math question

• Dec 10th 2013, 04:20 PM
surjective
math question
Hello,

I have the following question:

If the definition that $\displaystyle x^0=1$ was incorrect, would that have any impact on the different mathematical disciplines.

Or to put in other words: what would the consequences be on mathematics (in general) if $\displaystyle x^0 \neq 1$.
• Dec 10th 2013, 04:31 PM
Plato
Re: math question
Quote:

Originally Posted by surjective
Hello,

I have the following question:

If the definition that $\displaystyle x^0=1$ was incorrect, would that have any impact on the different mathematical disciplines.

Or to put in other words: what would the consequences be on mathematics (in general) if $\displaystyle x^0 \neq 1$.

• Dec 10th 2013, 07:20 PM
surjective
Re: math question

From the thread that you directed me to I understand how to derive at $\displaystyle x^0=1$.

My question, I feel, is not sufficiently answered. I would like to know whether there are any areas of mathematics that "break down" if this equation does not hold?

An analogy would be: What difficulties would mathematics suffer if $\displaystyle \sqrt{-1} \neq i$.

I had a discussion with a person and it got me thinking.
• Dec 10th 2013, 07:22 PM
Prove It
Re: math question
Well for one thing, you would have a direct contradiction to the other index laws. There is nothing else that \displaystyle \displaystyle \begin{align*} x^0 \end{align*} can possibly be besides 1, at least when \displaystyle \displaystyle \begin{align*} x \neq 0 \end{align*}.
• Dec 10th 2013, 09:33 PM
surjective
Re: math question
Hello,

I think I see what you mean but could you, for the sake of clarity, give one or two specific examples?

Are there other areas where things would "go wrong" ?
• Dec 10th 2013, 10:21 PM
romsek
Re: math question
Quote:

Originally Posted by surjective
Hello,

I think I see what you mean but could you, for the sake of clarity, give one or two specific examples?

Are there other areas where things would "go wrong" ?

the laws of exponents wouldn't work. You might as well ask what would happen if addition didn't work.

You would multiply things by 1 and not end up with what you started with. Everything would either blow up or implode depending on whether $\displaystyle x^0$ was greater or less than 1.

None of the laws of physics would work if the laws of exponents didn't work.
• Dec 10th 2013, 10:24 PM
topsquark
Re: math question
Quote:

Originally Posted by romsek
None of the laws of physics would work if the laws of exponents didn't work.

Awwwwww!!! Cease and desist!

Bad things would happen. Are you looking for something specific? If the exponent laws don't work then just about everything would fall apart.

-Dan
• Dec 11th 2013, 03:30 AM
surjective
Re: math question
I see. Could anyone show me an example of the exponent laws not working if you made the assumption that $\displaystyle x^0 \neq 1$?

This will be my last request. Thank you very much.
• Dec 11th 2013, 04:23 AM
topsquark
Re: math question
Quote:

Originally Posted by surjective
I see. Could anyone show me an example of the exponent laws not working if you made the assumption that $\displaystyle x^0 \neq 1$?

This will be my last request. Thank you very much.

It has been mentioned already, but it bears repeating:
$\displaystyle \frac{x^n}{x^n} = x^{n - n} = x^0$. If x^0 is not 1 then neither is $\displaystyle \frac{x^n}{x^n}$, which is kind of a disaster.

-Dan
• Dec 11th 2013, 04:33 AM
surjective
Re: math question
Thanks everyboby :)
• Dec 14th 2013, 04:09 PM
Plato
Re: math question
Quote:

Originally Posted by surjective
An analogy would be: What difficulties would mathematics suffer if $\displaystyle \sqrt{-1} \neq i$.

Actually for a whole school of modern analyst it is true that $\displaystyle \sqrt{-1} \neq i$.
We think that $\displaystyle \sqrt{x}$ is defined if and only if $\displaystyle x\ge 0$.

We believe that it far better to go with the ideas from modern non-standard analysis of a enlargement.
Add one new number, $\displaystyle i$, to the real number system.
We define the number $\displaystyle i$ as being a root of the equation $\displaystyle x^2+1=0$

There is a group centered is St Louis calling for a more geometric approach to complex numbers.
But I say that using the enlargement idea is just as natural.
Look at this. If $\displaystyle z=\sqrt7~ i$ then $\displaystyle z^2=-7$ and it is a root of $\displaystyle x^2+7=0$ !
Hence it seems as if $\displaystyle z=\sqrt{-7}$.