, they wouldn't deal the game.
September 17th, 2013 at 8:02:33 AM
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I vote "Other". Why? Because this is just a question of convention. The usual convention of the exponentiation operator does not provide for something like 0^0, any more than the division operator provides for x/0. By convention we call the latter "infinity". Others, also by convention, say it is "undefined". But one may decide to define it.
Well... not you or me, but some bunch of official mathematicians. The day they decide it is 1, or 0, or 42, I'll go along with it.
This is similar to 1 being excluded from the prime numbers (or Pluto from the planets).
Which reminds me of the blonde asking "How do they find out the names of the stars?"
Well... not you or me, but some bunch of official mathematicians. The day they decide it is 1, or 0, or 42, I'll go along with it.
This is similar to 1 being excluded from the prime numbers (or Pluto from the planets).
Which reminds me of the blonde asking "How do they find out the names of the stars?"
Reperiet qui quaesiverit
September 17th, 2013 at 8:14:15 AM
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I answered 1.
Scientific calculator says it's one.
Trying to think about it any more deeply than that hurts my head and makes me sad. j/k
Still, the scientific calculator says one, good enough for me.
Scientific calculator says it's one.
Trying to think about it any more deeply than that hurts my head and makes me sad. j/k
Still, the scientific calculator says one, good enough for me.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
September 17th, 2013 at 8:19:58 AM
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deleted
DUHHIIIIIIIII HEARD THAT!
September 17th, 2013 at 8:38:42 AM
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Quote: Mission146Scientific calculator says it's one.
My HP-15C (the greatest calculator ever) says it is "Error 0."
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 17th, 2013 at 8:39:16 AM
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Any Thing divided by itself equals 1.
Zero is the absence of value; in essence, it is No Thing.
No Thing, even if truly lacking in Every Thing, is still Some Thing, as we have to define the lack of Things.
Therefore, No Thing is Some Thing, and Any Thing divided by itself equals 1.
Zero is the absence of value; in essence, it is No Thing.
No Thing, even if truly lacking in Every Thing, is still Some Thing, as we have to define the lack of Things.
Therefore, No Thing is Some Thing, and Any Thing divided by itself equals 1.
The opinions of this moderator are for entertainment purposes only.
September 17th, 2013 at 8:45:11 AM
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deleted
DUHHIIIIIIIII HEARD THAT!
September 17th, 2013 at 8:52:14 AM
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Quote: beachbumbabsNareed, can we thank Windows 8 for the fact that your post is the only one I've ever seen beyond the margin of the thread?
I blame Windows 8 for the weather ;)
Donald Trump is a fucking criminal
September 17th, 2013 at 9:20:22 AM
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We all know 0^x is always zero but x^0 is always one. So 0^0 has contradictory values and is this undefined.
In the same vein, 1^(infinity) is undefined. Yes, 1^x is always 1, but as mentioned above, the formula (1 + 1/x)^x, as x approaches infinity, is e, or roughly 2.718. It is also 1^(infinity), so that formula would be both 1 and 2.718. This only makes sense if you're my junior year high school math teacher, who would spend entire class periods trying to prove that 1=2, so as such, it is undefined.
In the same vein, 1^(infinity) is undefined. Yes, 1^x is always 1, but as mentioned above, the formula (1 + 1/x)^x, as x approaches infinity, is e, or roughly 2.718. It is also 1^(infinity), so that formula would be both 1 and 2.718. This only makes sense if you're my junior year high school math teacher, who would spend entire class periods trying to prove that 1=2, so as such, it is undefined.
September 17th, 2013 at 9:22:55 AM
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Expanding on the Wizard's work from page 2, as I had suggested a little earlier:
Paraphrasing the Wizard, I submit for the consideration of the forum that as x approaches 0, x^x approaches 1, x^0 approaches 1, and 0^x approaches zero. If you want to know what the value of 0^0 really is, you just can't get the answer by looking at what values the functions approach. I don't think there is any answer to the original question other than by defining/assuming a value, and the value you choose determines which, if any, of these are continuous functions.
You can also choose to leave it as undefined, as some have suggested, but (as I said in my original post) I think that the convention is to define it as 0^0=1. You just can't prove it by looking at limits that are approached, unless you can prove that at least one of these is a continuous function.
