# Math Question!
**URL:** https://boards.straightdope.com/t/math-question/54872
**Category:** Factual Questions
**Created:** [February 17, 2001, 12:50pm UTC](https://boards.straightdope.com/t/math-question/54872 "2001-02-17T12:50:56Z")
**Posts on this page:** 9
**Page:** 1
### Author:  [@Spudo](https://boards.straightdope.com/u/Spudo)
#### Post date: [February 17, 2001, 12:50pm UTC](https://boards.straightdope.com/t/math-question/54872/1 "2001-02-17T12:50:56Z")
Does anyone know why any number to the power of 0 equals 1? I had it figured out a long time ago, and it’s so that another operation could work, but I can’t remember!!
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### Author:  [@Patty\_O\_Furniture](https://boards.straightdope.com/u/Patty_O_Furniture)
#### Post date: [February 17, 2001, 1:07pm UTC](https://boards.straightdope.com/t/math-question/54872/2 "2001-02-17T13:07:39Z")
There might be other reasons which aren’t coming to me at the moment since it’s 8:00 on a Saturday morning, but the first thing that came to me was powers of ten.
10[sup]4[/sup] = a 1 followed by 4 zeroes
10[sup]3[/sup] = a 1 followed by 3 zeroes
10[sup]2[/sup] = a 1 followed by 2 zeroes
10[sup]1[/sup] = a 1 followed by 1 zeroes
Therefore,
10[sup]0[/sup] = a 1 followed by no zeroes
Somebody else (possibly me) will be along shortly to present a more detailed reason.
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### Author:  [@Patty\_O\_Furniture](https://boards.straightdope.com/u/Patty_O_Furniture)
#### Post date: [February 17, 2001, 1:36pm UTC](https://boards.straightdope.com/t/math-question/54872/3 "2001-02-17T13:36:04Z")
Caffeine - taking - effect… eyes openeing… muscles flexing… I am- **AWAKE!** _(cape flapping in breeze)_
Now then. Flipping through some of my moldy oldies proof books for mathematics, this is all I could come up with:
> [@](#):
>
> Any number (except zero) raised to the power of zero is by convention, (not a mathematicla fact) equal to one, **not as the result of any proof**. This convention is adopted because raising any number to the power of zero is a meaningless operation. 3[sup]2[/sup] means 3 times itself (once). While 3[sup]0[/sup] means 3 times itself zero times, which is a meaningless statement.
In order to be able to generalize the algebraic rule that: a[sup](b + c)[/sup] = a[sup]b[/sup] × a[sup]c[/sup], for example, even when b = 0, we have to set a[sup]0[/sup] = 1. Therefore, it is the convention that a[sup]0[/sup] = 1. This convention is needed in order to have consistency in this algebraic rule of operations.
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### Author:  [@MilTan](https://boards.straightdope.com/u/MilTan)
#### Post date: [February 17, 2001, 2:04pm UTC](https://boards.straightdope.com/t/math-question/54872/4 "2001-02-17T14:04:36Z")
An easy way to think of it is this:
3[sup]3[/sup] = 27
3[sup]2[/sup] = 9 **or** 3[sup]2[/sup] = 3[sup]3[/sup]/3
This of course, holds for all numbers, giving the following generalization:
a[sup]x[/sup] = a[sup]x+1[/sup]/a
\*exception - when a = 0, weird stuff happens
So, 3[sup]0[/sup] = 3[sup]1[/sup]/3, or 3/3, or 1.
I think this qualifies as a proof, or at least a general property of powers, since by this, 3[sup]-1[/sup] = 3[sup]0[/sup]/3, or 1/3, which is quite correct.
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### Author:  [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)
#### Post date: [February 17, 2001, 9:31pm UTC](https://boards.straightdope.com/t/math-question/54872/5 "2001-02-17T21:31:27Z")
Here’s a threadfrom last year in which this point is discussed:
[http://boards.straightdope.com/sdmb/showthread.php?threadid=28327](http://boards.straightdope.com/sdmb/showthread.php?threadid=28327)
This should answer your questions.
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### Author:  [@Mikahw](https://boards.straightdope.com/u/Mikahw)
#### Post date: [February 17, 2001, 11:12pm UTC](https://boards.straightdope.com/t/math-question/54872/6 "2001-02-17T23:12:34Z")
What about negative and fractional exponents? I get negative exponents, but how are fractional exponents calculated?
of course, there is no such thing as multiplying 3 by itself -1 or 1.5 times, is there?
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### Author:  [@Mikahw](https://boards.straightdope.com/u/Mikahw)
#### Post date: [February 17, 2001, 11:14pm UTC](https://boards.straightdope.com/t/math-question/54872/7 "2001-02-17T23:14:02Z")
Wait, forgot. Fractional exponents have to do with square roots. Gee, it’s been so long since I’ve taken a math class. \
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### Author:  [@Cabbage](https://boards.straightdope.com/u/Cabbage)
#### Post date: [February 18, 2001, 3:26am UTC](https://boards.straightdope.com/t/math-question/54872/8 "2001-02-18T03:26:52Z")
I’ve always thought this was an intuitive way of thinking of it:
What do you get if you add three to itself zero times? I would argue that it’s zero, since zero is the additive identity (the “nothing” of addition). So an “empty sum” is zero.
What do you get if multiply three by itself zero times? Again, I would argue that it’s one, since one is the multiplicative identity (the “nothing” of multiplication). So an “empty product” is one.
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### Author:  [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)
#### Post date: [February 18, 2001, 6:12am UTC](https://boards.straightdope.com/t/math-question/54872/9 "2001-02-18T06:12:22Z")
If you think of X[sup]n[/sup] as the product of n copies of X multiplied together, then X[sup]0[/sup] makes no sense. The definition of exponentation must be extended to allow exponents that are not positive integers. We want to extend the definition so that X[sup]n[/sup] still follows as many of the same rules as the product defintion. One rule is:
X[sup]m[/sup]/X[sup]n[/sup] = X[sup]m-n[/sup], for m\>n
If we allow m = n we get,
X[sup]m[/sup]/X[sup]m[/sup] = X[sup]m-m[/sup]
1 = X[sup]0[/sup]
X[sup]0[/sup] must equal one so that the rules still apply.