# Number to the Zero Power Equals One?

1. Philosophy Forum
2. » Logic
3. » Number to the Zero Power Equals One?

Sat 8 Nov, 2008 05:49 am
Greetings to all...

When I was in gradeschool, I was taught that any number to the zero power equals one.

Logically this doesn't not make sense to me and I'm wondering if someone could explain this.

The definition of a number to a power that I have been using is: the number multiplied times itself that [the power] number of times. So:

10 to the 2nd power would be 10 times 10 two times.

However, if a number is multiplied times itself zero times, then it is not multiplied times itself and therefore should remain unchanged:

10 to the 0 power would be 10: Ten is being multiplied times itself zero times.

I cannot fathom how multiplying to the zero power would equal one. This seems to be some arcane mathematical "given" that is more akin to mysticism than to logic.

Any help would be appreciated. Thank you for your time.

Theaetetus

Sat 8 Nov, 2008 08:53 am
@OctoberMist,
Let's first look at an example. Let's look at the list of numbers 3^1, 3^2, 3^3, 3^4, ....
Finding the actual values, we get 3, 9, 27, 81, ....

So what is the pattern in the bottom sequence? Well, every time you move to the right in the list you multiply by 3, and every time you move to the left in the list you divide by 3. So we could take the bottom sequence and keep going to the left and dividing by 3, and we'd have the sequence that looks like this: ..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....

..., 1/27, 1/9, 1/3, 1, 3, 9, 27, 81, ....

So now we know what all the powers of 3 are! Actually, we just did the integer powers of 3. But that's probably enough for now.

OctoberMist

Mon 10 Nov, 2008 02:02 am
@Theaetetus,
That still doesn't answer my question....

jknilinux

Tue 18 Nov, 2008 03:35 am
@OctoberMist,
I'm not terribly familiar with set theory yet, but I'll give this a shot.

You say "10 to the 2nd power would be 10 times 10 two times." I think it's more like 10 times 10 once, aka 100.
So, your values are all shifted...
10^2 is 10 times itself once.
10^1 is 10 times itself zero times. So it's 10, just like you said.
10^0 is... wait a second... What the?

You have a point. I don't know- but now I'm curious as to what the answer is.

As for me, I gave up on logic long ago... Causality itself is just a pattern we noticed.

Deftil

Tue 18 Nov, 2008 07:40 am
@OctoberMist,
I suck at math and I've thought this same question myself. I've wondered if it's just a convention, rather than something rigorously and mathematically correct. (in other words, did someone one day say "Hey guys! When I put a number to the zero power, I really just mean 1! I know, it doesn't make any sense, but just humor me, alright!) But math doesn't normally work like that as far as I know, so there's probably a good (and likely difficult to understand) reason for it being like that.

I did some internetting to try to figure this out, but I guess b/c I suck at math the answers I found made no sense to me. Here's what I found.

First of all - BOOO to Theaetetus for copying and pasting directly from a website and not citing the source. At any rate, what he/she posted is supposed to explain why any number to the zero power equals one, according to a "Dr. Math."
Quote:
Let's first look at an example. Let's look at the list of numbers
3^1, 3^2, 3^3, 3^4, ....
Finding the actual values, we get 3, 9, 27, 81, ....

So what is the pattern in the bottom sequence? Well, every time you move to the right in the list you multiply by 3, and every time you move to the left in the list you divide by 3. So we could take the bottom sequence and keep going to the left and dividing by 3, and we'd have the sequence that looks like this:
..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....

..., 1/27, 1/9, 1/3, 1, 3, 9, 27, 81, ....

So now we know what all the powers of 3 are! Actually, we just did the integer powers of 3. But that's probably enough for now.

Math Forum: Ask Dr. Math FAQ: N to Zero Power

However, the page there goes on with further explanations. I'll copy and paste them here, but it seems they will be in a scroll box in my post, which may make them difficult to read, so of course you should feel free to go to the actual site and take a look at them there.

