Why ##a^0=1##?

[sophiecentaur]

I like this definition (on Wiki article).

There's quite a high level of sophistication about rational indices. Also even for all arithmetic with rational numbers; we tell kids about 'sharing' and division but even that is a matter of going through the motions and believing you got a right answer. As a lifetime Engineer, I'm used to a black box approach to Maths - as a tool and not a religion.

 

[Mike_bb]

For example, do you know the mechanics of deriving dy/dx for y=cos(x) and where the 'limit' is involved (plus some elementary trig identities).

Yes, I do. I studied full course of Math. But your example has another type of definition. Of course, if we are talking about derivatives we should use limit conception to define. "Derivative of a function" means that we "derive some function" and thus we use limit.

 

[Andrew Mason]

If $a^0\ne 1$ then $a^0\ne -e^{\pi i}$ and the beauty of mathematics would be gone.

AM

 

[sophiecentaur]

Yes, I do. I studied full course of Math. But your example has another type of definition. Of course, if we are talking about derivatives we should use limit conception to define. "Derivative of a function" means that we "derive some function" and thus we use limit.

Pure or applied Maths course? The Pure Maths guys look after the souls of Scientists and engineers etc. who use Maths for their work. Your terminology is a bit approximate and it's not getting us anywhere.

I'm afraid that, unless you are a 'good' pure mathematician, you need to trust their authority a bit blindly and follow the 'rules' really which have been derived carefully. You can't pick and choose between those rules to suit yourself. I gave up trying to do that.

 

[Mike_bb]

Pure or applied Maths course? The Pure Maths guys look after the souls of Scientists and engineers etc. who use Maths for their work. Your terminology is a bit approximate and it's not getting us anywhere.

Pure Maths.

 

[sophiecentaur]

Pure Maths

In which case you will understand how I'm using the concept of limits in this case. Afaik, my terminology is pretty well understood and accepted. Are you talking about something different?

 

[Roberto Pavani]

IMHO, $a^0 = 1$ is settled at post #8. If that's not satisfactory, then one must have in mind $a, x \notin \mathbb{R}$, or an algebra where $1$ is not the multiplicative identity, in which case, please define it precisely. Otherwise there is nothing left to discuss.
Probably a good matematician could have answered better than me here, but I'm a poor engineer.

 

[Mike_bb]

In which case you will understand how I'm using the concept of limits in this case. Afaik, my terminology is pretty well understood and accepted. Are you talking about something different?

Please make screenshots.

 

[sophiecentaur]

Please make screenshots.

Would this link help? It leads into the 'my terminology' link above and introduces the concept of Lim (Limit). In your pure Maths course, you must have come across that idea. That idea of finding a final value as the limiting value as another variable approaches (say) zero.
I really don;t see how this stuff is anything but basic. Is there a translation problem?

 

[Mike_bb]

Would this link help? It leads into the 'my terminology' link above and introduces the concept of Lim (Limit). In your pure Maths course, you must have come across that idea. That idea of finding a final value as the limiting value as another variable approaches (say) zero.
I really don;t see how this stuff is anything but basic. Is there a translation problem?

I visited this site but I can't understand what's wrong?

 

[sophiecentaur]

Limits are used when we want to check whether function continuous at the point or not but limits don't allow to define values at the point.

I think this is where our little 'skirmish' started. Can you not show that our function is continuous by finding the limit when x=0+∂x as ∂x →0. Why should the function be discontinuous only at (0,1) ?
ps No wonder ; we were both looking at the same site and both saying that it makes perfect sense.

if we are talking about derivatives we should use limit conception to define. "Derivative of a function" means that we "derive some function" and thus we use limit.

Is it not common terminology to describe dy/dx as a "derivative'? Maybe our two generations use different terminology.

 

[Mike_bb]

Why should the function be discontinuous only at (0,1) ?

Why not? You can define function as you want and then the function can be discontinuous. Following your logic, all functions should be continuous.

 

[sophiecentaur]

Why not? You can define function as you want and then the function can be discontinuous. Following your logic, all functions should be c

ontinuous.

I could go for that, except that wouldn't you have to test every function for continuity over all its range? for a function we know to be discontinuous at certain values - say tan(x), you can test near your π/2 point and the test would show up the discontinuity pretty quickly. You seem to be wanting to do this test at an arbitrary point (0,1) so wouldn't it pass a simple limits test? I'm no purist but why would that not be good enough? Maybe you were being a bit 'contrary' when you said that you don't blindly follow these rules. Maybe the onus is on you to (dis)prove the rule in this case.

