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# 0 Raised to 0 | Infinite versus Non-Defined

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Experts' Global Representative
Joined: 19 Feb 2010
Posts: 235
0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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23 Mar 2018, 09:37
Hello,

I found the following question in a different thread and felt it would be nice to start a new thread on this as 'infinite' and 'non-defined' are often considered the same while Mathematically, they are different.

Here we go...

ichadaram wrote:
According to Gmat what is the value of 0 raised to 0????

First thing first: This will not show up on GMAT.

However, for the numbers lovers here...

0^0 is 'not defined'.

Please understand the following:

1). 0^0 is not 1.

2). 0^0 is not 0.

3). 0^0 is not 'infinite'.

While we use the terms 'infinite' and 'not defined' interchangeably, there is a significant difference.

Let me try to explain:

'Infinite' is a number larger than all finite (or countable) numbers.
Examples:
1/0, 2/0 etc.

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

All the best!
Maxximus
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Joined: 02 Mar 2016
Posts: 18
Concentration: Technology, General Management
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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21 May 2019, 05:58
Maxximus wrote:
Hello,

I found the following question in a different thread and felt it would be nice to start a new thread on this as 'infinite' and 'non-defined' are often considered the same while Mathematically, they are different.

Here we go...

ichadaram wrote:
According to Gmat what is the value of 0 raised to 0????

First thing first: This will not show up on GMAT.

However, for the numbers lovers here...

0^0 is 'not defined'.

Please understand the following:

1). 0^0 is not 1.

2). 0^0 is not 0.

3). 0^0 is not 'infinite'.

While we use the terms 'infinite' and 'not defined' interchangeably, there is a significant difference.

Let me try to explain:

'Infinite' is a number larger than all finite (or countable) numbers.
Examples:
1/0, 2/0 etc.

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

All the best!
Maxximus

can you please throw some light on the following questions?

1.Is 0 a multiple of all numbers? (My answer: Yes)
2. Are all numbers multiple of 0? (my answer: No)
3. Is 0 a factor of all numbers? (my answer: No)
4. Are all numbers factors of 0? (my answer: Yes)
Math Expert
Joined: 02 Sep 2009
Posts: 59269
0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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21 May 2019, 06:13
1
NA_JS wrote:
Maxximus wrote:
Hello,

I found the following question in a different thread and felt it would be nice to start a new thread on this as 'infinite' and 'non-defined' are often considered the same while Mathematically, they are different.

Here we go...

ichadaram wrote:
According to Gmat what is the value of 0 raised to 0????

First thing first: This will not show up on GMAT.

However, for the numbers lovers here...

0^0 is 'not defined'.

Please understand the following:

1). 0^0 is not 1.

2). 0^0 is not 0.

3). 0^0 is not 'infinite'.

While we use the terms 'infinite' and 'not defined' interchangeably, there is a significant difference.

Let me try to explain:

'Infinite' is a number larger than all finite (or countable) numbers.
Examples:
1/0, 2/0 etc.

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

All the best!
Maxximus

can you please throw some light on the following questions?

1.Is 0 a multiple of all numbers? (My answer: Yes)
2. Are all numbers multiple of 0? (my answer: No)
3. Is 0 a factor of all numbers? (my answer: No)
4. Are all numbers factors of 0? (my answer: Yes)

1. Yes.
2. No. 0 is not a divisor of any number.
3. 0 is not a factor of any number.
4. Yes. 0 is divisible by every number, except 0 itself.

ZERO.

1. 0 is an integer.

2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

5. $$0^0$$, in some sources equals to 1, some mathematicians say it's undefined. Anyway you won't need this for GMAT.

"During the past decade, mathematicians argued extensively about the value of 0^0. Some answer that 0^0 = 1, while others answer that 0^0 is undefined. In the unlikely event that this question appears in some format or is a required intermediary calculation, the correct answer is more likely that 0^0 = 1."
http://www.platinumgmat.com/gmat_study_ ... ial_powers

and:
"Note: the case of 0^0 is not tested on the GMAT."
http://www.manhattangmat.com/np-exponents.cfm

Check for more below threads:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
_________________
Intern
Joined: 02 Mar 2016
Posts: 18
Concentration: Technology, General Management
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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21 May 2019, 06:26
Thanks, it helped. I do not find any methof to prove 0^0 as 1 (however I can prove 0^0 as undefined assuming 0 divided by 0 is undefined) - anyway as you suggests and I take it as a word of caution that this won't appear in GMAT so I need not to worry. Cool
Experts' Global Representative
Joined: 19 Feb 2010
Posts: 235
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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25 May 2019, 08:51
Sorry, just noticed this post.

Thanks Bunuel, for answering this on my behalf.

Kudos!

Bunuel wrote:
NA_JS wrote:
Maxximus wrote:
Hello,

I found the following question in a different thread and felt it would be nice to start a new thread on this as 'infinite' and 'non-defined' are often considered the same while Mathematically, they are different.