Quote:
Code:```While the above argument might help convince your intuitive side that any number to the zero power is 1, the following argument is a little more rigorous. This proof uses the laws of exponents. One of the laws of exponents is: n^x --- = n^(x-y) n^y for all n, x, and y. So for example, 3^4 --- = 3^(4-2) = 3^2 3^2 3^4 --- = 3^(4-3) = 3^1 3^3 Now suppose we have the fraction: 3^4 --- 3^4 This fraction equals 1, because the numerator and the denominator are the same. If we apply the law of exponents, we get: 3^4 1 = --- = 3^(4-4) = 3^0 3^4 So 3^0 = 1. We can plug in any in number in the place of three, and that number raised to the zero power will still be 1. In fact, the whole proof works if we just plug in x for 3: x^4 x^0 = x^(4-4) = --- = 1 x^4 Here's another explanation from Aldo Daniel Completa, a contributor from Argentina: In the FAQ there is an explanation about " n^0 (any number to zero power) ." I think another is this: Let's begin with examples. Ex.1: 5^3 / 5^2 = 125 / 25 = 5 5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5 Ex.2: 5^3 / 5^3 = 125 / 125 = 1 5^3 / 5^3 = 5 ^(3-3) = 5^0 = ... and the result must be 1. So 5^0 =1. The rule is: x^b / x^c = x^(b-c). In order to generalize this rule for the case b=c, it must be defined that x^0 = 1 (x is any number different from 0). In mathematics, usefulness and consistency are very important. This convention allows us to extend definitions of power that would otherwise require treating 0 as a special case. This method can also explain the definition: x^(-b) = 1 / x^b. Ex. 5^2 / 5^4 = 25 / 625 = 1 / 5^2 5^2 / 5^4 = 5 ^(2-4) = 5^(-2)```

Here's another attempt at an answer; it also does nothing for me.
Quote:
Code:```Q: Why does any number to the zero power equal one? Consider this. a^b. it is natural to restrict a > 0, but we'll only assume that number b is any real number. We'll use the natural exponential function defined by the derivative of the exponential function. Now we have a^r=e^rln(a). And we know that e^rln(a)=e^((ln(a))^r), where a >0 and r is in the domain of all real numbers negative infinity to infinity. We can apply this definition to any number a to any power r. Particularly, a^0. By the provided definition, a^0=e^(0*ln(a))=e^0=1. Furthermore, a^1=e^(1*ln(a))=e(ln(a))=a. And a^2=(e^(ln(a))^2)=a^2. ```

WikiAnswers - Why does any number to the zero power equal one

Yet another attempt to explain is here:
Quote:
Code:``` Question : Why is it that any number raised to the power zero is equal to 1 and not zero ? Answer : When a number is raise to the power 0, we are not actually multiplying the particular number by 0. For example, let us take 20. In this case we are not actually multiplying the number 2 by 0. We define 20 = 1, so that each power of 2 is one factor of 2 larger than the last, e.g., 1,2,4,8,16,32... This involves the rules of exponents particularly division. If a is a number and x and y are also numbers, then according to the rule of division for powers with the same base, a^x/a^y = a^(x - y). It says the quotient of two powers with the same base is equal to the common base raise to the exponent equal to the difference between x and y. So, if x = y, then a^x/a^y = a^(x -y) = a^0 But a^x is equal to a^y, since x = y; hence a^x/a^y = 1 Therefore, by Transitive property of Equality, a^0 = 1 Thus, this result says that number raised to the power zero is equal to 1. ```

Math Logic - Number raised to the power zero is equal to 1

You may also want to look at what Wikipedia has to say about this - Exponentiation - Wikipedia, the free encyclopedia - it seems like it might be a relatively simple and accurate answer, but what do I know, b/c (yes, I'll say it again) I suck at math.

For more on this question just google somthing like "why is a number raised to the zero power always equal to one"; it turns out that particular search returns 2,750,000 results. :Not-Impressed:

jknilinux

Tue 18 Nov, 2008 03:10 pm
@Deftil,
Is anything wrong with the "(x^y/x^z) = x^(y-z)" solution? I think it looks good.

Zetetic11235

Fri 5 Dec, 2008 03:34 pm
@jknilinux,
I think that there is a better way to think of this. The answer is immediately obvious from the viewpoint of limits. As n approaches 0, so does x^n, that is, taking n to be a real number, as we take progressively smaller values for n, x^n gets closer to 1.

e.g 5^2=25, 5^1=5, 5^(1/2) = 2.23607(approx).... 5^(1/1000); i.e., the thousandth root of 5, =1.00161....., the millionth root of 5 equals 1.0000016 and so on. Notice that the corresponding powers are 1/1000=.001 and 1/1000000=..000001 which are getting progressively closer to 0 as five raised to them is getting progressively closer to 1.

ACB

Mon 29 Dec, 2008 04:13 pm
@OctoberMist,
OctoberMist wrote:
The definition of a number to a power that I have been using is: the number multiplied times itself that [the power] number of times.

OK, forget that definition; it's not very good. Try this one instead - it works better!

The definition of a number to a power is: one multiplied by the number that [the power] number of times.

So:
10 to the power 2 = 1 multiplied by 10 twice = 1 x 10 x 10 = 100.
10 to the power 1 = 1 multiplied by 10 once = 1 x 10 = 10.
10 to the power 0 = 1 not multiplied by 10 at all = 1.

10 to the power -1 = 1 'un-multiplied' [i.e. divided] by 10 once = 1/10 = 0.1
and so on.