| x | x^x | x^0 | 0^x |
|---|---|---|---|
| 8.00000000 | 16777216.00000000 | 1.00000000 | 0.00000000 |
| 4.00000000 | 256.00000000 | 1.00000000 | 0.00000000 |
| 2.00000000 | 4.00000000 | 1.00000000 | 0.00000000 |
| 1.00000000 | 1.00000000 | 1.00000000 | 0.00000000 |
| 0.50000000 | 0.70710678 | 1.00000000 | 0.00000000 |
| 0.25000000 | 0.70710678 | 1.00000000 | 0.00000000 |
| 0.12500000 | 0.77110541 | 1.00000000 | 0.00000000 |
| 0.06250000 | 0.84089642 | 1.00000000 | 0.00000000 |
| 0.03125000 | 0.89735454 | 1.00000000 | 0.00000000 |
| 0.01562500 | 0.93708382 | 1.00000000 | 0.00000000 |
| 0.00781250 | 0.96280297 | 1.00000000 | 0.00000000 |
| 0.00390625 | 0.97857206 | 1.00000000 | 0.00000000 |
| 0.00195313 | 0.98788970 | 1.00000000 | 0.00000000 |
| 0.00097656 | 0.99325384 | 1.00000000 | 0.00000000 |
| 0.00048828 | 0.99628396 | 1.00000000 | 0.00000000 |
| 0.00024414 | 0.99797136 | 1.00000000 | 0.00000000 |
| 0.00012207 | 0.99890064 | 1.00000000 | 0.00000000 |
| 0.00006104 | 0.99940789 | 1.00000000 | 0.00000000 |
| 0.00003052 | 0.99968275 | 1.00000000 | 0.00000000 |
| 0.00001526 | 0.99983079 | 1.00000000 | 0.00000000 |
| 0.00000763 | 0.99991010 | 1.00000000 | 0.00000000 |
| 0.00000381 | 0.99995241 | 1.00000000 | 0.00000000 |
| 0.00000191 | 0.99997488 | 1.00000000 | 0.00000000 |
| 0.00000095 | 0.99998678 | 1.00000000 | 0.00000000 |
| 0.00000048 | 0.99999306 | 1.00000000 | 0.00000000 |
| 0.00000024 | 0.99999636 | 1.00000000 | 0.00000000 |
| 0.00000012 | 0.99999810 | 1.00000000 | 0.00000000 |
| 0.00000006 | 0.99999901 | 1.00000000 | 0.00000000 |
| 0.00000003 | 0.99999948 | 1.00000000 | 0.00000000 |
| 0.00000001 | 0.99999973 | 1.00000000 | 0.00000000 |
Paraphrasing the Wizard, I submit for the consideration of the forum that as x approaches 0, x^x approaches 1, x^0 approaches 1, and 0^x approaches zero. If you want to know what the value of 0^0 really is, you just can't get the answer by looking at what values the functions approach. I don't think there is any answer to the original question other than by defining/assuming a value, and the value you choose determines which, if any, of these are continuous functions.
You can also choose to leave it as undefined, as some have suggested, but (as I said in my original post) I think that the convention is to define it as 0^0=1. You just can't prove it by looking at limits that are approached, unless you can prove that at least one of these is a continuous function.
September 17th, 2013 at 9:26:11 AM
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Our whole Universe was in a hot dense state then nearly fourteen-billion years ago expansion started.Quote: NewToCrapsQuote: WizardOr what came first, the chicken OF the egg?
Did water vapor condense to liquid water or did liquid water evaporate to water vapor first, back at that BIG BANG?
PS - still inputting 9's.
Someday, joor goin' to see the name of Googie Gomez in lights and joor goin' to say to joorself, "Was that her?" and then joor goin' to answer to joorself, "That was her!" But you know somethin' mister? I was always her yuss nobody knows it! - Googie Gomez
September 17th, 2013 at 9:29:36 AM
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Quote: WizardMy HP-15C (the greatest calculator ever) says it is "Error 0."
Well, I probably wouldn't even know how to turn on the greatest calculator ever, so I reiterate that my scientific calculator says it's one!
EDIT: It has an, "On," button, of course. The above was just a joke.
https://wizardofvegas.com/forum/off-topic/gripes/11182-pet-peeves/120/#post815219
September 17th, 2013 at 9:56:59 AM
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Anorher question in terms of roulette, what is 00^00 ?
I am now starting to think the answer to both equations is 0.
That is how much I win when betting at roulette and one of them is the winning number ( I don't ever cover 0 or 00) .
I am now starting to think the answer to both equations is 0.
That is how much I win when betting at roulette and one of them is the winning number ( I don't ever cover 0 or 00) .
Learned Craps in 2013 .... Developed and have a PATENT on Craps "Back On Bet" side bet ...
Working on Craps game variations hope to have patents in 2018 - Second Chance Craps and Sub-Crap-tion ...
A completely new dice game idea is next - D.. Dice D......
September 17th, 2013 at 10:21:41 AM
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Quote: NewToCrapsSo the number 0.999 etc. with a googolplexian of 9's is not the same as 1?
With a "googolplexian" of nines, no.
With infinite nines, yes.
Again, you can make 0^0 approach any number c by taking the limit of the function x^(-ln c / ln |csc x - cot x|)
The trick to poker is learning not to beat yourself up for your mistakes too much, and certainly not too little, but just the right amount.