Here we go...

According to Gmat what is the value of 0 raised to 0????

First thing first: This will not show up on GMAT.

However, for the numbers lovers here...

0^0 is 'not defined'.

Please understand the following:

1). 0^0 is not 1.

2). 0^0 is not 0.

3). 0^0 is not 'infinite'.

While we use the terms 'infinite' and 'not defined' interchangeably, there is a significant difference.

Let me try to explain:

'Infinite' is a number larger than all finite (or countable) numbers.
Examples:
1/0, 2/0 etc.

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

All the best!
Maxximus

can you please throw some light on the following questions?

1.Is 0 a multiple of all numbers? (My answer: Yes)
2. Are all numbers multiple of 0? (my answer: No)
3. Is 0 a factor of all numbers? (my answer: No)
4. Are all numbers factors of 0? (my answer: Yes)

1. Yes.
2. No. 0 is not a divisor of any number.
3. 0 is not a factor of any number.
4. Yes. 0 is divisible by every number, except 0 itself.

ZERO.

1. 0 is an integer.

2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

5. $$0^0$$, in some sources equals to 1, some mathematicians say it's undefined. Anyway you won't need this for GMAT.

"During the past decade, mathematicians argued extensively about the value of 0^0. Some answer that 0^0 = 1, while others answer that 0^0 is undefined. In the unlikely event that this question appears in some format or is a required intermediary calculation, the correct answer is more likely that 0^0 = 1."
http://www.platinumgmat.com/gmat_study_ ... ial_powers

and:
"Note: the case of 0^0 is not tested on the GMAT."
http://www.manhattangmat.com/np-exponents.cfm

Check for more below threads:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.

_________________
Experts' Global Representative
Joined: 19 Feb 2010
Posts: 235
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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25 May 2019, 08:54
NA_JS wrote:
Thanks, it helped. I do not find any methof to prove 0^0 as 1 (however I can prove 0^0 as undefined assuming 0 divided by 0 is undefined) - anyway as you suggests and I take it as a word of caution that this won't appear in GMAT so I need not to worry. Cool

Yes, logically 0^0 is not defined. Once again, the reasoning is-

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

Nonetheless, as Bunuel mentioned, this won't appear on GMAT.

All the best!
Experts' Global
_________________
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1827
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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25 May 2019, 10:19
Maxximus wrote:
'Infinite' is a number larger than all finite (or countable) numbers.
Examples:
1/0, 2/0 etc.

This is not correct. 1/0 and 2/0 are undefined, just as 0^0 is undefined. 1/0 and 2/0 are not "infinite". Any mathematical reference at all will confirm this.

One way to see why this is the case: for 0^0, say, if we take the expression 0^x, where x is positive, we know 0^x = 0. So if we imagine making x infinitely close to 0, 0^x should be 0. But if we do the same for x^0, the answer should be 1. There's no way to decide whether 0^0 should be 0 or 1, so it is undefined. The same is true for 1/x. If we imagine x getting infinitely close to 0, from above, i.e. when x is positive, 1/x gets larger and larger. But if we imagine x approaching zero from below (so x is negative), 1/x gets smaller and smaller. So it does not make sense to say 1/0 is "infinite", since it makes as much sense to say it is "negatively infinite".

These are really calculus questions, so they're out of the scope of the GMAT, but for GMAT purposes, test takers should consider x/0 to be undefined (or in other words "mathematically nonsensical"). Everything Bunuel says above about 0 is correct, though the external link he cites about 0^0 is not correct; that expression is undefined on the GMAT, and there will absolutely never be a circumstance where it "appears in some format or is a required intermediary calculation".
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Intern
Joined: 02 Mar 2016
Posts: 18
Concentration: Technology, General Management
Re: 0 Raised to 0 | Infinite versus Non-Defined  [#permalink]

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26 May 2019, 05:50
Maxximus wrote:
NA_JS wrote:
Thanks, it helped. I do not find any methof to prove 0^0 as 1 (however I can prove 0^0 as undefined assuming 0 divided by 0 is undefined) - anyway as you suggests and I take it as a word of caution that this won't appear in GMAT so I need not to worry. Cool

Yes, logically 0^0 is not defined. Once again, the reasoning is-

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

Nonetheless, as Bunuel mentioned, this won't appear on GMAT.

All the best!
Experts' Global

Nicely put. However, if I am gonna prove 0^0 as undefined via equations it would be as following:
0^0 = 0^(1-1) {1-1=0; hence putting 0 as 1-1}
0^(1-1) = (0^1 * 0^-1) = (0^1/0^1) (using power operations)
hence 0^0 = 0/0 and which is undefined.
Again, not relevant to GMAT
Re: 0 Raised to 0 | Infinite versus Non-Defined   [#permalink] 26 May 2019, 05:50
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# 0 Raised to 0 | Infinite versus Non-Defined

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