Does that make it clearer?

mxmm

Fri 2 Jan, 2009 10:55 pm
@OctoberMist,
The solution is rather simple. It is because the multiplicative identity is 1. Exponentiation can be stated as follows:

All numbers can be stated as a^x, where a is a non-zero base and x is the exponent. Multiplication can then be stated in terms of addition because (a^x)*(a^y)=a^(x+y). The additive identity is zero, so since we have now fitted multiplication to a type of adding, the multiplicative identity would be a^0. Therefore, for any non-zero number a, a^0=1, since (a^x)*(a^0)=a^(x+0)=a^x, 1 being the only number we can substitute for a^0. I hope that cleared things up.

Ceilidh169

Wed 18 Feb, 2009 03:09 pm
@OctoberMist,
(X^3)(X^-3)=n
put any number in for x

(2^3)(2^-3)

in algebra when you multiply 2 numbers with exponents you add the exponents

(2^3)(2^-3)=2^0

(1x2x2x2)(1/2/2/2)=(8)(1/8)=2^0

(8)(1/8)=8/8=1

if 2^0=(8)(1/8)=8/8=1
then 2^0=1

fit any other number in for X and the result will be the same
changing exponents to a different set of integers will also work

odenskrigare

Thu 26 Feb, 2009 02:20 pm
@OctoberMist,
Just follow the pattern:

...
2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 0
2^-1 = 1/2
2^-2 = 1/4
2^-3 = 1/8
...

And then ... if you have 2^3 * 2^-3 ... you have to add the exponents, so the most reasonable value for 2^0 is 1. So it is with any other base

Ceilidh169

Thu 26 Feb, 2009 03:02 pm
@odenskrigare,
Quote:
Just follow the pattern:

...
2^3 = 8
2^2 = 4
2^1 = 2
2^2 = 0
2^-1 = 1/2
2^-2 = 1/4
2^-3 = 1/8
...

And then ... if you have 2^3 * 2^-3 ... you have to add the exponents, so the most reasonable value for 2^0 is 1. So it is with any other base

I am picky and had to correct you.

2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1
2^-1 = 1/2
2^-2 = 1/4
2^-3 = 1/8

odenskrigare

Thu 26 Feb, 2009 03:15 pm
@OctoberMist,
DOI! Dumb odenskrigare post

YumClock

Mon 20 Apr, 2009 08:56 pm
@OctoberMist,
Yes, but what is infinity multiplied by zero?

Ceilidh169

Tue 21 Apr, 2009 01:43 pm
@OctoberMist,
Math only works for finite numbers so there is no answer to that seing as infinity, as obvious by its name, is not a finite ammount.

YumClock

Tue 21 Apr, 2009 11:06 pm
@OctoberMist,
Math only works for finite numbers...
That's new. Never heard of that rule before.
These are philosophy forums. The only rule is that there are no rules.

Owen phil

Sat 2 Jan, 2010 08:05 am
@OctoberMist,
If ~(a=0) then...

a^0 = a^(1-1) = (a^1)*(a^-1) = (a^1)/(a^1) = 1

0^0 = 0/0, but, 0/0 does not exist, therefore 0^0 does not exist.

Emil

Sat 2 Jan, 2010 09:01 am
@ACB,
ACB;40066 wrote:
OK, forget that definition; it's not very good. Try this one instead - it works better!

The definition of a number to a power is: one multiplied by the number that [the power] number of times.

So:
10 to the power 2 = 1 multiplied by 10 twice = 1 x 10 x 10 = 100.
10 to the power 1 = 1 multiplied by 10 once = 1 x 10 = 10.
10 to the power 0 = 1 not multiplied by 10 at all = 1.

10 to the power -1 = 1 'un-multiplied' [i.e. divided] by 10 once = 1/10 = 0.1
and so on.

Does that make it clearer?

I recall hearing this before and it makes sense to me. The problem seems to be the buggy definition that one learned in early school.

Arjuna

Sat 2 Jan, 2010 10:48 am
@YumClock,
YumClock;59337 wrote:
Math only works for finite numbers...
That's new. Never heard of that rule before.
These are philosophy forums. The only rule is that there are no rules.
Or maybe the rule is that there are thousands of rules each of which conflicts with its opposite.

You can use 0 and infinity in multiplication if you want to. It would just be good to keep in mind that you're not really making sense.... which is maybe more obvious when you try to divide by 0 or infinity.

ACB

Sat 2 Jan, 2010 12:57 pm
@Arjuna,
Arjuna;116356 wrote:
You can use 0 and infinity in multiplication if you want to. It would just be good to keep in mind that you're not really making sense.... which is maybe more obvious when you try to divide by 0 or infinity.

Agreed, except that multiplication by 0 makes sense. If I have no boxes of 12 pencils, I have 0 x 12 = 0 pencils. If I have 12 boxes, each containing no pencils, I have 12 x 0 = 0 pencils.

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