A^0

[Sleeth]

I was arguing with my girlfriend’s mum and dad, who teach maths in
secondary school in Scotland.

It was mainly her dad that was saying that a^0=1
I asked how this can be as there is nothing there to get 1 in the first place
as something to the power of 0 means that there is nothing there.

So he gave me an example...

10^3=1000
10^2=100
10^1=10
10^0=1

He says that with any number, in this case ten, then to get the previous
term you divide by the variable, which is 10, so...

   10^3=1000 10^3/10=1000/10=100=10^2
   10^2=100  10^2/10=100/10=10=10^1
   10^1=10   10^1/10=10/10=1=10^0
So  10^0=1

Then I gave him an example...
  
         10^3=10*10*10
         10^2=10*10
         10^1=10
Therefore 10^0=0 (or a^0=0) as there is nothing to times anything by!

Then her dad started talking about gradients, he said that the gradient
of y=10 is 0, so does that not mean that a^0=0 and not 1? 
He said that this was correct, but then this contradicted his first point.  
This means that either he was wrong about something otherwise the
laws of maths contradict themselves also.  
Yet, I seriously doubt that and feel that I was wrong.

So, what does a^o equal?
Does it equal 0 or does it equal 1?
And if possible, can you tell me why this is so that I can go back
to him about this.

Cheers :)

[CraigD]

So, what does a^o equal?

Does it equal 0 or does it equal 1?

Your math teacher friends are right: [math]a^0 = 1[/math]

And if you can, can you tell me why this is so that I can go back to him about this.

The examples your girlfriend’s dad gave are pretty good, but here’s another that may further your understanding:

 

By definition, [math]a^b * a^c = a^{b+c}[/math]

 

For example, [math]10^2 *10^3 = 10^{2+3} = 10^5 = 100*1000= 100000[/math].

 

So [math]10^0 *10^3 = 10^{0+3} = 10^3 = 1 *1000 = 1000[/math].

 

If [math]10^0 = 0[/math], then [math]10^0 *10^3 = 10^{0+3} = 10^3 = 0 *1000 = 1000[/math]. But [math]0 * x = 0[/math], for any finite number [math]x[/math]. So [math]10^0 = 1[/math], not [math]0[/math].

[Sleeth]

Ahh, I see, I don't feel that he explained it well enough for me to understand properly, but the example you have given me pretty much sums it up.

 

So thanks :)