Thing is, you could be one of 'those' Mathematicians who I used to work with and very often, they knew best about these things. Life with them could be fun - a bit like at Hogwarts.
PS I happened upon this thread ,just now https://www.physicsforums.com/threa...e-limit-is-the-limit-of-the-integral.1085205/

 

[Mike_bb]

You seem to be wanting to do this test at an arbitrary point (0,1) so wouldn't it pass a simple limits test?

Maybe the onus is on you to (dis)prove the rule in this case.

This case: we know that $a^n$ is defined exactly for all natural N. And that's all. Next step is to define $a^x$ not only for natural values. Do you agree with this step? If you agree then we define $a^x$ for all real values such way that laws of exponents should not be broken.

If you want to explore a function for continuity you must follow the rules:

 

[sophiecentaur]

Do you agree with this step? I

Yes. It's fine with me. I believe the the relationships between all the axioms in the Maths we use have been sorted out and are consistent. It always makes me uneasy when trying to think just how this relates to the real world, though. But that's all Philosophy and PF doesn't deal with it. It's hard when people ask for 'easy' answers about Physics - as if they actually exist. The 'answers' to questions about advanced maths seldom have short cut answers - so you don't tend to get a guy on the TV trying to explain Calculus cos it's hard from the get go and people hate that.

 

[sophiecentaur]

Good to get a tick / check. I only recently found that smiley. Have I been unobservant for years?

 

[Mike_bb]

Have I been unobservant for years?

Moments turn into years before you even know it.

 

[bhobba]

I'm afraid that, unless you are a 'good' pure mathematician, you need to trust their authority a bit blindly and follow the 'rules' really which have been derived carefully.

True.

Fortunately, however, the question asked is not one of those cases. See my previous post for the gory details of why the definition of the natural logarithm implies a^0 = 1

Thanks
Bill

 

[Dale]

we define ax for all real values such way that laws of exponents should not be broken

What are the laws of exponents that you refer to here? Please be explicit.

 

[Mike_bb]

What are the laws of exponents that you refer to here? Please be explicit.

Product rule/Quotient rule

 

[Dale]

Product rule/Quotient rule

Please be explicit and write it out. Usually the product rule is $$\frac{d(xy)}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}$$ but I don’t think that is what you mean here.

And is that the only “laws of exponents” you intended? You used plural “laws” so I assumed there would be at least two. Perhaps you intended the product rule and quotient rule to be two separate rules. I cannot tell since you were not explicit as I had asked

 

[Mike_bb]

Please be explicit and write it out. Usually the product rule is $$\frac{d(xy)}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}$$ but I don’t think that is what you mean here.

And is that the only “laws of exponents” you intended? You used plural “laws” so I assumed there would be at least two. Perhaps you intended the product rule and quotient rule to be two separate rules. I cannot tell since you were not explicit as I had asked

1. $a^m \cdot a^n=a^{m+n}$
2. $\frac{a^m}{a^n}=a^{m-n}$
3.$(a^m)^n=a^{m \cdot n}$
4.$(ab)^m=a^m \cdot b^m$
5.$\frac {a}{b}^m = \frac{a^m}{b^m}$

 

[Mike_bb]

True.

Fortunately, however, the question asked is not one of those cases. See my previous post for the gory details of why the definition of the natural logarithm implies a^0 = 1

Thanks
Bill

I reread your post. Very interesting and rigorous proof. Big thanks!

 

[Dale]

we define $a^x$ for all real values such way that laws of exponents should not be broken

So we define $a^x$ such that

2. $\frac{a^m}{a^n}=a^{m-n}$

holds for all real $n$ and $m$, not just natural numbers.

Then we immediately have $$1=\frac{a^x}{a^x}=a^{x-x}=a^0$$

So, your property 2 implies that $a^0=1$. And as you said we define $a^x$ such that it obeys property 2, among others.

1. $a^m \cdot a^n=a^{m+n}$

I will also point out that @Ibix used your property 1 all the way back in post #3 to show this. So, since you understand that we are defining $a^x$ such that all of those properties hold, and since you understood property 1 itself, then I am a little surprised that you continued to struggle beyond post #3.

 

[Roberto Pavani]

4.

is wrong, probably you meant: $(a b)^m = a^m b^m$

 

[jedishrfu]

Moments turn into years before you even know it.

And Years turn into legend.

It's a good time to close this thread, which has been spinning around the notion of $a^0=1$ since time immemorial, and move on to new fields.

Thank you all for participating here.

Jedi