Does A^0 = 1?
 

I've seen several so called proofs such as this:

A^0/A^0 = 1/1 therefore A^0 = 1

I would counter that:

Infinity/Infinity = 1/1 therefore Infinity = 1

This is mostly just a tongue-in-cheek question for a late April night.

DarJones

Did you see any examples on the Web?

Of course, the converse (A^0 = 1 => A^0/A^0 = 1/1) is true ...

... as is (Infinity = 1 => Infinity/Infinity = 1/1). :-)

So, now there's just the contrapositive (one of my favorite words) and inverse to consider.

[QUOTE=Fusion_power;131459]I've seen several so called proofs such as this:

A^0/A^0 = 1/1 therefore A^0 = 1

I would counter that:

Infinity/Infinity = 1/1 therefore Infinity = 1[/QUOTE]Yes, indeed:

(any number)/(any number)=1/1, therefore (any number)=1

Curiously, what if (any number) is 0?

[QUOTE=Fusion_power;131459]I've seen several so called proofs such as this:

A^0/A^0 = 1/1 therefore A^0 = 1

I would counter that:

Infinity/Infinity = 1/1 therefore Infinity = 1

This is mostly just a tongue-in-cheek question for a late April night.

DarJones[/QUOTE]A more mature proof would be consider the value of a^x as a and x independently tend to zero. from above and below. For all non-zero a, a^x tends to 1 as x tends to zero from either direction. Accordingly, a^0 may be defined as 1.

Note that 0^x = 0 for all x>0 but is undefined for x<0. Accordingly, 0^0 is undefined.


Paul

a^x*a^-x = a^0
This is what convinces me.

[QUOTE=davieddy;131465]a^x*a^-x = a^0
This is what convinces me.[/QUOTE]Even when a=0?


Paul

[QUOTE=Fusion_power;131459]I've seen several so called proofs such as this:

A^0/A^0 = 1/1 therefore A^0 = 1

I would counter that:

Infinity/Infinity = 1/1 therefore Infinity = 1

This is mostly just a tongue-in-cheek question for a late April night.

DarJones[/QUOTE]I noted the "tongue-in-cheek" comment but there have been people here wov'e previously been confused by the word "infinity". In particular, division by infinity is not a well-defined operation.

Anyway, 2/2 = 1/1 therefore, by your reasoning, 2=1. I suggest that your reasoning has a serious problem (which can be remedied by taking limits).


Paul

[quote=xilman;131468]
Anyway, 2/2 = 1/1 therefore, by your reasoning, 2=1. I suggest that your reasoning has a serious problem (which can be remedied by taking limits).


Paul[/quote]
Limits on alcohol intake?

Xilman's comments contain the grain of truth needed to see the light on this.

10 ^ 0.9001 = 7.945111566
10 ^ 0.8001 = 6.311026445
10 ^ 0.7001 = 5.013026495
10 ^ 0.6001 = 3.981988487
10 ^ 0.5001 = 3.163005885
10 ^ 0.4001 = 2.512464881
10 ^ 0.3001 = 1.995721794
10 ^ 0.2001 = 1.58525817
10 ^ 0.1001 = 1.259215323
10 ^ 0.0001 = 1.000230285

If you look at the above, it shows A^X for X declining toward 0 and as it does, the result approaches 1. I might quibble that for any value of X no matter how minute, the actual answer only approches 1 but never actually equals 1. Still, the clear result is that A^0 must equal 1.

Something about this reminds me of Sines and Cosines

DarJones

Paul rarely says anything wrong. That's his failing....
Can someone complete/place this misquote of Oscar Wilde?

Bugger limits. If I ask my calculator for 10^0 it tells
me that it is 1.000000000 E0