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## #1 2015-12-30 16:07:45

NakulG
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Registered: 2014-09-02
Posts: 186

### What is 0^0

Is it right to ask .... What is 0^0?

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## #2 2015-12-30 16:45:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: What is 0^0

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2015-12-30 17:03:39

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: What is 0^0

This is going to sound really wacky and unschooled, but if there are two limits that predict 0^x = 0 if x = 0 and y^0 = 1 if y = 0, instead of throwing one's hands up, could they average the two?

Last edited by Relentless (2015-12-30 17:04:10)

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## #4 2015-12-30 21:07:07

zetafunc
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Registered: 2014-05-21
Posts: 2,123
Website

### Re: What is 0^0

The link bobbym posted covers this quite well. For simplicity we take 0^0 = 1, although it is really undefined.

Relentless wrote:

This is going to sound really wacky and unschooled, but if there are two limits that predict 0^x = 0 if x = 0 and y^0 = 1 if y = 0, instead of throwing one's hands up, could they average the two?

In the context of limits of functions, I've only seen this practice adopted when dealing with jump discontinuities in Fourier series, in which one takes the average of the limits from the left and right at a jump discontinuity. However, a similar idea has been used with sequences, called Cesaro summation.

Last edited by zetafunc (2015-12-30 21:10:53)

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## #5 2015-12-30 21:35:37

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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### Re: What is 0^0

0^0 is the empty product, which is 1 by definition

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #6 2015-12-31 01:14:20

Monox D. I-Fly
Member
Registered: 2015-12-02
Posts: 888

### Re: What is 0^0

But... 0 ^ 0 = 0 ^ (1 - 1) = (0 ^ 1) / (0 ^ 1) = 0/0

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## #7 2015-12-31 02:42:19

NakulG
Member
Registered: 2014-09-02
Posts: 186

### Re: What is 0^0

Thanks for all the responses, these are very insightful.
But
1. why would a 0^0 situation arise, that it would need to be answered.
2. 0 represents the 'lack of anything'. So 'lack of anything' times 'lack of anything' would be something or even a set or is this gibberish. That is why I wanted to know what is it to try to answer 0^0?

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## #8 2015-12-31 03:12:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: What is 0^0

For one thing we would have to modify the binomial theorem and some other series. It is more convenient to define it like this 0^0 = 1.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2015-12-31 13:02:13

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: What is 0^0

Hi NakulG,
Somebody else is free to correct me, but I don't think there is any practical use for an answer to 0^0, even in quantum theory ... except for making certain theorems more elegant, as mentioned, if it equals 1.

I don't think your second point is gibberish at all. One way of thinking about exponentiation is 'multiply the base by itself exponent number of times", and an extension to the zeroth power is often given as "It is not multiplied by itself at all, so we are just left with the empty product", which is 1. I.e. 0^2 = 1*0*0, 0^1 = 1*0, 0^0 = 1.
The problem is that you can't prove this using the laws of exponentiation, or by using limits, so it's more of an answer for the sake of convenience. The problem is that that answer requires dividing by the base, and division by zero is not allowed (how can we say that 1*0 / 0 = 1?)

Last edited by Relentless (2015-12-31 13:13:31)

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## #10 2015-12-31 19:38:11

NakulG
Member
Registered: 2014-09-02
Posts: 186

### Re: What is 0^0

I think, equating 0^0 to anything is our construct, I think we should leave 0^0 as it is without equating it to anything that we understand today.

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## #11 2016-01-01 00:39:56

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: What is 0^0

It's actually quite enlightening to graph the various functions and see why it's so ambiguous what happens at 0.

For 0^0, if you graph:
x^x you get 1,
x^0 you get 1, and
0^x you get 0.

For 0/0, if you graph:
x/x you get 1, and
0/x you get 0.

For ordinary division by 0, if you graph y/x you get plus or minus infinity. Plus the implication that if, for instance 1/0 = 2/0, then 1 = 2

Last edited by Relentless (2016-01-01 01:00:19)

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## #12 2016-01-01 03:28:06

bob bundy
Registered: 2010-06-20
Posts: 8,153

### Re: What is 0^0

hi NakulG

Numbers have the value that is defined for them.  For counting numbers it is straight forward, but as you delve into the number system it becomes harder to decide on a value.

Most mathematicians work with the definition zero to the power zero equals 1.  This maintains consistency across several theorems and gives completeness.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #13 2016-02-12 02:01:50

mathaholic
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From: Earth
Registered: 2012-11-29
Posts: 3,251

### Re: What is 0^0

Yeah. It's better to use 0^0 as 1 rather than undefined.

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## #14 2017-08-30 04:36:25

Samuel.lagodam
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Registered: 2017-08-30
Posts: 1

### Re: What is 0^0

It's more appropriate to say 0^0 is  1 than saying its undefined

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## #15 2017-08-30 04:53:27

zetafunc
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Registered: 2014-05-21
Posts: 2,123
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### Re: What is 0^0

Hi Samuel.lagodam,

Welcome to the forum. Why would that be?

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## #16 2017-09-24 06:37:49

greg1313
Member
Registered: 2016-12-19
Posts: 17

### Re: What is 0^0

Samuel.lagodam wrote:

It's more appropriate to say 0^0 is  1 than saying its undefined

No, it is incorrect to say that 0^0 is 1, because it is undefined.  For instance, in the graph of
y = x^x, there is a "hole" at (0, 1), because there is no function value associated with x = 0.

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